1.1 Notation

1.1.1 Elliptic curves

We keep the notation of [Reference Green34, Reference Green33, Reference Green and Papanikolas36] and work on elliptic curves. Throughout this paper, let $\mathbb F_q$ be a finite field of characteristic p, having q elements. Let X be an elliptic curve defined over $\mathbb F_q$ given by

(1.1) $$ \begin{align} y^2+c_1ty+c_3y=t^3+c_2t^2+c_4t+c_6, \quad c_i \in \mathbb F_q. \end{align} $$

It is equipped with the rational point $\infty \in X(\mathbb F_q)$ at infinity, which we designate as the neutral element for the group law on X. We set $\textbf {A}=\mathbb F_q[t,y]$ the affine coordinate ring of X, which is the set of functions on X regular outside $\infty $ and $\textbf {K}=\mathbb F_q(t,y)$ , its fraction field. We also fix other variables $\theta , \eta $ so that $A=\mathbb F_q[\theta ,\eta ]$ and $K=\mathbb F_q(\theta ,\eta )$ are isomorphic to $\textbf {A}$ and $\textbf {K}$ . We denote the canonical isomorphism $\iota : \textbf {K} \longrightarrow K$ such that $\iota (t)=\theta $ and $\iota (y)=\eta $ . Let $\lambda = \frac {dt}{2y + c_1 t + c_3}$ be the invariant differential on X.

The $\infty $ -adic completion $K_{\infty }$ of K is equipped with the normalized $\infty $ -adic valuation $v_{\infty }:K_{\infty } \rightarrow \mathbb Z\cup \{+\infty \}$ and has residue field $\mathbb F_q$ . We set $\deg :=-v_{\infty }$ so that $\deg \theta =2$ and $\deg \eta =3$ . The completion $\mathbb C_{\infty }$ of a fixed algebraic closure $\overline K_{\infty }$ of $K_{\infty }$ comes with a unique valuation extending $v_{\infty }$ , which we also denote by $v_{\infty }$ . We define the Frobenius $\tau : \mathbb C_{\infty } \rightarrow \mathbb C_{\infty }$ as the $\mathbb F_q$ -algebra homomorphism which sends x to $x^q$ . Similarly, we can define $\textbf {K}_{\infty }$ equipped with $v_{\infty }$ and $\deg $ .

We set $\Xi =(\theta ,\eta )$ which is a K-rational point of the elliptic curve X. We define a sign function $\mathbf {sgn}:\textbf {A}\setminus \{0\} \rightarrow \mathbb F_q^{\times }$ as follows. For any $a \in \textbf {A}\setminus \{0\}$ , there is a unique way to write

$$\begin{align*}a=\sum_{i \geq 0} a_i t^i + \sum_{i \geq 0} b_i t^i y, \quad a_i,b_i \in \mathbb F_q. \end{align*}$$

Recall that $\deg t=2$ and $\deg y=3$ . The sign of a is defined to be the coefficient of the term of highest degree. It is easy to see that it extends to a group homomorphism

$$\begin{align*}\mathbf{sgn}:\textbf{K}_{\infty}^{\times} \rightarrow \mathbb F_q^{\times}. \end{align*}$$

Similarly, we can define the sign function

$$\begin{align*}\operatorname{\mathrm{sgn}}:K_{\infty}^{\times} \rightarrow \mathbb F_q^{\times}. \end{align*}$$

For any field extension $L/\mathbb {F}_q$ , the coordinate ring of X over L is $L[t,y] = L\otimes _{\mathbb {F}_q} \textbf {A}$ . We extend the sign function to such rings $L[t,y]\setminus \{0\} $ by using the same notion of leading term, namely, by writing

$$\begin{align*}a=\sum_{i \geq 0} a_i t^i + \sum_{i \geq 0} b_i t^i y, \quad a_i,b_i \in L,\end{align*}$$

and then defining $\widetilde {\operatorname {\mathrm {sgn}}}:L[t,y]\setminus \{0\} \to L^{\times }$ to be the coefficient of the leading term. This extends naturally to $L(t,y)^{\times }$ by taking quotients.

1.1.2 Tate algebras

We denote by $\mathbb T$ the Tate algebra in the variable t with coefficients in $\mathbb C_{\infty }$ ,

(1.2) $$ \begin{align} \mathbb{T} = \biggl\{ \sum_{i=0}^{\infty} b_i t^i \in {\mathbb{C}_{\infty} [[ t ]]} \biggm| \big\lvert b_i \big\rvert_{\infty} \to 0 \biggr\}, \end{align} $$

where $|\cdot |_{\infty }$ is the norm on $\mathbb {C}_{\infty }$ given by $|g|_{\infty } = q^{\deg (g)}$ . The Gauss norm on $\mathbb {T}$ is given by

$$\begin{align*}\lVert f \rVert:=\max_{i} \left\{ |b_{i}|_{\infty} \right\} \end{align*}$$

for $f=\sum _{i \geq 0} b_{i}t^{i}\in \mathbb {T}.$ Let $\mathbb L$ be its fraction field.

Further, we define the Tate algebra $\mathbb T_{\theta }$ as the space of power series in t with coefficients in $\mathbb C_{\infty }$ on the disc of radius $|\theta |_{\infty }$ ,

(1.3) $$ \begin{align} \mathbb{T}_{\theta} = \biggl\{ \sum_{i=0}^{\infty} b_i t^i \in {\mathbb{C}_{\infty} [[ t ]]} \biggm| q^i \big\lvert b_i \big\rvert_{\infty} \to 0 \biggr\}. \end{align} $$

Similarly, we define the norm $\lVert \cdot \rVert _{\theta }$ on $\mathbb T_{\theta }$ : if $f=\sum _{i=0}^{\infty } b_i t^i \in \mathbb {T}_{\theta }$ , then

$$\begin{align*}\lVert f \rVert_{\theta}=\max_{i} \left\{ q^i |b_{i}|_{\infty} \right\}. \end{align*}$$

We note that $\mathbb {T}_{\theta } \subset \mathbb {T}$ .

1.2 Anderson $\textbf {A}$ -modules on elliptic curves

We briefly review the basic theory of Anderson $\textbf {A}$ -modules and dual $\textbf {A}$ -motives and the relation between them. This material follows closely to [Reference Green34, §3-4], and the reader is directed there for proofs.

For R an $\mathbb {F}_p$ -algebra, we let $R[\tau ]$ denote the (non-commutative) skew-polynomial ring with coefficients in R, subject to the relation for $r\in R$ ,

$$\begin{align*}\tau r = r^q\tau.\end{align*}$$

We similarly define $R[\sigma ]$ , but subject to the restriction that R must be an algebraically closed field and subject to the relation

$$\begin{align*}\sigma r = r^{1/q} \sigma.\end{align*}$$

We define the ith Frobenius twisting automorphism of $\mathbb {C}_{\infty }[t,y]$ by

$$\begin{align*}g \mapsto g^{(i)} : \sum_{i,j} c_{ij} t^i y^j \mapsto \sum_{i,j} c_{ij}^{q^i} t^i y^j. \end{align*}$$

We extend twisting to matrices $\operatorname {\mathrm {Mat}}_{i\times j}(\mathbb {C}_{\infty }[t,y])$ by twisting coordinatewise. We also define Frobenius twisting on points $P\in X(\mathbb {C}_{\infty })$ , also denoted by $P^{(i)}$ , by raising each coordinate to the $q^i$ power. We extend this to divisors on X in the natural way.

Definition 1.1. 1) An n-dimensional Anderson $\textbf {A}$ -module is an $\mathbb {F}_q$ -algebra homomorphism $E:\textbf {A}\to \operatorname {\mathrm {Mat}}_n(\overline K_{\infty })[\tau ]$ , such that for each $a\in \textbf {A}$ ,

$$\begin{align*}E_a = d[a] + A_1 \tau + \dots ,\quad A_i \in \operatorname{\mathrm{Mat}}_n(\overline K_{\infty}),\end{align*}$$

where $d[a] = \iota (a)\text {Id}_n + N$ for some nilpotent matrix $N\in \operatorname {\mathrm {Mat}}_n(\overline K_{\infty })$ (depending on a).

2) A Drinfeld module is a nontrivial one-dimensional Anderson $\textbf {A}$ -module $\rho :\textbf {A} \to \overline K_{\infty }[\tau ]$ .

We note that the map $a\mapsto d[a]$ is a ring homomorphism.

Let E be an $\textbf {A}$ -Anderson module of dimension n. We introduce the exponential and logarithm functions attached to E, denoted $\operatorname {\mathrm {Exp}}_E$ and $\operatorname {\mathrm {Log}}_E$ , respectively. The exponential function is the unique function on $\mathbb {C}_{\infty }^n$ such that for all $a\in \textbf {A}$ and $\mathbf {z}\in \mathbb {C}_{\infty }^n$ ,

(1.4) $$ \begin{align} \operatorname{\mathrm{Exp}}_E(d[a]\mathbf{z}) = E_a(\operatorname{\mathrm{Exp}}_E(\mathbf{z})) \end{align} $$

and $\operatorname {\mathrm {Exp}}_E(\textbf {0}) = \text {Id}_n$ .

The function $\operatorname {\mathrm {Log}}_E$ is then defined as the formal power series inverse of $\operatorname {\mathrm {Exp}}_E$ . We note that as functions on $\mathbb {C}_{\infty }^n$ , the function $\operatorname {\mathrm {Exp}}_E$ is everywhere convergent, whereas $\operatorname {\mathrm {Log}}_E$ has a finite domain of convergence.

We briefly set out some notation regarding points and divisors on the elliptic curve X. We will denote addition of points using the group law of X by adding the points without parenthesis. For example, for $R_1,R_2 \in X$ ,

$$\begin{align*}R_1 + R_2 \in X,\end{align*}$$

and we will denote formal sums of divisors involving points on X using the points inside parenthesis. For example, for $g \in K(t,y)$ ,

$$\begin{align*}\operatorname{\mathrm{div}}(g) = (R_1) - (R_2).\end{align*}$$

Further, multiplication on the curve X will be denoted with square brackets. For example,

$$\begin{align*}[2]R_1 \in X,\end{align*}$$

whereas formal multiplication of points in a divisor will be denoted with simply a number where possible, or by an expression inside parenthesis; for example, for $h \in K(t,y)$ ,

$$\begin{align*}\operatorname{\mathrm{div}}(h) = 3(R_1) - (n+2)(R_2).\end{align*}$$

1.3 Tensor powers of Drinfeld-Hayes modules on elliptic curves

We now construct the canonical rank 1 sign-normalized Drinfeld module associated to the ring $\textbf {A}$ to which we will attach zeta values (see [Reference Green and Papanikolas36] for a detailed account). For curves of general genus, we refer the interested reader to Hayes’s work [Reference Hayes39, Reference Hayes40] (see also [Reference Anderson3, Reference Anglès, Dac and Tavares Ribeiro6, Reference Thakur52] or [Reference Goss32, §7]) for more details on sign-normalized rank one Drinfeld modules.

For the sign function defined in §1.1, a rank 1 sign-normalized Drinfeld module is a Drinfeld module $\rho :\textbf {A} \to \overline K_{\infty }[\tau ]$ such that for $a\in \textbf {A}$ , we have

$$\begin{align*}\rho_a = \iota(a) + a_1\tau + \dots + \mathbf{sgn}(a) \tau^{\deg(a)}.\end{align*}$$

By Drinfeld’s seminal work [Reference Drinfeld27], there exists a unique effective divisor V on X such that the divisor $V^{(1)} - V + (\Xi ) - (\infty )$ is principal. In our setting of elliptic curves, the situation is much more concrete. Recall that $H\subset K_{\infty }$ is the Hilbert class field of A. Then the Drinfeld divisor is the unique point $V \in X(H)$ whose coordinates have positive degree that verifies the equation on X

$$\begin{align*}V - V^{(1)} = \Xi.\end{align*}$$

We remind the reader that a divisor $\sum n_P P$ on X is principal if and only if $\sum n_p P = \infty $ on X and $\sum n_p = 0$ (see [Reference Silverman49, Cor. III.3.5]). Thus, in our situation, we conclude that the divisor $(\Xi )+(V^{(1)}) - (V) - (\infty )$ is principal, and we denote the function with that divisor $f \in H(t,y)$ (recall H is the Hilbert class field of A), normalized so that $\widetilde {\operatorname {\mathrm {sgn}}}(f)=1$ . We call this the shtuka function associated to X; that is,

(1.5) $$ \begin{align} \operatorname{\mathrm{div}}(f) = (\Xi)+(V^{(1)}) - (V) - (\infty). \end{align} $$

We will denote the denominator and numerator of the shtuka function as

(1.6) $$ \begin{align} f := \frac{\nu(t,y)}{\delta(t)} := \frac{y - \eta - m(t-\theta)}{t-\alpha}, \end{align} $$

where $m\in H$ is the slope on X (in the sense of [Reference Silverman49, III.2.3]) between the collinear points $V^{(1)}, -V$ and $\Xi $ . From [Reference Green and Papanikolas36, (19)-(20)], we get $\deg (m) = q$ , and

(1.7) $$ \begin{align} \operatorname{\mathrm{div}}(\nu) = (V^{(1)}) + (-V) + (\Xi) - 3(\infty), \quad \operatorname{\mathrm{div}}(\delta) = (V) + (-V) - 2(\infty). \end{align} $$

Definition 1.2. 1) An abelian $\textbf {A}$ -motive is a $\overline K[t,y,\tau ]$ -module M which is a finitely generated projective $\overline K[t,y]$ -module and free finitely generated $\overline K[\tau ]$ -module such that for $\ell \gg 0$ , we have

$$\begin{align*}(t-\theta)^{\ell}(M/\tau M) = \{0\},\quad (y-\eta)^{\ell}(M/\tau M) = \{0\}.\end{align*}$$

2) An $\textbf {A}$ -finite dual $\textbf {A}$ -motive is a $\overline K[t,y,\sigma ]$ -module N which is a finitely generated projective $\overline K[t,y]$ -module and free finitely generated $\overline K[\sigma ]$ -module such that for $\ell \gg 0$ , we have

$$\begin{align*}(t-\theta)^{\ell}(N/\sigma N) = \{0\},\quad (y-\eta)^{\ell}(N/\sigma N) = \{0\}.\end{align*}$$

Note that our definitions here are in line with [Reference Brownawell, Papanikolas, Böckle, Goss, Hartl and Papanikolas15, §1.5.4], rather than the more general definition given in [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38, Def. 2.4.1].

We then let $U = \operatorname {\mathrm {Spec}} \overline K[t,y]$ (i.e., the affine curve $(\overline K \times _{\mathbb {F}_q} X) \setminus \{ \infty \}$ ). For a divisor D on the curve X, we let $\mathcal L(D)$ be the $\overline K$ -vector space of rational functions g on X with $\operatorname {\mathrm {div}}(g) \geq -D$ . The (geometric) $\textbf {A}$ -motive associated to X is given by

$$\begin{align*}M_1 = \Gamma(U, \mathcal O_{X}(V)) = \bigcup_{i \geq 0} \mathcal L((V) + i(\infty)). \end{align*}$$

We make $M_1$ into a left $\overline K[t,y,\tau ]$ -module by letting $\tau $ act by

$$\begin{align*}\tau g = f g^{(1)}, \quad g \in M_1, \end{align*}$$

and letting $\overline K[t,y]$ act by left multiplication.

The (geometric) dual $\textbf {A}$ -motive associated to X is given by

(1.8) $$ \begin{align} N_1 = \Gamma\bigl(U, \mathcal O_X(-(V^{(1)}))\bigr) = \bigcup_{i \geq 1} \mathcal L(-(V^{(1)}) + i(\infty)) \subseteq \overline K[t,y]. \end{align} $$

We make $N_1$ into a left $\overline K[t,y,\sigma ]$ -module by letting $\sigma $ act by

$$\begin{align*}\sigma g = f g^{(-1)}, \quad g \in N_1, \end{align*}$$

and letting $\overline K[t,y]$ act by left multiplication.

We find that $M_1$ and $N_1$ are projective $\overline K[t,y]$ -module of rank $1$ , that $M_1$ is as a free $\overline K[\tau ]$ -module of rank $1$ and that $N_1$ is as a free $\overline K[\sigma ]$ -module of rank $1$ (see [Reference Green and Papanikolas36, §3] for proofs of these facts). A quick check shows that $M_1$ (resp. $N_1$ ) is indeed an abelian $\textbf {A}$ -motive (resp. $\textbf {A}$ -finite dual $\textbf {A}$ -motive).

We now follow as in [Reference Green34, §3] and form the nth tensor power of $M_1$ and of $N_1$ and denote these as

$$\begin{align*}M_n = M_1^{\otimes n} = M_1\otimes_{\overline K[t,y]} \dots \otimes_{\overline K[t,y]}M_1,\end{align*}$$

$$\begin{align*}N_n = N_1^{\otimes n} = N_1\otimes_{\overline K[t,y]} \dots \otimes_{\overline K[t,y]}N_1,\end{align*}$$

with $\tau $ and $\sigma $ action on $a\in M_n$ and $b\in N_n$ given respectively by

$$\begin{align*}\tau a = f^n b^{(1)},\quad \sigma b = f^n b^{(-1)}.\end{align*}$$

Observe that

$$\begin{align*}M_n = \Gamma (U,\mathcal O_X(nV)),\quad N_n \cong \Gamma (U,\mathcal O_X(-nV^{(1)}))\end{align*}$$

and that $M_n$ (resp. $N_n$ ) is also an $\textbf {A}$ -motive (resp. a dual $\textbf {A}$ -motive). Again, $M_n$ and $N_n$ are projective $\overline K[t,y]$ -modules of rank 1. Further, $M_n$ is a free $\overline K[\tau ]$ -module of rank n, and $N_n$ is a free $\overline K[\sigma ]$ -module of rank n.

We write down convenient bases for $M_n$ and $N_n$ as free $\overline K[\tau ]$ - and $\overline K[\sigma ]$ -modules, respectively (see [Reference Green34, Prop. 3.3]). Define functions $g_i \in M_n$ for $1\leq i\leq n$ with $\widetilde {\operatorname {\mathrm {sgn}}}(g_i)=1$ and with divisors

(1.9) $$ \begin{align} \operatorname{\mathrm{div}}(g_{j}) = -n(V) + (n-j)(\infty) + (j-1)(\Xi) + ([j-1]V^{(1)} + [n-(j-1)]V), \end{align} $$

and similarly define functions $h_i \in N_n$ for $1\leq i\leq n$ , each with $\widetilde {\operatorname {\mathrm {sgn}}}(h_i) = 1$ and with divisor

(1.10) $$ \begin{align} \operatorname{\mathrm{div}}(h_j) = n(V^{(1)}) - (n+j)(\infty) + (j-1)(\Xi) + (-[n - (j - 1)]V^{(1)}-[j-1]V). \end{align} $$

Then we have

$$\begin{align*}M_n = \overline K[\tau]\{g_1,\dots,g_n\},\quad N_n = \overline K[\sigma]\{h_1,\dots,h_n\}.\end{align*}$$

For $g\in N_n$ , we set $m = \lfloor \deg (g)/n \rfloor $ and define two maps

$$\begin{align*}\delta_0,\delta_1: N_n \to \overline K^n \end{align*}$$

in the following way. We write g in the $\overline K[\sigma ]$ -basis for $N_n$ described in (1.10),

(1.11) $$ \begin{align} g = \sum_{i=0}^{m} \sum_{j=1}^n b_{j,i}^{(-i)} \sigma^i h_{n-j+1}, \end{align} $$

then denote $\mathbf {b}_i = (b_{1,i},b_{2,i},\dots ,b_{n,i})^{\top }$ , and set

(1.12) $$ \begin{align} \delta_0(g) = \mathbf{b}_0,\quad \delta_1(g) = \mathbf{b}_0 + \mathbf{b}_1 + \dots + \mathbf{b}_m. \end{align} $$

We then observe that the kernel of $\delta _1$ equals $(\sigma -1)N_n$ and that $N_n/(\sigma -1)N_n \xrightarrow {\delta _1} \overline K^n$ is an isomorphism of $\mathbb {F}_q$ -vector spaces. Thus, we can write the commutative diagram of $\mathbb {F}_q$ -vector spaces

where the left vertical arrow is multiplication by a and the right vertical arrow is the map induced by multiplication by a, which we denote by $\rho ^{\otimes n}_a$ .

We can write down the matrices defining $\rho _t^{\otimes n}$ using the coefficients $a_i\in H$ from Proposition 1.4, for $n\geq 2$ :

(1.14)

The t-action of the Drinfeld $\textbf {A}$ -module $\rho $ is given by

$$\begin{align*}\rho_t=\theta+x_1 \tau + \tau^2, \quad x_1 \in B, \end{align*}$$

(see [Reference Green and Papanikolas36, §3] for more details on this construction).

The logarithm and exponential functions associated to $\rho ^{\otimes n}$ will be denoted $\operatorname {\mathrm {Log}}_{\rho }^{\otimes n}$ and $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}$ , respectively, and the kernel of $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}$ will be denoted by $\Lambda _{\rho }^{\otimes n}$ , which we call the period lattice of $\rho ^{\otimes n}$ .

We recall the Tate algebra $\mathbb T$ in the variable t with coefficients in $\mathbb C_{\infty }$ as in §1.1.2,

$$ \begin{align*} \mathbb{T} = \biggl\{ \sum_{i=0}^{\infty} b_i t^i \in {\mathbb{C}_{\infty} [[ t ]]} \biggm| \big\lvert b_i \big\rvert_{\infty} \to 0 \biggr\}, \end{align*} $$

where $|\cdot |_{\infty }$ is the norm on $\mathbb {C}_{\infty }$ given by $|g|_{\infty } = q^{\deg (g)}$ and its fraction field $\mathbb L$ . We now give a brief review of the functions $\omega _{\rho }$ , $E_{\mathbf {u}}^{\otimes n}$ and $G_{\mathbf {u}}^{\otimes n}$ defined in [Reference Green34, §5-6]. Let

(1.15) $$ \begin{align} \omega_{\rho} = \xi^{1/(q-1)} \prod_{i=0}^{\infty} \frac{\xi^{q^i}}{f^{(i)}} \in \mathbb T[y]^{\times}, \end{align} $$

where $f \in H(t,y)$ is the shtuka function defined above, and we refer the reader to [Reference Green and Papanikolas36, Thm. 4.6] for the definition of $\xi \in H$ and for details on convergence. Observe that $\omega _{\rho }$ satisfies the functional equation $\omega _{\rho }^{(1)} = f \omega _{\rho }$ . For $\mathbf {u} = (u_1,...,u_n)^{\top } \in \mathbb {C}_{\infty }^n$ , define

(1.16) $$ \begin{align} E_{\mathbf{u}}^{\otimes n}(t) = \sum_{i=0}^{\infty} \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}\left (d[\theta]^{-i-1} \mathbf{u}\right ) t^i, \end{align} $$

(1.17) $$ \begin{align} G_{\mathbf{u}}^{\otimes n} (t,y) = E_{d[\eta]\mathbf{u}}^{\otimes n}(t) + (y+c_1t + c_3) E_{\mathbf{u}}^{\otimes n}(t). \end{align} $$

For $n=1$ and $\mathbf {u} = u\in \mathbb {C}_{\infty }$ , we will simplify notation by setting $E_u(t):=E_{\mathbf {u}}^{\otimes 1}(t)$ and $G_u(t,y):=G_{\mathbf {u}}^{\otimes 1}(t,y)$ .

Define $\mathcal {M}$ to be the submodule of $\mathbb {T}[y]$ consisting of all elements in $\mathbb {T}[y]$ which have a meromorphic continuation to all of $U = \operatorname {\mathrm {Spec}} \overline K[t,y]$ in the sense of [Reference Fresnel and van der Put29, §4.6] (we comment that the sheaf of meromorphic functions on a rigid space is a sheaf which locally looks like $\mathbb {L}$ , and its affine pieces can be glued together coherently). Now define the map $\operatorname {\mathrm {RES}}_{\Xi }:\mathcal {M}^n\to \mathbb {C}_{\infty }^n$ for a vector of functions $(z_1,...,z_n)^{\top }\in \mathcal {M}^n$ as

(1.18) $$ \begin{align} \operatorname{\mathrm{RES}}_{\Xi}((z_1,\dots, z_n)^{\top}) = (\operatorname{\mathrm{Res}}_{\Xi}(z_1\lambda),\dots, \operatorname{\mathrm{Res}}_{\Xi}(z_n\lambda))^{\top}, \end{align} $$

where $\lambda $ is the invariant differential on E (defined in §1.1).

We define a map $T:\mathbb {T}[y]\to \mathbb {T}[y]^n$ by

(1.19) $$ \begin{align} T( h(t,y)) = \left (\begin{matrix} h(t,y)\cdot g_1 \\ h(t,y)\cdot g_2 \\ \vdots \\ h(t,y)\cdot g_n \\ \end{matrix}\right ). \end{align} $$

We collect the following facts from [Reference Green34, §5-6] about the above functions.

1.4 A generalization of Anderson-Thakur’s theorem on elliptic curves

Recall that $\rho : \textbf {A}\rightarrow \mathbb C_{\infty }\{\tau \}$ is the canonical sign-normalized rank one Drinfeld module associated to the elliptic curve X constructed in the previous section and that H is the Hilbert class field of A. Let B (or $\mathcal O_H$ ) be the integral closure of A in H. We denote by G the Galois group $\mathrm {Gal}(H/K)$ .

We denote by $\mathcal {I}(A)$ the group of fractional ideals of A. For $I \in \mathcal I(A)$ , denote its Artin symbol by

(1.20) $$ \begin{align} \sigma_I:=(I,H/K) \in G. \end{align} $$

By [Reference Goss32], Proposition 7.4.2 and Corollary 7.4.9, the subfield of $\mathbb C_{\infty }$ generated by K and the coefficients of $\rho _a$ is H. Furthermore, by [Reference Goss32], Lemma 7.4.5, we get

$$\begin{align*}\forall a\in A, \quad \rho_a\in B\{\tau\}. \end{align*}$$

Let I be a nonzero ideal of A. We define $\rho _I$ to be the monic element in $H\{\tau \}$ such that

$$\begin{align*}H\{ \tau \} \rho_I=\sum_{a\in I} H\{\tau\} \rho_a. \end{align*}$$

We have

$$\begin{align*}{\ker} \, \rho_I=\bigcap _{a\in I} {\ker}\, \rho_a, \end{align*}$$

$$\begin{align*}\rho_I\in B\{\tau\}, \end{align*}$$

$$\begin{align*}\deg_{\tau} \rho_I= \deg I. \end{align*}$$

We write $\rho _I=\rho _{I,0}+\cdots +\rho _{I,\deg I} \tau ^{\deg I}$ with $\rho _{I,\deg I}=1$ and denote by $\psi (I) \in B\setminus \{0\}$ the constant coefficient $\rho _{I,0}$ of $\rho _I$ . Thus, the map $\psi $ extends uniquely into a map $\psi : \mathcal I(A)\rightarrow H^{\times }$ with the following properties (proved in [Reference Hayes39, Prop. 3.2 and Thm. 8.5]):

• 1) for all $I, J\in \mathcal I(A), \psi (IJ)= \sigma _J(\psi (I))\, \psi (J)$ ,

• 2) for all $I\in \mathcal I(A), IB=\psi (I)B$ ,

• 3) for all $x\in K^{\times }, \psi (xA)= \frac {x}{\operatorname {\mathrm {sgn}} (x)}$ .

Finally, for $n \in \mathbb N$ and $b\in B$ , we define Anderson’s zeta value at n attached to $\rho $ as follows:

(1.21) $$ \begin{align} \zeta_{\rho}(b,n)=\sum_{I \subseteq A} \frac{\sigma_I(b)}{\psi(I)^n} \in K_{\infty}. \end{align} $$

By the work of Anderson (see [Reference Anderson2], [Reference Anderson3]), for any $b \in B$ , we have

(1.22) $$ \begin{align} \exp_{\rho}(\zeta_{\rho}(b,1)) \in B. \end{align} $$

The following theorem is a generalization of the celebrated Anderson-Thakur theorem for tensor powers of the Carlitz module (see [Reference Anderson and Thakur5], Theorem 3.8.3).

1.5 Linear relations among Anderson’s zeta values

In this section, we determine completely linear relations among Anderson’s zeta values and powers of $\pi _{\rho }$ associated to an elliptic curve over $\mathbb {F}_p$ . In the genus 0 case, this was done by Yu (see [Reference Yu55], Theorem 3.1 and [Reference Yu56], Theorem 4.1). His works are built on two main ingredients. The first one is Yu’s theory where he developed an analogue of Wüstholz’s analytic subgroup theorem for function fields, and the second one is the Anderson-Thakur theorem mentioned in the previous section. The main result of this section extends Yu’s work to elliptic curves.

Recall that $\textbf {A}=\mathbb F_q[t,y]$ , where t and y satisfy the Weierstrass equation (1.1). Following Green (see [Reference Green33], Section 7), we still denote by $\rho :\mathbb F_q[t] \longrightarrow \mathbb C_{\infty } \{\tau \}$ the Drinfeld $\mathbb F_q[t]$ -module induced by forgetting the y-action of the sign-normalized rank 1 Drinfeld module $\rho $ of the previous sections. Similarly, we denote by $\rho ^{\otimes n}:\mathbb F_q[t] \longrightarrow \operatorname {\mathrm {Mat}}_n(\mathbb C_{\infty }) \{\tau \}$ the Anderson $\mathbb F_q[t]$ -module defined by forgetting the y-action. Basic properties of this Anderson module are given below.

We slightly generalize [Reference Green33], Theorem 7.1 to obtain the following theorem which settles the problem of determining linear relations among Anderson’s zeta values and periods attached to $\rho $ , which generalizes the work of Yu.

Theorem 1.10. Let $\{b_1,\ldots ,b_h\}$ be a K-basis of H with $b_i \in B$ . We consider the following sets for $m,s\geq 1$ :

$$ \begin{align*} & \mathcal R :=\{\pi_{\rho}^k, 0 \leq k \leq m\} \cup \{\zeta_{\rho}(b_i,n): 1 \leq i \leq h, 1 \leq n\leq s \text{ such that } q-1 \nmid n \}, \\ & \mathcal R' :=\{\pi_{\rho}^k, 0 \leq k \leq m\} \cup \{\zeta_{\rho}(b_i,n): 1 \leq i \leq h, 1 \leq n\leq s \}. \end{align*} $$

Then

1) The $\overline K$ -vector space generated by the elements in $\mathcal {R}$ and that generated by those in $\mathcal {R}'$ are the same.

2) The elements in $\mathcal {R}$ are linearly independent over $\overline K$ .

Proof. The proof follows the same lines as Yu’s celebrated theorem [Reference Yu56, Th.m 4.1] (see also [Reference Green33, Thm. 7.1]). We provide a proof for the convenience of the reader.

Recall that for $n \in \mathbb N$ , the Anderson $\textbf {A}$ -module $\rho ^{\otimes n}$ induces the structure of an $\mathbb F_q[t]$ -module which, by abuse of notation, we also denote by $\rho ^{\otimes n}$ . Also recall that h is the class number of A. We consider the product of t-modules

$$\begin{align*}G = \mathbb G_a \times \left (\prod_{k=1}^{m} \rho^{\otimes k} \right ) \times \left (\prod_{\substack{n=1\\q-1 \nmid n}}^{s} \left( \rho^{\otimes n} \right)^{\oplus h} \right ),\end{align*}$$

where we view $\mathbb G_a$ in the first coordinate as a trivial t-module with the scalar A-action and with exponential function $\exp _{G_L}(z) = z$ .

For $1 \leq n \leq s$ , set $Z_n(b_i) =(*,\ldots ,*,C_n \zeta _{\rho }(b_i,n))^{\top } \in \mathbb C_{\infty }^n$ to be a vector from Theorem 1.8 such that $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(Z_n(b_i)) \in H^n$ . For $1 \leq k \leq m$ , let $\Pi _k \in \mathbb C_{\infty }^k$ be a fundamental period of $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes k}$ so that the bottom coordinate of $\Pi _k$ is a multiple of $\pi _{\rho }^k$ by a nonzero element of H (see [Reference Green34, Thm. 6.7] for the exact multiple). Define the vector

$$\begin{align*}\mathbf{u} = 1 \times \left (\prod_{k=1}^{m} \Pi_k \right ) \times \left (\prod_{\substack{n=1\\q-1 \nmid n}}^{s} \prod_{i=1}^h Z_n(b_i) \right ) \in G(\mathbb C_{\infty}) \end{align*}$$

and note $\operatorname {\mathrm {Exp}}_G(\mathbf {u}) \in G(H)$ , where $\operatorname {\mathrm {Exp}}_G$ is the exponential function on G, defined coordinatewise. Suppose, by contradiction, that there is a $\overline K$ -linear relation among the $\zeta _{\rho }(b_i,n)$ and $\pi _{\rho }^k$ . This implies that $\mathbf {u}$ is contained in a $d[\mathbb F_q[t]]$ -invariant hyperplane of $G(\mathbb C_{\infty })$ defined over $\overline K$ . This allows us to apply [Reference Yu56, Thm. 3.3], which says that $\mathbf {u}$ is in $\text {Lie}_{G'}(\overline K)$ for a proper t-submodule $G' \subset G$ . Then Proposition 1.9 together with [Reference Yu56, Thm. 1.3] implies that there exist $1 \leq n \leq s$ with $q-1 \nmid n$ and a linear relation of the form

$$\begin{align*}\sum_{i=1}^h a_i \zeta_{\rho}(b_i,n) + b \pi_{\rho}^n=0 \end{align*}$$

for some $a_i,b \in A$ not all zero. Since $\zeta _{\rho }(b_i,n) \in K_{\infty }$ and since $H \subset K_{\infty }$ , this implies that $b \pi _{\rho }^n \in K_{\infty }$ . Since $q-1 \nmid n$ , we know that $\pi _{\rho }^n \notin K_{\infty }$ . It follows that $b=0$ and hence $\sum _{i=1}^h a_i \zeta _{\rho }(b_i,n)=0$ . Since $a_i \in A$ , we get

$$\begin{align*}0=\sum_{i=1}^h a_i \zeta_{\rho}(b_i,n)=\zeta_{\rho} \left( \sum_{i=1}^h a_i b_i,n \right).\end{align*}$$

We deduce that $\sum _{i=1}^h a_i b_i=0$ . Since $\{b_i\}$ is a K-basis of H, this forces $a_i=0$ for all i. This provides a contradiction and proves the theorem.

As explained by B. Anglès, Footnote ² the following result is attributed to Carlitz and Goss (see [Reference Goss31, Thm. 3.2.2]) which improves [Reference Anglès, Dac and Tavares Ribeiro7], Theorem 5.7 and [Reference Green33], Corollary 7.4.

1.6 Papanikolas’s work

We review Papanikolas’ theory [Reference Papanikolas47] (see also [Reference Anderson1, Reference Anderson, Brownawell and Papanikolas4]) and work with t-motives. Let $\overline K[t,\sigma ]$ be the polynomial ring in variables t and $\sigma $ with the rules

$$\begin{align*}at=ta, \ \sigma t=t \sigma, \ \sigma a=a^{1/q} \sigma, \quad a \in \overline K. \end{align*}$$

Definition 1.12. An Anderson dual t-motive is a left $\overline K[t,\sigma ]$ -module N which is free and finitely generated both as a left $\overline K[t]$ -module and as a left $\overline K[\sigma ]$ -module and which satisfies

$$\begin{align*}(t-\theta)^d N \subset \sigma N \end{align*}$$

for some integer d sufficiently large.

We consider $\overline K(t)[\sigma ,\sigma ^{-1}]$ the ring of Laurent polynomials in $\sigma $ with coefficients in $\overline K(t)$ .

The category of pre-t-motives is abelian, and there is a natural functor from the category of Anderson dual t-motives to the category of pre-t-motives

$$\begin{align*}N \mapsto M:=\overline K(t) \otimes_{\overline K[t]} N, \end{align*}$$

where $\sigma $ acts diagonally on M.

We now consider pre-t-motives M which are rigid analytically trivial, which we describe here. Let $\{\mathbf {m} \} \in \operatorname {\mathrm {Mat}}_{r \times 1}(M)$ be a $\overline K(t)$ -basis of M and let $\Phi \in \text {GL}_r(\overline K[t])$ be the matrix representing the multiplication by $\sigma $ on M:

$$\begin{align*}\sigma({\mathbf{m}})=\Phi {\mathbf{m}}. \end{align*}$$

We recall that $\mathbb T$ is the Tate algebra (Definition 1.2) in variable t with coefficients in $\mathbb C_{\infty }$ and that $\mathbb L$ is the fraction field of the Tate algebra $\mathbb T$ . We say that M is rigid analytically trivial if there exists $\Psi \in \text {GL}_r(\mathbb L)$ such that

$$\begin{align*}\Psi^{(-1)}=\Phi \Psi. \end{align*}$$

We set $M^{\dagger }:=\mathbb L \otimes _{\overline K(t)} M$ on which $\sigma $ acts diagonally and define $H_{\text {Betti}}(M)$ to be the sub $\mathbb F_q(t)$ -vector space of the elements of $M^{\dagger }$ which are fixed by $\sigma $ . We call $H_{\text {Betti}}(M)$ the Betti cohomology of M. It is shown in [Reference Papanikolas47, Prop. 3.3.9] that M is rigid analytically trivial if and only if the natural map

$$\begin{align*}\mathbb L \otimes_{\mathbb F_q(t)} H_{\text{Betti}}(M) \rightarrow M^{\dagger} \end{align*}$$

is an isomorphism. We then call $\Psi $ a rigid analytical trivialization for the matrix $\Phi $ .

The category of pre-t-motives which are rigid analytically trivial is a neutral Tannakian category over $\mathbb F_q(t)$ with the fiber functor $\omega $ which maps $M \mapsto H_{\text {Betti}}(M)$ (see [Reference Papanikolas47], Theorem 3.3.15).

By Tannakian duality, for each (rigid analytically trivial) t-motive M, the Tannakian subcategory generated by M is equivalent to the category of finite dimensional representations over $\mathbb F_q(t)$ of some algebraic group $\Gamma _M$ called the (motivic) Galois group of the t-motive M. Further, we have a faithful representation $\Gamma _M \hookrightarrow \text {GL}(H_{\text {Betti}}(M))$ , which is called the tautological representation of M.

Papanikolas proved an analogue of Grothendieck’s period conjecture which unveils a deep connection between Galois groups of t-motives and transcendence.

Theorem 1.15 (Papanikolas [Reference Papanikolas47], Theorem 1.1.7)

Let M be a t-motive and let $\Gamma _M$ be its Galois group. Suppose that $\Phi \in \text {GL}_n(\overline K(t)) \cap \operatorname {\mathrm {Mat}}_{n \times n}(\overline K[t])$ represents the multiplication by $\sigma $ on M and that $\det \Phi =c(t-\theta )^s, c \in \overline K^{\times }$ . If $\Psi \in \text {GL}_n(\mathbb T)$ is a rigid analytic trivialization for $\Phi $ , then the entries of $\Psi $ may be evaluated at $\theta $ and

$$\begin{align*}\text{tr.} \deg_{\overline K} \overline K(\Psi(\theta))=\dim \Gamma_M. \end{align*}$$

Papanikolas also shows that $\Gamma _M$ equals the Galois group $\Gamma _{\Psi }$ of the Frobenius difference equation corresponding to M (see [Reference Papanikolas47], Theorem 4.5.10). This provides a method to explicitly compute the Galois groups for t-motives in many cases. This is a very powerful tool and has led to major transcendence results in the last decade. We refer the reader to [Reference Anderson, Brownawell and Papanikolas4, Reference Chang, Böckle, Goss, Hartl and Papanikolas17, Reference Chang and Papanikolas18, Reference Chang and Papanikolas19, Reference Chang, Papanikolas, Thakur and Yu20, Reference Chang, Papanikolas and Yu21, Reference Chang, Papanikolas and Yu22, Reference Chang and Yu23] for more details about transcendence applications.

Papanikolas proved that $\Gamma _M$ is an affine algebraic groupe scheme over $\mathbb F_q(t)$ which is absolutely irreducible and smooth over $\overline {\mathbb F_q(t)}$ (see [Reference Papanikolas47], Theorems 4.2.11, 4.3.1 and 4.5.10). Further, for any $\mathbb F_q(t)$ -algebra R, the map

$$\begin{align*}\Gamma_M(R) \rightarrow \text{GL}(R \otimes_{\mathbb F_q(t)} H_{\text{Betti}}(M)) \end{align*}$$

is given by the tautological map

(1.23) $$ \begin{align} \gamma \mapsto (1 \otimes \Psi^{-1} {\mathbf{m}} \mapsto (\gamma^{-1} \otimes 1) \cdot (1 \otimes \Psi^{-1} {\mathbf{m}})). \end{align} $$

1.7 Hardouin’s work

In this section, we review the work of Hardouin [Reference Hardouin37] on unipotent radicals of Tannakian groups in positive characteristic. Let F be a field and $(\mathcal T,\omega )$ be a neutral Tannakian category over F with fiber functor $\omega $ . We denote by $\mathbb G_m$ the multiplicative group over F. For an object $\mathcal U \in \mathcal T$ , we denote by $\Gamma _{\mathcal U}$ the Galois group of $\mathcal U$ . Let $\mathbf 1$ be the unit object for the tensor product and $\mathcal Y$ be a completely reducible object, which means that $\mathcal Y$ is a direct sum of finitely many irreducible objects. We consider extensions $\mathcal U \in \text {Ext}^1(\mathbf 1,\mathcal Y)$ of $\mathbf 1$ by $\mathcal Y$ , which means that we have a short exact sequence

$$\begin{align*}0 \rightarrow \mathcal Y \rightarrow \mathcal U \rightarrow \mathbf 1 \rightarrow 0. \end{align*}$$

For such an extension $\mathcal U$ , the Galois group $\Gamma _{\mathcal U}$ of U is isomorphic to the semi-direct product

$$\begin{align*}\Gamma_{\mathcal U}=R_u({\mathcal U}) \rtimes \Gamma_{\mathcal Y}, \end{align*}$$

where $R_u({\mathcal U})$ stands for the unipotent part of $\Gamma _{\mathcal U}$ . Therefore, with the knowledge of $\Gamma _{\mathcal Y}$ , we reduce the computation of $\Gamma _{\mathcal U}$ to that of its unipotent part. In [Reference Hardouin37], Hardouin proves several fundamental results which characterize $R_u({\mathcal U})$ in terms of the extension group $\text {Ext}^1(\mathbf 1,\mathcal Y)$ . In the next result, we keep the same notation as above, and we remind the reader that the action of a group G on a module V is isotypic if the module V is the direct sum of irreducible isomorphic G-modules.

As a consequence, Hardouin proves the following corollary which states that algebraic relations between the extensions are exactly given by the linear relations. We continue with the above notation.

We remark that we will apply this theorem and its corollary in §3.6 to the t-motives $\mathcal {X}_{\rho }^{\otimes n}$ and $\mathcal {X}_n(b)$ , which are defined in §2.2 and §3.2.

2.1 The t-motive associated to $\rho $

We follow the construction given by Chang-Papanikolas [Reference Chang and Papanikolas18], Sections 3.3 and 3.4. Recall that B is the integral closure of A in the Hilbert class field H. We consider $\rho :\mathbb F_q[t] \longrightarrow B\{\tau \}$ from Definition 1.3 as a Drinfeld $\mathbb F_q[t]$ -module of rank $2$ by forgetting the y action. We recall

$$\begin{align*}\rho_t=\theta+x_1 \tau + \tau^2, \quad x_1 \in B \end{align*}$$

(see [Reference Green and Papanikolas36, §3] for more details on this construction). For $\mathbf {u} = u \in \mathbb C_{\infty }$ , we denote the associated Anderson generating function $E_u(t):= E_{\mathbf {u}}^{\otimes 1}(t)$ given by Equation (1.16). This function extends meromorphically to all of $\mathbb C_{\infty }$ with simple poles at $t=\theta ^{q^i}, i \geq 0$ . Further, it satisfies the functional equation

$$\begin{align*}\rho_t(E_u(t))=\exp_{\rho}(u)+t E_u(t). \end{align*}$$

In other words, we have

$$\begin{align*}\theta E_u(t)+x_1 E_u(t)^{(1)}+E_u(t)^{(2)}=\exp_{\rho}(u)+t E_u(t). \end{align*}$$

Now we fix an $\mathbb F_q[t]$ -basis $u_1 = \pi _{\rho }$ and $u_2 = \eta \pi _{\rho }$ of the period lattice $\Lambda _{\rho }$ of $\rho $ . We set $E_i:=E_{u_i}$ for $i=1,2$ . We define the following matrices:

$$ \begin{align*} \Phi_{\rho} &= \begin{pmatrix} 0 & 1 \\ t-\theta & -x_1^{(-1)} \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K[t]), \quad \Upsilon = \begin{pmatrix} E_1 & E_1^{(1)} \\ E_2 & E_2^{(1)} \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\mathbb T), \\ \Theta &= \begin{pmatrix} 0 & t-\theta \\ 1 & -x_1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K[t]), \quad V = \begin{pmatrix} x_1 & 1 \\ 1 & 0 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K). \end{align*} $$

Then we set

$$\begin{align*}\Psi_{\rho}:=V^{-1} (\Upsilon^{(1)})^{-1}. \end{align*}$$

Since $V^{(-1)} \Phi _{\rho }=\Theta V$ and $\Upsilon ^{(1)}=\Upsilon \Theta $ , we get

$$\begin{align*}\Psi^{(-1)}_{\rho}=\Phi_{\rho} \Psi_{\rho}. \end{align*}$$

2.2 The t-motive associated to $\rho ^{\otimes n}$

Let $n \geq 2$ be an integer. Recall the definition of $\rho ^{\otimes n}$ from Definition 1.3. By forgetting the y-action, the Anderson $\textbf {A}$ -module $\rho ^{\otimes n}$ can be considered as an Anderson $\mathbb F_q[t]$ -module given by

$$\begin{align*}\rho^{\otimes n}_t=d[\theta]+E_{\theta} \tau \end{align*}$$

(see (1.14) for explicit formulas of $d[\theta ]$ and $E_{\theta }$ ). For $\mathbf {u}=(u_1,\ldots ,u_n)^{\top } \in \operatorname {\mathrm {Mat}}_{n \times 1}(\mathbb C_{\infty })$ , recall the associated Anderson generating function $E^{\otimes n}_{\mathbf {u}}(t)$ given by Equation (1.16). It satisfies the functional equation

$$\begin{align*}\rho_t(E^{\otimes n}_{\mathbf{u}}(t))=\operatorname{\mathrm{Exp}}^{\otimes n}_{\rho}(\mathbf{u})+t E^{\otimes n}_{\mathbf{u}}(t). \end{align*}$$

As n is fixed throughout this section, to simplify notation, we will suppress the dependence on n and denote the coordinates of

(2.1) $$ \begin{align} E^{\otimes n}_{\mathbf{u}}(t) := (E_{\mathbf{u},1},\dots,E_{\mathbf{u},n})^{\top}, \end{align} $$

and similarly for other vector valued functions. Recall the definitions of the functions $h_i$ and the coefficients $a_i$ and $b_i$ from Proposition 1.4. For $1 \leq i \leq n$ , we set

$$\begin{align*}\Theta_i = \begin{pmatrix} 0 & 1\\ t-\theta & -a_i \end{pmatrix},\quad \phi_i = \begin{pmatrix} 0 & 1\\ t-\theta & -b_i \end{pmatrix}. \end{align*}$$

From Proposition 1.4, we have

$$ \begin{align*} \begin{pmatrix} h_{i+1} \\ h_{i+2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ t-\theta & -b_i \end{pmatrix} \begin{pmatrix} h_i \\ h_{i+1} \end{pmatrix} = \phi_i \begin{pmatrix} h_i \\ h_{i+1} \end{pmatrix}. \end{align*} $$

We define

$$ \begin{align*} \Phi_{\rho}^{\otimes n}=\phi_n \ldots \phi_1 = \begin{pmatrix} 0 & 1 \\ t-\theta & -b_n \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -b_1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K[t]). \end{align*} $$

It follows that

$$ \begin{align*} \begin{pmatrix} h_1 \\ h_2 \end{pmatrix}^{(-1)} = \begin{pmatrix} h_{n+1} \\ h_{n+2} \end{pmatrix} = \Phi_{\rho}^{\otimes n} \begin{pmatrix} h_1 \\ h_2 \end{pmatrix}. \end{align*} $$

Now we fix $\mathbf {u}_1 = \Pi _n$ and $\mathbf {u}_2 = d[\eta ]\Pi _n$ such that $\{\mathbf {u}_1,\mathbf {u}_2\}$ is a basis of the period lattice $\Lambda _{\rho }^{\otimes n}$ of $\rho ^{\otimes n}$ , where $\Pi _n$ is defined in Proposition 1.6(d). We denote by $E_i^{\otimes n} = E_{\mathbf {u}_i}^{\otimes n}$ for $i=1,2$ . When the dimension n is fixed, we will often drop the $\otimes n$ from our notation to avoid clutter. Then

$$\begin{align*}\rho_t(E^{\otimes n}_i)=\operatorname{\mathrm{Exp}}^{\otimes n}_{\rho}(\mathbf{u}_i)+t E^{\otimes n}_i=t E^{\otimes n}_i. \end{align*}$$

If we set

$$ \begin{align*} \Theta=\Theta_n \ldots \Theta_1= \begin{pmatrix} 0 & 1 \\ t-\theta & -a_n \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -a_1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K[t]), \end{align*} $$

and

(2.2) $$ \begin{align} \Upsilon = \begin{pmatrix} E_{1,1} & E_{2,1} \\ E_{1,2} & E_{2,2} \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\mathbb{T}), \end{align} $$

then we obtain

$$\begin{align*}\Upsilon^{(1)}=\Theta \Upsilon. \end{align*}$$

We define

$$ \begin{align*} V = \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\overline K). \end{align*} $$

Note that V is symmetric and

$$ \begin{align*} V^{(-1)} = \begin{pmatrix} b_n & 1 \\ 1 & 0 \end{pmatrix}. \end{align*} $$

We claim that

(2.3) $$ \begin{align} V^{(-1)} \Phi_{\rho}^{\otimes n}=(\Theta^{\top}) V. \end{align} $$

In fact, recall from Proposition 1.4 that $b_i=a_{n-i}$ for $1 \leq i \leq n-1$ and $a_n=b_n^q$ . It is clear that for any $x \in \mathbb C_{\infty }$ , we have

$$ \begin{align*} \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ t-\theta & x \end{pmatrix} = \begin{pmatrix} 0 & t-\theta \\ 1 & x \end{pmatrix} \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix}. \end{align*} $$

Then the claim follows immediately:

$$ \begin{align*} V^{(-1)} \Phi_{\rho}^{\otimes n} &= \begin{pmatrix} b_n & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ t-\theta & -b_n \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -b_1 \end{pmatrix} \\ &= \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ t-\theta & -b_{n-1} \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -b_1 \end{pmatrix} \\ &= \begin{pmatrix} 0 & t-\theta \\ 1 & -b_{n-1} \end{pmatrix} \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ t-\theta & -b_{n-2} \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -b_1 \end{pmatrix} \\ &= \cdots \\ &= \begin{pmatrix} 0 & t-\theta \\ 1 & -b_{n-1} \end{pmatrix} \cdots \begin{pmatrix} 0 & t-\theta \\ 1 & -b_1 \end{pmatrix} \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix} \\ &= \begin{pmatrix} 0 & t-\theta \\ 1 & -a_1 \end{pmatrix} \cdots \begin{pmatrix} 0 & t-\theta \\ 1 & -a_{n-1} \end{pmatrix} \begin{pmatrix} t-\theta & 0 \\ 0 & 1 \end{pmatrix} \\ &= \begin{pmatrix} 0 & t-\theta \\ 1 & -a_1 \end{pmatrix} \cdots \begin{pmatrix} 0 & t-\theta \\ 1 & -a_{n-1} \end{pmatrix} \begin{pmatrix} 0 & t-\theta \\ 1 & -a_n \end{pmatrix} \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix} \\ &= (\Theta^{\top}) V. \end{align*} $$

We set

$$\begin{align*}\Psi_{\rho}^{\otimes n}:=V^{-1} ((\Upsilon^{\top})^{(1)})^{-1} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\mathbb{L}). \end{align*}$$

Thus, we get

$$\begin{align*}(\Psi^{\otimes n}_{\rho})^{(-1)}=\Phi_{\rho}^{\otimes n} \Psi_{\rho}^{\otimes n}. \end{align*}$$

Remark 2.1. From the previous discussion, we have

$$ \begin{align*} (\Psi_{\rho}^{\otimes n})^{-1}=(\Upsilon^{\top})^{(1)} V= \begin{pmatrix} a_n E_{1,1}^{(1)}+E_{1,2}^{(1)} & E_{1,1}^{(1)} \\ a_n E_{2,1}^{(1)}+E_{2,2}^{(1)} & E_{2,2}^{(1)} \end{pmatrix}. \end{align*} $$

By direct calculations (see Lemma 3.3), we show that

$$\begin{align*}[\Psi_{\rho}^{\otimes n}]^{-1}_{i,1}(\theta)=\mathbf{u}_{i,n} \in K \pi^n_{\rho}, \end{align*}$$

where $\mathbf {u}_{i,n}$ is the nth coordinate of the period $\mathbf {u}_i$ for $i=1,2$ .

2.3 Galois groups

We denote by $\mathcal {X}^{\otimes n}_{\rho }$ the pre t-motive associated to $\rho ^{\otimes n}$ . The following proposition gives some basic properties of this pre t-motive (compare to [Reference Chang and Papanikolas18], Theorem 3.5.4).

Proof. For Part 1, since $\mathcal {X}^{\otimes n}_{\rho }$ is an A-motive of rank $1$ , it follows from [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Proposition 2.3.24, Part b that $\mathcal {X}^{\otimes n}_{\rho }$ is uniformizable as an A-motive. This implies that $\mathcal {X}^{\otimes n}_{\rho }$ is uniformizable as an $\mathbb F_q[t]$ -motive. Thus, Part 1 follows immediately.

For Part 2, if $n=1$ , then $\rho $ is a Drinfeld $\mathbb F_q[t]$ -module. By [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Example 2.2.5, we know that $\rho $ is pure. Thus, the motive N (viewed as a t-motive) is pure, which implies $N_n$ is also pure as a t-motive, by [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Proposition 2.3.11(e). Hence, we get the purity of $\mathcal {X}^{\otimes n}_{\rho }$ .

We now prove Part 3. To calculate the Galois group $\Gamma _{\mathcal {X}_{\rho }^{\otimes n}}$ associated to the t-motive $\mathcal {X}^{\otimes n}_{\rho }$ , we will use [Reference Juschka41], Theorem 5.1.2. We claim that the t-motive $\mathcal {X}^{\otimes n}_{\rho }$ verifies all the conditions of this theorem. In fact, it is a pure uniformizable dual $\mathbb F_q[t]$ -motive thanks to Part 2. Further, it has complex multiplication since $\operatorname {\mathrm {End}}_{\mathbb C_{\infty }}(\mathcal {X}^{\otimes n}_{\rho })=\operatorname {\mathrm {End}}_{\mathbb C_{\infty }}(\rho ^{\otimes n})=A$ by [Reference Green33], Lemma 7.3. Thus, we apply [Reference Juschka41], Theorem 5.1.2 to the t-motive $\rho ^{\otimes n}$ to obtain

$$\begin{align*}\Gamma_{\mathcal{X}_{\rho}^{\otimes n}}=\textrm{Res}_{K/\mathbb F_q[t]} \mathbb{G}_{m,K}.\\[-40pt] \end{align*}$$

Remark 2.4. 1) We note that the Galois group associated to the A-motive $\rho ^{\otimes n}$ is also equal to $\textrm {Res}_{K/\mathbb F_q[t]} \mathbb {G}_{m,K}$ (see [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Example 2.3.29).

2) We should mention a similar result of Pink and his collaborators that completely determines the Galois group of a Drinfeld A-module (see [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Theorem 2.6.3 and also [Reference Chang and Papanikolas19], Theorem 3.5.4 for more details). It states that if M is the t-motive associated to a Drinfeld A-module defined over $\overline K$ , then

$$\begin{align*}\Gamma_M=\text{Cent}_{\text{GL}(H_{\text{Betti}}(M))} \operatorname{\mathrm{End}}_{\mathbb C_{\infty}}(M). \end{align*}$$

Proposition 2.5. The entries of $\Psi _{\rho }^{\otimes n}$ are regular at $t=\theta $ , and we have

$$\begin{align*}\text{tr. deg}_{\overline K} \overline K(\Psi_{\rho}^{\otimes n}(\theta))=\dim \Gamma_{\mathcal{X}_{\rho}^{\otimes n}}. \end{align*}$$

Proof. By [Reference Papanikolas47], Proposition 3.3.9 (c) and Section 4.1.6, there exists a matrix $U \in \text {GL}_2(\mathbb F_q(t))$ such that $\widetilde {\Psi }=\Psi _{\rho }^{\otimes n} U$ is a rigid analytic trivialization of $\Phi _{\rho }^{\otimes n}$ and $\widetilde {\Psi } \in \text {GL}_2(\mathbb T)$ . By [Reference Anderson, Brownawell and Papanikolas4], Proposition 3.1.3, the entries of $\widetilde {\Psi }$ are entire functions in the variable t. Thus, in particular, the entries of $\widetilde {\Psi }$ and $U^{-1}$ are regular at $t=\theta $ . This implies that the entries of $\Psi _{\rho }^{\otimes n}$ are regular at $t=\theta $ .

Further, since $\overline K(\Psi _{\rho }^{\otimes n}(\theta ))=\overline K(\widetilde {\Psi }(\theta ))$ , by Theorem 1.15, we have

$$\begin{align*}\text{tr. deg}_{\overline K} \overline K(\Psi_{\rho}^{\otimes n}(\theta))=\text{tr. deg}_{\overline K} \overline K(\widetilde{\Psi}(\theta))=\dim \Gamma_{\mathcal{X}_{\rho}^{\otimes n}}. \end{align*}$$

The proof is finished.

2.4 Endomorphisms of t-motives

We write down explicitly the endomorphism of $\mathcal {X}^{\otimes n}_{\rho }$ given by $y \in A$ . In [Reference Green34], Proposition 4.2 gives a formula for $yg_i$ for the basis elements $g_i$ of Proposition 1.4. Using the same strategy, and replacing the $g_i$ by $h_i$ , a short argument shows that there exist $y_i, z_i \in H$ such that for $1 \leq i \leq n$ , we have

$$\begin{align*}yh_i=\eta h_i + y_i h_{i+1}+z_i h_{i+2}+h_{i+3}.\end{align*}$$

We deduce that there exists $M_y \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\overline K[t])$ such that the endomorphism y expressed in terms of the basis $\{h_1,h_2\}$ is represented by $M_y$ :

$$ \begin{align*} y \begin{pmatrix} h_1 \\ h_2 \end{pmatrix} =M_y \begin{pmatrix} h_1 \\ h_2 \end{pmatrix}. \end{align*} $$

Since $A \cong \mathbb {F}_q[t] + y\mathbb {F}_q[t]$ , using the formulas above for any element $a \in A$ , we can find a matrix $M_a \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\overline K[t])$ such that the endomorphism induced by multiplication by a on $\mathcal {X}^{\otimes n}_{\rho }$ in the basis $\{h_1,h_2\}$ is represented by $M_a$ :

$$ \begin{align*} a \begin{pmatrix} h_1 \\ h_2 \end{pmatrix} =M_a \begin{pmatrix} h_1 \\ h_2 \end{pmatrix}. \end{align*} $$

3.1 Logarithms attached to $\rho $

We keep the notation of Section 2.1. Following Chang-Papanikolas (see [Reference Chang and Papanikolas18], Section 4.2), for $u \in \mathbb C_{\infty }$ with $\exp _{\rho }(u)=v \in \overline K$ , we consider $E_u$ the associated Anderson generating function associated to u given by (1.16). We set

$$ \begin{align*} h_v = \begin{pmatrix} v \\ 0 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 1}(\overline K), \quad \Phi_v = \begin{pmatrix} \Phi_{\rho} & 0 \\ h_v^{\top} & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]). \end{align*} $$

We define

$$ \begin{align*} g_v &=V \begin{pmatrix} -E_u^{(1)} \\ -E_u^{(2)} \end{pmatrix} = \begin{pmatrix} -(t-\theta)E_u-v \\ -E_u^{(1)} \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 1}(\mathbb{T}), \\ \Psi_v &= \begin{pmatrix} \Psi_{\rho} & 0 \\ g_v^{\top} \Psi_{\rho} & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\mathbb{T}). \end{align*} $$

Then we get

$$\begin{align*}\Phi_{\rho}^{\top} g_v^{(-1)}=g_v+h_v,\end{align*}$$

and

$$\begin{align*}\Psi_v^{(-1)}=\Phi_v \Psi_v.\end{align*}$$

The associated pre-motive $\mathcal {X}_v$ is, in fact, a t-motive in the sense of Papanikolas as is proved in [Reference Chang and Papanikolas19, §4.2].

For $v:=\exp _{\rho }(\zeta _{\rho }( b,1)) \in H$ , we call the corresponding t-motive $\mathcal {X}_v$ the zeta motive associated to $\zeta _{\rho }( b,1)$ .

3.2 Logarithms attached to $\rho ^{\otimes n}$

We now switch to the case for $n\geq 2$ and use freely the notation of Section 2.2. We fix $\mathbf {u} \in \operatorname {\mathrm {Mat}}_{n \times 1}(\mathbb C_{\infty })$ with $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(\mathbf {u})=\mathbf {v} \in \operatorname {\mathrm {Mat}}_{n \times 1}(\overline K)$ , and we consider $E_{\mathbf {u}}^{\otimes n}$ the Anderson generating function associated to $\mathbf {u}$ given by Equation (1.16). Recall that

(3.1) $$ \begin{align} \rho_t(E^{\otimes n}_{\mathbf{u}}(t))=\operatorname{\mathrm{Exp}}^{\otimes n}_{\rho}(\mathbf{u})+t E^{\otimes n}_{\mathbf{u}}(t)=\mathbf{v}+t E^{\otimes n}_{\mathbf{u}}(t). \end{align} $$

Thus, we get

$$ \begin{align*} \begin{pmatrix} E_{\mathbf{u},i+1} \\ E_{\mathbf{u},i+2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ t-\theta & -a_i \end{pmatrix} \begin{pmatrix} E_{\mathbf{u},i} \\ E_{\mathbf{u},i+1} \end{pmatrix}+ \begin{pmatrix} 0 \\ -v_i \end{pmatrix}= \Theta_i \begin{pmatrix} E_{\mathbf{u},i} \\ E_{\mathbf{u},i+1} \end{pmatrix}+ \begin{pmatrix} 0 \\ -v_i \end{pmatrix}. \end{align*} $$

We define the vector $f_{\mathbf {v}}:= (f_{\mathbf {v},1},f_{\mathbf {v},2})^{\top }$ given by

(3.2) $$ \begin{align} f_{\mathbf{v}} = \Theta_n \cdots \Theta_2 \begin{pmatrix} 0 \\ v_1 \end{pmatrix} + \Theta_n \cdots \Theta_3 \begin{pmatrix} 0 \\ v_2 \end{pmatrix} + \cdots + \begin{pmatrix} 0 \\ v_n \end{pmatrix}. \end{align} $$

It follows that

$$ \begin{align*} \begin{pmatrix} E_{\mathbf{u},1} \\ E_{\mathbf{u},2} \end{pmatrix}^{(1)} = \Theta \begin{pmatrix} E_{\mathbf{u},1} \\ E_{\mathbf{u},2} \end{pmatrix}- \begin{pmatrix} f_{\mathbf{v},1} \\ f_{\mathbf{v},2} \end{pmatrix}. \end{align*} $$

Here, we recall

$$ \begin{align*} \Theta= \begin{pmatrix} 0 & 1 \\ t-\theta & -a_n \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ t-\theta & -a_1 \end{pmatrix} \end{align*} $$

and

$$ \begin{align*} \Upsilon = \begin{pmatrix} E_{1,1} & E_{2,1} \\ E_{1,2} & E_{2,2} \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{2 \times 2}(\mathbb{T}). \end{align*} $$

They verify

$$\begin{align*}\Upsilon^{(1)}=\Theta \Upsilon.\end{align*}$$

We set

(3.3) $$ \begin{align} \Theta_{\mathbf{v}} = \begin{pmatrix} \Theta & -(f_{\mathbf{v},1},f_{\mathbf{v},2})^{\top} \\ 0 & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]), \quad \Upsilon_{\mathbf{v}} = \begin{pmatrix} \Upsilon & (E_{\mathbf{u},1},E_{\mathbf{u},2})^{\top} \\ 0 & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\mathbb{T}). \end{align} $$

Then we get

$$\begin{align*}\Upsilon^{(1)}_{\mathbf{v}}=\Theta_{\mathbf{v}} \Upsilon_{\mathbf{v}}.\end{align*}$$

Now we are ready to construct the associated rigid analytic trivialization. Recall that

$$ \begin{align*} V = \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix}. \end{align*} $$

Note that V is symmetric. We set

$$ \begin{align*} W &= \text{diag}(V,1)= \begin{pmatrix} V & 0 \\ 0 & 1 \end{pmatrix} \in \text{GL}_3(\overline K), \\ \Phi_{\mathbf{v}} &=(W^{(-1)})^{-1} (\Theta_{\mathbf{v}}^{\top}) W =\begin{pmatrix} \Phi_{\rho}^{\otimes n} & 0 \\ (h_{\mathbf{v},1},h_{\mathbf{v},2}) & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]), \\ \Psi_{\mathbf{v}} &=W^{-1} ((\Upsilon_{\mathbf{v}}^{\top})^{(1)})^{-1}=\begin{pmatrix} \Psi_{\rho}^{\otimes n} & 0 \\ (g_{\mathbf{v},1},g_{\mathbf{v},2})\Psi_{\rho}^{\otimes n} & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\mathbb{T}). \end{align*} $$

Remark 3.1. Note that by direct calculations, we obtain

$$\begin{align*}(g_{\mathbf{v},1},g_{\mathbf{v},2})\Psi_{\rho}^{\otimes n}=-(a_n E_{\mathbf{u},1}^{(1)}+E_{\mathbf{u},2}^{(1)}, E_{\mathbf{u},1}^{(1)}).\end{align*}$$

Further, we will show below that $g_{\mathbf {v},1}(\theta )=u_n-v_n$ , where $u_n$ and $v_n$ are the last coordinate of $\mathbf {u}$ and $\mathbf {v}$ , respectively (see Lemma 3.3).

Thus, we get

$$\begin{align*}\Psi_{\mathbf{v}}^{(-1)}=\Phi_{\mathbf{v}} \Psi_{\mathbf{v}}.\end{align*}$$

The associated pre-motive $\mathcal {X}_{\mathbf {v}}$ is in fact a t-motive because it is an extension of two t-motives (see for example [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38], Lemma 2.3.25).

3.3 Period calculations

In this section, we show explicitly how to obtain the periods and zeta values discussed in the previous section from evaluations of the entries of the rigid analytic trivialization $\Psi _{\mathbf {v}}$ , for fixed $\mathbf {u}$ and $\mathbf {v}$ as in the previous section.

Lemma 3.2. Let $E_{\mathbf {u}}^{\otimes n}$ be defined as above. Then

$$\begin{align*}\operatorname{\mathrm{Res}}_{\theta}(E_{\mathbf{u}}^{\otimes n} dt) = \left (\begin{matrix} \operatorname{\mathrm{Res}}_{\theta}(E_{\mathbf{u},1}dt) \\ \vdots \\ \operatorname{\mathrm{Res}}_{\theta}(E_{\mathbf{u},n}dt) \\ \end{matrix}\right ) = -\mathbf{u}.\end{align*}$$

Proof. Write $\mathbf {u} = (u_1,\dots ,u_n)^{\top } \in \mathbb C_{\infty }^n$ . As in the proof of [Reference Green34, Prop. 6.5], we have the identity

$$\begin{align*}E_{\mathbf{u}}^{\otimes n}= \sum_{j=0}^{\infty} Q_j\left (d[\theta]^{(j)} - t\text{Id}_n \right )^{-1}\mathbf{u}^{(j)},\end{align*}$$

and we find that only the $j=0$ term contributes to the residue. Then, using the cofactor expansion of $(d[\theta ]^{(j)} - t\text {Id}_n )^{-1}$ from [Reference Green34, Pg. 26], we find that

$$\begin{align*}\operatorname{\mathrm{Res}}_{\theta}(E_{\mathbf{u}}^{\otimes n} dt) = \left (\begin{matrix} \operatorname{\mathrm{Res}}_{\theta}\left((\frac{u_1}{\theta-t} + r_1(t))dt\right)\\ \vdots\\ \operatorname{\mathrm{Res}}_{\theta}\left((\frac{u_n}{\theta-t} + r_n(t))dt\right) \end{matrix}\right) = - \left(\begin{matrix} u_1\\ \vdots\\ u_n \end{matrix}\right), \end{align*}$$

where $r_i(t)$ is some function in powers of $(\theta -t)^k$ for $k\leq -2$ , and hence does not contribute to the residue (see [Reference Green34, (57)] and preceding discussion for more details).

Lemma 3.3. Let $E_{\mathbf {u}}^{\otimes n} = (E_{\mathbf {u},1},\dots ,E_{\mathbf {u},n})^{\top }$ as above and let $a_i$ be the defining coefficients for the Drinfeld t-module action as in Proposition 1.4. Then if we write $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(\mathbf {u}) := \mathbf {v} = (v_1,\dots ,v_n)^{\top }$ ,

$$\begin{align*}a_nE_{\mathbf{u},1}^{(1)}(\theta) + E_{\mathbf{u},2}^{(1)}(\theta) = -u_n + v_n .\end{align*}$$

Proof. As in (3.1), we have that

$$\begin{align*}\rho_t^{\otimes n}(E_{\mathbf{u}}^{\otimes n}) = (d[\theta] + E_{\theta} \tau)(E_{\mathbf{u}}^{\otimes n}) = \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\mathbf{u}) + tE_{\mathbf{u}}^{\otimes n}.\end{align*}$$

Rearranging terms gives

$$ \begin{align*} E_{\theta} (E_{\mathbf{u}}^{\otimes n})^{(1)} &= (t-d[\theta])E_{\mathbf{u}}^{\otimes n} + \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\mathbf{u}). \end{align*} $$

Both sides of the above equation are regular at $t=\theta $ in each coordinate, so evaluating at $t=\theta $ , using the formula for $E_{\theta }$ and taking the last coordinate gives

$$\begin{align*}a_nE_{\mathbf{u},1}^{(1)}(\theta) + E_{\mathbf{u},2}^{(1)}(\theta) = E_{\mathbf{u},n}(t-\theta)\big|_{t=\theta}+v_n.\end{align*}$$

Finally, from our analysis in Lemma 3.2, we see that $E_{\mathbf {u},n}$ has a simple pole at $t=\theta $ , and thus the right-hand side of the above equation is simply the residue at $t=\theta $ . Thus,

$$\begin{align*}a_nE_{\mathbf{u},1}^{(1)}(\theta) + E_{\mathbf{u},2}^{(1)}(\theta) = \operatorname{\mathrm{Res}}_{\theta}(E_{\mathbf{u},n}dt)+v_n = -u_n+v_n.\\[-40pt]\end{align*}$$

Proof. As above, we have that $\overline K(\Psi _{\mathbf {v}}(\theta )) = \overline K(\Upsilon _{\mathbf {v}}(\theta ))$ . Then note that

$$\begin{align*}E^{(1)}_{i,j}(\theta),E^{(1)}_{\mathbf{u},i}(\theta)\in \overline K(\Upsilon_{\mathbf{v}}(\theta))\end{align*}$$

for $1\leq i,j\leq 2$ by construction. Then by Lemma 3.3, we see that the last coordinates of $\mathbf {u}_1,\mathbf {u}_2,\mathbf {u}$ are contained in $\overline K(\Upsilon _{\mathbf {v}}(\theta ))$ , where $\mathbf {u}_1,\mathbf {u}_2$ are an $\mathbb F_q[t]$ -basis for the period lattice of $\rho ^{\otimes n}$ and $\mathbf {u}$ is a vector such that $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(\mathbf {u}) = \mathbf {v} \in \overline K$ . From [Reference Green34, Thm. 6.7], we know that the last coordinate of $\Pi _n$ is an algebraic multiple of $\pi _{\rho }^n$ , and thus it follows that $\pi _{\rho }^n$ and $u_n$ are contained in $\overline K(\Psi _{\mathbf {v}}(\theta ))$ .

We next prove a lemma about the linear relations between the entries of $\overline K(\Psi _{\rho }^{\otimes n}(\theta ))$ . Let $\pi _{\rho }$ and $\Pi _n$ be generators of the period lattices of $\exp _{\rho }$ and $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}$ , respectively, as $\textbf {A}$ -modules. Then recall that $\{\pi _{\rho },\eta \pi _{\rho }\}$ and $\{\Pi _n,d[y]\Pi _n\}$ are bases for $\Lambda _{\rho }$ and $\Lambda _{\rho }^{\otimes n}$ , respectively, as $\mathbb {F}_q[t]$ -modules. Also for $\mathbf {u} \in \mathbb {C}_{\infty }^n$ , we recall the definition of $G_{\mathbf {u}}$ from 1.17 and define

$$\begin{align*}\overline G_{\mathbf{u}}^{\otimes n} = -yE_{\mathbf{u}}^{\otimes n} + E_{\eta\mathbf{u}}^{\otimes n},\end{align*}$$

and note that this equals $[-1]G_{\mathbf {u}}^{\otimes n}$ , where $[-1]$ represents the inverse of the group law on the elliptic curve X. We will denote the coordinates of $G_{\mathbf {u}}^{\otimes n}:= (G_{\mathbf {u},1},\dots ,G_{\mathbf {u},n})$ and similarly for $\overline G_{\mathbf {u}}$ .

Recall that $\Psi _{\rho }$ and $\Psi _{\rho }^{\otimes n}$ are the $2\times 2$ rigid analytic trivialization matrices from §2.1–2.2. The following lemma gives $\overline K$ -linear relations between the entries of these matrices evaluated at $t=\theta $ and thus allows us to find a smaller set of generators for the following fields, which notably include powers of the period.

Proof. To ease the exposition, we will assume that $p=\text {char}(\mathbb F_q) \geq 3$ , so that we may assume the elliptic curve X has Weierstrass equation given by $y^2 = t^3+at+b$ with $a,b\in \mathbb {F}_q$ and such that inversion on X is given by $[-1]:(t,y)\mapsto (t,-y)$ . The lemma is also true for $p=\text {char}(\mathbb F_q) =2$ , and the proof is similar, but calculations are more cumbersome. Particularly, the negation map is more complicated, and this requires more sophisticated analysis. Additionally, we give the full details for the case $n\geq 2$ . The details for the case for $n=1$ are similar, and we leave them to the interested reader. By our definition of $G_{\Pi _n}^{\otimes n}$ , we can write

$$\begin{align*}\begin{pmatrix} y & 1 & 0 & 0\\ -y & 1 & 0 & 0\\ 0 & 0 & y & 1\\ 0 & 0 & -y & 1\\ \end{pmatrix}\hspace{6pt} \begin{pmatrix} E_{\Pi_n,1}^{(1)}\\ E_{d[y]\Pi_n,1}^{(1)}\\ E_{\Pi_n,2}^{(1)}\\ E_{d[y]\Pi_n,2}^{(1)}\\ \end{pmatrix} = \begin{pmatrix} G_{\Pi_n,1}^{(1)}\\ \overline G_{\Pi_n,1}^{(1)}\\ G_{\Pi_n,2}^{(1)}\\ \overline G_{\Pi_n,2}^{(1)}\\ \end{pmatrix}, \end{align*}$$

and, in particular, we note that the above matrix is invertible in $\operatorname {\mathrm {Mat}}_{4\times 4}(\textbf {K})$ . By inverting the above matrix, this allows us to write each of the Anderson generating functions $E_{d[y]\Pi _n,k}^{(1)}$ for $k=1,2$ as a $\textbf {K}$ -linear combination of the functions $G_{\Pi _n,k}^{(1)}$ and $\overline G_{\Pi _n,k}^{(1)}$ . Further, from Proposition 1.6(d), we get the formula $g_2\cdot G_{\Pi _n,1} = g_1 \cdot G_{\Pi _n,2}$ . Note that the functions $g_i\in \overline K(t,y)$ . Taken together, these two facts allow us to write the functions $E_{d[\eta ]\Pi _n,1}^{(1)}(\theta )$ and $E_{d[\eta ]\Pi _n,2}^{(1)}(\theta )$ as $\overline K$ -linear combinations of $E_{\Pi _n,1}^{(1)}(\theta )$ and $E_{\Pi _n,2}^{(1)}(\theta )$ . As $\overline K(\Psi _{\rho }^{\otimes n}(\theta )) = \overline K(\Upsilon (\theta ))$ ( $\Upsilon $ is defined in (2.2)), and this allows us to conclude that $\overline K(\Psi _{\rho }^{\otimes n}(\theta )) = \overline K(E_{\Pi _n,1}^{(1)}(\theta ),E_{\Pi _n,2}^{(1)}(\theta ))$ . Finally, setting $\mathbf {u} = \Pi _n$ and $\mathbf {v} = 0$ in Lemma 3.3 shows that a $\overline K$ -linear combination of $E_{\Pi _n,1}^{(1)}(\theta )$ and $E_{\Pi _n,2}^{(1)}(\theta )$ equals the last coordinate of $\Pi _n$ , which is an algebraic multiple of $\Pi _n^n$ by Proposition 1.6(e). It remains to take an additional linearly independent combination of $E_{\Pi _n,1}^{(1)}(\theta )$ and $E_{\Pi _n,2}^{(1)}(\theta )$ , which we set equal to $W_n$ , to finish the proof.

3.4 An application of Hartl-Juschka’s work to period calculations

In this section, we maintain the notation from the previous section of $\mathbf {u}_1,\mathbf {u}_2$ being a generating set for the period lattice $\Lambda _{\rho }^{\otimes n}$ and $\mathbf {u}\in \mathbb {C}_{\infty }^n$ . We wish to apply Corollaries 2.5.23 and 2.5.24 from Hartl and Juschka [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38] (originally due to Anderson in unpublished work) to give a more conceptual method for period calculations with an aim towards generalizing these arguments to curves of arbitrary genus. We restate [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38, Cors. 2.5.23, 2.5.24] here for the convenience of the reader, but we first translate their notation into our setting. By definition, $\sigma $ acts by the matrix $\Phi _{\rho }^{\otimes n}$ on a $\mathbb {C}_{\infty }[t]$ -basis $\{h_1,h_2\}$ of $N_n$ . Explicitly, for $z \in N_n$ , we express $z = ah_1 + bh_2$ with $a,b\in \mathbb C_{\infty }[t]$ and we get

$$\begin{align*}\sigma(z) = \sigma(ah_1 + bh_2) = \sigma\left ((a,b) \begin{pmatrix} h_1\\ h_2 \end{pmatrix}\right ) = (a,b)^{(-1)} \Phi_{\rho}^{\otimes n} \begin{pmatrix} h_1\\ h_2 \end{pmatrix}.\end{align*}$$

Thus, we see that if we view $N_n$ as a free $\mathbb C_{\infty }[t]$ -module, then $\sigma $ acts by inverse twisting and right multiplication by $\Phi _{\rho }^{\otimes n}$ . Transposing, we get a left multiplication:

$$\begin{align*}\sigma\begin{pmatrix} a\\ b \end{pmatrix} = \left (\Phi_{\rho}^{\otimes n}\right )^{\top} \begin{pmatrix} a\\ b \end{pmatrix}^{(-1)},\quad a,b\in \mathbb{F}_q[t].\end{align*}$$

By [Reference Namoijam and Papanikolas45, Lemma 3.4.1], $\delta _0$ extends to $N_n \otimes _{\mathbb C_{\infty }[t]} \mathbb {T}_{\theta }$ , where $\mathbb {T}_{\theta }$ is a Tate algebra of functions with radius of convergence $|\theta |_{\infty }$ (see §1.1.2).

Corollary 3.6. (This is [Reference Hartl, Juschka, Böckle, Goss, Hartl and Papanikolas38, Cor. 2.5.23 and 2.5.24]) Let $N_n$ and $\Phi _{\rho }^{\otimes n}$ be as above. Further, let $\mathbf {w}\in \mathbb {T}_{\theta }^2$ satisfy

(3.4) $$ \begin{align} (\sigma-1)(\mathbf{w}) = (\Phi_{\rho}^{\otimes n})^{\top}\mathbf{w}^{(-1)} - \mathbf{w} = \mathbf{z} \in N_n. \end{align} $$

Then

$$\begin{align*}\operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\delta_0(\mathbf{w}+\mathbf{z})) = \delta_1(\mathbf{z}) .\end{align*}$$

Further, if $\mathbf {z}=0$ , then $\delta _0(\mathbf {w}) \in \Lambda _{\rho }^{\otimes n}$ and the set of all such $\mathbf {w}$ forms a spanning set for the periods.

So, we wish to look for vectors $\mathbf {w}\in \mathbb {T}_{\theta }^2$ which satisfy (3.4) for some $z\in N_n$ .

Lemma 3.7. For $\Upsilon $ as in (2.2), we have

$$\begin{align*}(\Phi_{\rho}^{\otimes n})^{\top}\left (V^{\top} \Upsilon^{(1)} \right )^{(-1)} = V^{\top} \Upsilon^{(1)}.\end{align*}$$

Proof. From (2.3), we have that $(\Phi _{\rho }^{\otimes n}) = (V^{(-1)})^{-1} \Theta ^{\top } V$ , and then substituting in gives

$$ \begin{align*} (\Phi_{\rho}^{\otimes n})^{\top}(V^{\top} \Upsilon^{(1)})^{(-1)} &= ((V^{(-1)})^{-1} \Theta^{\top} V)^{\top}(V^{\top} \Upsilon^{(1)})^{(-1)}\\ &= V^{\top}\Theta ((V^{(-1)})^{-1})^{\top})(V^{\top} \Upsilon^{(1)})^{(-1)}\\ &= V^{\top} (\Theta\Upsilon)\\ &= V^{\top} (\Upsilon^{(1)}).\\[-39pt] \end{align*} $$

We comment that $V^{\top } \Upsilon ^{(1)} = ((\Psi _{\rho }^{\otimes n})^{-1})^{\top } \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {T}_{\theta })$ by Remark 2.2, but to save on notation, we shall denote $P := V^{\top } \Upsilon ^{(1)} \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {T}_{\theta })$ and denote the columns of P by $P_i \in \mathbb {T}_{\theta }^2$ . Thus, for $i=1,2$ , we have $(\Phi _{\rho }^{\otimes n})^{\top } P_i^{(-1)} - P_i = 0$ , and therefore the vectors $P_i$ satisfy the conditions of Lemma 3.6. So we get

(3.5) $$ \begin{align} \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\delta_0(P_i+0)) = \delta_1(0) = 0. \end{align} $$

Lemma 3.8. For $\mathbf {u} \in \operatorname {\mathrm {Mat}}_{n \times 1}(\mathbb C_{\infty })$ with $\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(\mathbf {u})=\mathbf {v} \in \operatorname {\mathrm {Mat}}_{n \times 1}(\overline K)$ , we let $E_{\mathbf {u}}^{\otimes n}$ be the Anderson generating function (with coordinates denoted as in (2.1)) associated to $\mathbf {u}$ and let $E_{\mathbf {u},*} = (E_{\mathbf {u},1},E_{\mathbf {u},2})^{\top }$ and $P_{\mathbf {u}} := V^{\top } E_{\mathbf {u},*}^{(1)} \in \mathbb {T}_{\theta }^2$ . Then $P_{\mathbf {u}}$ satisfies the conditions of Lemma 3.6; that is,

$$\begin{align*}(\Phi_{\rho}^{\otimes n})^{\top}(P_{\mathbf{u}}) - P_{\mathbf{u}} = V^{\top} f_{\mathbf{v}},\end{align*}$$

where $f_{\mathbf {v}}$ is the vector defined in Section 3.2 and $V^{\top } f_{\mathbf {v}} \in \overline K[t]^2$ .

Proof. As stated as in Section 3.2, we find that $\Theta E_{\mathbf {u},*} = E_{\mathbf {u},*}^{(1)} + f_{\mathbf {v}}$ . We then calculate that

$$ \begin{align*} (\Phi_{\rho}^{\otimes n})^{\top} (V^{\top} E_{\mathbf{u},*}^{(1)})^{(-1)} &= V^{\top}\Theta ((V^{(-1)})^{-1})^{\top} (V^{(-1)})^{\top} E_{\mathbf{u},*}\\ &= V^{\top} \Theta E_{\mathbf{u},*} \\ &= V^{\top} E_{\mathbf{u},*}^{(1)} + V^{\top} f_{\mathbf{v}}.\\[-40pt] \end{align*} $$

Thus, $P_{\mathbf {u}}$ satisfies the conditions for Lemma 3.6, and we can write

(3.6) $$ \begin{align} \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\delta_0(P_{\mathbf{u}}+V^{\top} f_{\mathbf{v}})) = \delta_1(V^{\top} f_{\mathbf{v}}). \end{align} $$

Proof. By definition, we have that $\overline K(\Psi _{\mathbf {v}}(\theta )) = \overline K(\Upsilon _{\mathbf {v}}(\theta ))$ , and we further see that $\overline K(\Upsilon _{\mathbf {v}}(\theta )) = \overline K(P(\theta ),E_{\mathbf {u},1}^{(1)}(\theta ),E_{\mathbf {u},2}^{(1)}(\theta ))$ , for $P \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {T}_{\theta })$ defined in the proof of Lemma 3.7. Lemma 3.6 implies that $\delta _0(P_i)$ for $i=1,2$ is in the period lattice, and we deduce that the vectors $\delta _0(P_i)$ must form a generating set for the period lattice over $\mathbb {F}_q[t]$ . This implies that some $\textbf {A}$ -linear combination of $\delta _0(P_1)$ and $\delta _0(P_2)$ (via the $\text {Lie}_{\rho }^{\otimes n}(\textbf {A})$ -action of $\textbf {A}$ ) equals $\Pi _n$ , the fundamental period. Since the bottom coordinate of $\Pi _n$ is a $\overline K$ -multiple of $\pi _{\rho }^n$ , it implies that $\pi _{\rho }^n \in \overline K(\Psi _{\mathbf {v}}(\theta ))$ .

We now perform a similar analysis on Equation (3.6). To proceed, we need to better understand the vector $f_{\mathbf {v}}:= (f_{\mathbf {v},1},f_{\mathbf {v},2})^{\top }$ for $\mathbf {v}$ chosen as in Lemma 3.8.

We recall from Section 2.2 that

$$\begin{align*}\Theta_i = \begin{pmatrix} 0 & 1\\ t-\theta & a_i \end{pmatrix},\quad \phi_i = \begin{pmatrix} 0 & 1\\ t-\theta & b_i \end{pmatrix},\quad V = \begin{pmatrix} a_n & 1\\ 1 & 0 \end{pmatrix}. \end{align*}$$

If we denote

$$\begin{align*}X = \begin{pmatrix} t-\theta & 0\\ 0 & 1 \end{pmatrix}, \end{align*}$$

then we have the following equalities from (2.3):

$$\begin{align*}\Theta_n^{\top} V=V^{(-1)} \phi_n=X, \quad \Theta_{n-i}^{\top} X = X \phi_i, \quad 1 \leq i \leq n-1. \end{align*}$$

By the definition of $f_{\mathbf {v}}$ (3.2), we find that

$$\begin{align*}V^{\top} f_{\mathbf{v}} = V^{\top}\Theta_n \cdots \Theta_2 \begin{pmatrix} 0 \\ v_1 \end{pmatrix} + V^{\top}\Theta_n \cdots \Theta_3 \begin{pmatrix} 0 \\ v_2 \end{pmatrix} + \cdots + V^{\top}\begin{pmatrix} 0 \\ v_n \end{pmatrix}.\end{align*}$$

Then, in anticipation of calculating $\delta _0(V^{\top } f_{\mathbf {v}})$ , using the above equalities, we find that

$$ \begin{align*} (h_1, h_2)V^{\top} f_{\mathbf{v}} &= (h_1, h_2)\left (V^{\top}\Theta_n \cdots \Theta_2 \begin{pmatrix} 0 \\ v_1 \end{pmatrix} + V^{\top}\Theta_n \cdots \Theta_3 \begin{pmatrix} 0 \\ v_2 \end{pmatrix} + \cdots + V^{\top}\begin{pmatrix} 0 \\ v_n \end{pmatrix}\right )\\ &= \left ( (0,v_1) \Theta_2^{\top} \cdots \Theta_n^{\top} V + \cdots + (0,v_{n-1})\Theta_n^{\top} V + (0,v_n) V\right )\begin{pmatrix} h_1 \\ h_2 \end{pmatrix}\\ &= \left ( (0,v_1) X \phi_{n-2}\cdots \phi_1 + \cdots + (0,v_{n-2}) X \phi_1 +(0,v_{n-1}) X + (0,v_n) V\right )\begin{pmatrix} h_1 \\ h_2 \end{pmatrix}. \end{align*} $$

Now recall that (see Section 2.2)

$$\begin{align*}\begin{pmatrix} h_{i+1} \\ h_{i+2} \end{pmatrix} = \phi_i \begin{pmatrix} h_i \\ h_{i+1} \end{pmatrix}. \end{align*}$$

Since $(0,v_i)X=(0,v_i)$ , it follows that

$$ \begin{align*} (h_1, h_2)V^{\top} f_{\mathbf{v}} &= \left ( (0,v_1) X \phi_{n-2}\cdots \phi_1 + \cdots + (0,v_{n-2}) X \phi_1 +(0,v_{n-1}) X + (0,v_n) V\right )\begin{pmatrix} h_1 \\ h_2 \end{pmatrix}\\ &= (0,v_1)\begin{pmatrix} h_{n-1} \\ h_{n} \end{pmatrix} + \cdots + (0,v_{n-2})\begin{pmatrix} h_2 \\ h_3 \end{pmatrix} + (0,v_{n-1})\begin{pmatrix} h_1 \\ h_2 \end{pmatrix} + (v_{n},0)\begin{pmatrix} h_1 \\ h_2 \end{pmatrix}\\ &= v_1h_n + \cdots + v_nh_1. \end{align*} $$

Thus, we find that

$$\begin{align*}\delta_0(V^{\top} f_{\mathbf{v}}) = \delta_1(V^{\top} f_{\mathbf{v}}) = \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}= \mathbf{v}.\end{align*}$$

Then, returning to (3.6), we find that

$$\begin{align*}\operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\delta_0(P_{\mathbf{u}}) + \mathbf{v}) = \mathbf{v} = \operatorname{\mathrm{Exp}}_{\rho}^{\otimes n}(\mathbf{u}).\end{align*}$$

Thus, the two quantities in the exponential functions in the above equality differ by a period, so there exists some $a\in A$ such that

$$\begin{align*}\delta_0(P_{\mathbf{u}}) + \mathbf{v} = \mathbf{u} + d[a]\Pi_n,\end{align*}$$

where $d[a]$ denotes the action of a under $\text {Lie}(\rho ^{\otimes n})$ . By our above analysis of $\delta _0$ , we conclude that the bottom coordinate $\alpha $ of $\delta _0(P_{\mathbf {u}})$ is equal to

$$\begin{align*}\alpha = a_nE_{\mathbf{u},1}^{(1)}(\theta) + E_{\mathbf{u},2}^{(1)}(\theta) = u_n-v_n + a \pi_{\rho}^n.\end{align*}$$

Since we have proved above that $\pi _{\rho }^n \in \overline K(\Psi _{\mathbf {v}}(\theta ))$ and since $v_n \in \overline K$ , it follows that $u_n\in \overline K(\Psi _{\mathbf {v}}(\theta ))$ as well.

3.5 Independence in $\text {Ext}^1_{\mathcal {T}}(\mathbf 1,\mathcal {X}_{\rho }^{\otimes n})$

For $b\in B$ and $\mathbf {v}:=\operatorname {\mathrm {Exp}}_{\rho }^{\otimes n}(Z_n(b)) \in \operatorname {\mathrm {Mat}}_{n \times 1}(H)$ , where $Z_n(b)$ is the log-algebraic vector of Theorem 1.8, we call the corresponding t-motive $\mathcal {X}_n(b):=\mathcal {X}_{\mathbf {v}}$ the zeta t-motive associated to $\zeta _{\rho }( b,n)$ . We also denote by $\Phi _n(b):=\Phi _{\mathbf {v}}$ and $\Psi _n(b):=\Psi _{\mathbf {v}}$ the corresponding matrices. As in §3.2, we put

$$ \begin{align*} \Phi_{\mathbf{v}}=\begin{pmatrix} \Phi_{\rho}^{\otimes n} & 0 \\ (h_{\mathbf{v},1},h_{\mathbf{v},2}) & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]). \end{align*} $$

We have a short exact sequence

$$\begin{align*}0 \longrightarrow \mathcal{X}^{\otimes n}_{\rho} \longrightarrow \mathcal{X}_n(b) \longrightarrow \mathbf{1} \longrightarrow 0,\end{align*}$$

where $\mathbf {1}$ is the trivial pre-t-motive, equal to $\overline K[t]$ with $\sigma $ -action given by the inverse Frobenius twist.

We follow closely [Reference Chang and Papanikolas19], Section 4.2. The group $\text {Ext}^1_{\mathcal {T}}(\mathbf 1,\mathcal {X}_{\rho }^{\otimes n})$ has the structure of a K-vector space by pushing along $\mathcal {X}_{\rho }^{\otimes n}$ . With the above notation, for $a \in A$ whose corresponding matrix is $M_a \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\overline K[t])$ as in §2.4, the extension $a_* \mathcal {X}_{\mathbf {v}}$ is represented by the matrix

$$ \begin{align*} \begin{pmatrix} \Phi_{\rho}^{\otimes n} & 0 \\ (h_{\mathbf{v},1},h_{\mathbf{v},2}) M_a & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]). \end{align*} $$

We will show the following proposition (compare to [Reference Chang and Papanikolas18], Theorem 4.4.2):

Proof. The proof follows closely that of [Reference Chang and Papanikolas18], Theorem 4.4.2. Suppose that there exist $e_1,\ldots ,e_m \in K$ not all zero so that $N=e_{1*} X_1+\ldots +e_{m*} X_m$ is trivial in $\text {Ext}^1_{\mathcal {T}}(\mathbf 1,\mathcal {X}_{\rho }^{\otimes n})$ . We can suppose that for $1 \leq i \leq m$ , $e_i$ belongs to A and is represented by $M_i \in \operatorname {\mathrm {Mat}}_{2 \times 2}(\overline K[t])$ . Then the extension $N=e_{1*} X_1+\ldots +e_{m*} X_m$ is represented by $h_{\mathbf {v}_1} M_1+\ldots +h_{\mathbf {v}_m} M_m$ and we have

$$ \begin{align*} \Phi_N &=\begin{pmatrix} \Phi_{\rho}^{\otimes n} & 0 \\ \sum_{i=1}^m h_{\mathbf{v}_i} M_i & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]), \\ \Psi_N &=\begin{pmatrix} \Psi_{\rho}^{\otimes n} & 0 \\ (\sum_{i=1}^m g_{\mathbf{v}_i} M_i)\Psi_{\rho}^{\otimes n} & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\mathbb{T}). \end{align*} $$

Since this extension is trivial in $\text {Ext}^1_{\mathcal {T}}(\mathbf 1,\mathcal {X}_{\rho }^{\otimes n})$ , there exists a matrix

$$ \begin{align*} \gamma &=\begin{pmatrix} \text{Id}_2 & 0 \\ (\gamma_1,\gamma_2) & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\overline K[t]) \end{align*} $$

such that $\gamma ^{(-1)} \Phi _N=\text {diag}(\Phi _{\rho }^{\otimes n},1) \gamma $ . By [Reference Papanikolas47], Section 4.1.6, there exists

$$ \begin{align*} \delta &=\begin{pmatrix} \text{Id}_2 & 0 \\ (\delta_1,\delta_2) & 1 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{3 \times 3}(\mathbb F_q(t)) \end{align*} $$

such that $\gamma \Psi _N=\text {diag}(\Psi _{\rho }^{\otimes n},1) \delta $ . It follows that

$$\begin{align*}(\gamma_1,\gamma_2)+(\sum_{i=1}^m g_{\mathbf{v}_i} M_i)=(\delta_1,\delta_2) (\Psi_{\rho}^{\otimes n})^{-1}.\end{align*}$$

Specializing the first coordinates at $t=\theta $ and recalling that by Lemma 2.6, we have $(M_i)_2(\theta ) \in K^{\times }$ and $(M_i)_{2,1}(\theta )=0$ , we obtain

$$\begin{align*}\gamma_1(\theta)+\sum_{i=1}^m g_{\mathbf{v}_i,1}(\theta) (M_i)_2(\theta)=\delta_1(\theta) [(\Psi_{\rho}^{\otimes n})^{-1}]_2(\theta)+\delta_2(\theta) [(\Psi_{\rho}^{\otimes n})^{-1}]_{2,1}(\theta).\end{align*}$$

By Lemma 3.3, we have $g_{\mathbf {v}_i,1}(\theta )=\mathbf {u}_{i,n}-\mathbf {v}_{i,n}$ and $[(\Psi _{\rho }^{\otimes n})^{-1}]_2(\theta ),[(\Psi _{\rho }^{\otimes n})^{-1}]_{2,1}(\theta ) \in K \pi _{\rho }^n$ . Since $(M_i)_2(\theta ) \in K^{\times }$ (see Section 2.4), we get a nontrivial $\overline K$ -linear relation between $1, \mathbf {u}_{1,n},\ldots ,\mathbf {u}_{m,n}, \pi _{\rho }^n$ . By [Reference Green33], page 29 (proof of Theorem 7.1), it implies a nontrivial K-linear relation between $\mathbf {u}_{1,n},\ldots ,\mathbf {u}_{m,n}, \pi _{\rho }^n$ . Thus, we get a contradiction.

3.6 An application of Hardouin’s work

We now apply Hardouin’s work to our context to determine the Galois groups of the t-motives defined in the previous sections. We work with the neutral Tannakian category of t-motives $\mathcal T$ over $F=\mathbb F_q(t)$ defined in §1.6 endowed with the fiber functor $\omega :M \mapsto H_{\text {Betti}}(M)$ . By the proof of [Reference Green33], Lemma 7.2, we know that $\mathcal {X}^{\otimes n}_{\rho }$ is irreducible. Then we consider this irreducible object $\mathcal Y=\mathcal {X}^{\otimes n}_{\rho }$ and extensions of $\mathbf 1$ by $\mathcal {X}^{\otimes n}_{\rho }$ . Hardouin’s work turns out to be a powerful tool and allows us to prove the proposition below which generalizes the results of Papanikolas [Reference Papanikolas47] (for the Carlitz module C) and Chang-Yu [Reference Chang and Yu23] (for the tensor powers $C^{\otimes n}$ of the Carlitz module).

Proposition 3.11. Let $n \geq 1$ be an integer with $(q-1)\nmid n$ and b be an element in B. Then the unipotent radical of $\Gamma _{\mathcal {X}_n(b)}$ is equal to the $\mathbb F_q(t)$ -vector space $\mathbb F_q(t)^2$ of dimension 2. In particular,

$$\begin{align*}\dim \Gamma_{\mathcal{X}_n(b)}=\dim \Gamma_{\mathcal{X}_{\rho}^{\otimes n}}+2=4.\end{align*}$$

As a consequence, we obtain a generalization of [Reference Chang and Yu23], Theorem 4.4.

We obtain the following theorem which could be considered as a partial generalization of [Reference Chang and Papanikolas19], Theorem 5.1.5 in our context.

4.1 Direct sums of t-motives

Let $m \in \mathbb N, m \geq 1$ . To study Anderson’s zeta values $\zeta _{\rho }(b,n)$ for $1 \leq n \leq m$ and $\pi _{\rho }$ simultaneously, we set

$$ \begin{align*} \mathcal S:=\{n \in \mathbb N: 1 \leq n \leq m \text{ such that } p \nmid n \text{ and } (q-1) \nmid n \}, \end{align*} $$

and consider the direct sum t-motive

$$\begin{align*}\mathcal{X}(b):=\bigoplus_{n \in \mathcal S} \mathcal{X}_n(b) \end{align*}$$

and define block diagonal matrices

$$\begin{align*}\Phi(b):=\bigoplus_{n \in \mathcal S} \Phi_n(b), \quad \Psi(b):=\bigoplus_{n \in \mathcal S} \Psi_n(b).\end{align*}$$

Then $\Phi (b)$ represents multiplication by $\sigma $ on $\mathcal {X}(b)$ and $\Psi (b)$ is a rigid analytic trivialization of $\Phi (b)$ . We would like to understand the Galois group $\Gamma _{\mathcal {X}(b)}$ of the t-motive $\mathcal {X}(b)$ and to calculate the dimension of this Galois group.

We first have

$$\begin{align*}\Gamma_{\mathcal{X}(b)} \subseteq \bigoplus_{n \in \mathcal S} \Gamma_{\mathcal{X}_n(b)}=\bigoplus_{n \in \mathcal S} \begin{pmatrix} \textrm{Res}_{K/\mathbb F_q[t]} \mathbb{G}_{m,K} & 0\\ * & 1 \end{pmatrix}\end{align*}$$

For $n=1$ , the t-motive $\mathcal {X}_1(b)$ contains $\mathcal {X}_{\rho }$ . It follows that $\mathcal {X}_{\rho }$ is also contained in $\mathcal {X}(b)$ . We consider $\mathcal T_{\mathcal {X}(b)}$ and $\mathcal T_{\mathcal {X}_{\rho }}$ , the strictly full Tannakian subcategories of the category $\mathcal T$ of t-motives which are generated by $\mathcal {X}(b)$ and $\mathcal {X}_{\rho }$ respectively. Thus we get a functor from $\mathcal T_{\mathcal {X}_{\rho }}$ to $\mathcal T_{\mathcal {X}(b)}$ . By Tannakian duality, we have a surjective map of algebraic groups over $\mathbb F_q(t)$

$$\begin{align*}\pi:\Gamma_{\mathcal{X}(b)} \twoheadrightarrow \Gamma_{\mathcal{X}_{\rho}}=\textrm{Res}_{K/\mathbb F_q[t]} \mathbb{G}_{m,K} \end{align*}$$

where we have the last equality by Proposition 2.3. By Equation (1.23), this map $\pi $ is in fact the projection on the upper left-most corner of elements of $\Gamma _{\mathcal {X}(b)}$ . We denote by $U(b)$ the kernel of $\pi $ . It follows that $U(b)$ is contained in the unipotent group

$$\begin{align*}U:=\bigoplus_{n \in \mathcal S} \begin{pmatrix} \text{Id}_2 & 0\\ * & 1 \end{pmatrix}.\end{align*}$$

We prove the following result similar to [Reference Chang and Yu23], Section 4.3.

Proposition 4.1. We keep the previous notation. Then we have

$$\begin{align*}U(b)=\bigoplus_{n \in \mathcal S} \begin{pmatrix} \textrm{Id}_2 & 0\\ * & 1 \end{pmatrix}.\end{align*}$$

Proof. In fact, the strategy of Chang-Yu (see [Reference Chang and Yu23], Section 4.3) based on a weight argument indeed carries over without much modification. For completeness, we sketch a proof of this Proposition.

We introduce a $\mathbb G_{m,\mathbb F_q(t)}$ -action on $U(b)$ and on the direct sum of unipotent groups

$$\begin{align*}U=\bigoplus_{n \in \mathcal S} \begin{pmatrix} \textrm{Id}_2 & 0\\ * & 1 \end{pmatrix}.\end{align*}$$

On the matrix indexed by $n \in \mathcal S$ , it is defined by

$$\begin{align*}a \cdot \begin{pmatrix} \text{Id}_2 & 0\\ \mathbf u & 1 \end{pmatrix} \mapsto \begin{pmatrix} \text{Id}_2 & 0\\ a^n \mathbf u & 1 \end{pmatrix}, \quad a \in \mathbb G_{m,\mathbb F_q(t)}. \end{align*}$$

Note that this action on $U(b)$ agrees with the conjugation of $\mathbb G_{m,\mathbb F_q(t)}$ on $U(b)$ .

For each $n \in \mathcal S$ , we recall that

$$\begin{align*}\Gamma_{\mathcal{X}_n(b)}=\begin{pmatrix} \textrm{Res}_{K/\mathbb F_q[t]} \mathbb{G}_{m,K} & 0\\ * & 1 \end{pmatrix}.\end{align*}$$

We denote by $U_n(b)$ the unipotent part of this Galois group. Thus

$$\begin{align*}U_n(b)=\begin{pmatrix} \text{Id}_2 & 0\\ * & 1 \end{pmatrix}\end{align*}$$

and we have a short exact sequence

$$\begin{align*}1 \rightarrow U_n(b) \rightarrow \Gamma_{\mathcal{X}_n(b)} \rightarrow \textrm{Res}_{K/\mathbb F_q[t]} \mathbb{G}_{m,K} \rightarrow 1. \end{align*}$$

Since $\mathcal {X}_n(b)$ is contained in $\mathcal {X}(b)$ , by Tannakian duality, we obtain a commutative diagram

Here the middle vertical arrow is surjective by Tannakian duality and the map $\chi _n$ is the character $a \mapsto a^n$ . We deduce that the induced map $U(b) \rightarrow U_n(b)$ is also surjective.

We suppose now that $U(b)$ is of codimension $r>0$ in U. We identify U with the product

$$\begin{align*}U \simeq \prod_{n \in \mathcal S} \mathbb G_{a,\mathbb F_q(t)}^2. \end{align*}$$

Chang and Yu proved that there exist an integer $n \in \mathcal S$ and a set J of r double indices $ij$ with $i \in \mathcal S$ and $j \in \{1,2\}$ such that if we denote by $W_{(J)}$ the linear subspace of U of codimension r consisting of points $(x_{ij})$ satisfying $x_{ij}=0$ whenever $ij \in J$ , then $W_{(J)} \cap U_n(b) \subsetneq U_n(b)$ and the composed map

$$\begin{align*}f_n:W_{(J)} \hookrightarrow U(b) \overset{\varphi_n}{\longrightarrow} U_n(b) \end{align*}$$

is surjective.

Recall that for $k \in \mathcal S$ , the action of $\mathbb G_{m,\mathbb F_q(t)}$ on $U_k(b)$ is of weight k. Since $p \nmid n$ , by [Reference Chang and Yu23], Lemma 4.7, $f_n$ maps $W_{(J)} \cap U_k(b)$ to zero for all $k \neq n$ in $\mathcal S$ . Thus, it maps $W_{(J)} \cap U_n(b)$ onto $U_n(b)$ which has strictly greater dimension. We obtain a contradiction.

As a consequence, we get $U(b)=U$ as required. The proof is complete.