2.1 Supersingular reduction of elliptic curves of varying level

It is natural to ask what happens if we let p vary with $d_K $ as has been considered by Liu–Masri–Young [Reference Liu, Masri and YoungLMY15]. In the above terminology, they consider the convergence with respect to the basis corresponding to the connected components $\{e_1,\ldots , e_n\}$ of $X^{p,\infty }$ . Then [Reference Liu, Masri and YoungLMY15, Theorem 1.1] amounts to the following:

(2.5) $$ \begin{align} \left| \left|\frac{\sum_{A \in \mathrm{Cl}_K} r_p(A)}{|\!|\sum_{A \in \mathrm{Cl}_K} r_p(A)|\!|_{B,1}}-\frac{e_0}{|\!|e_0|\!|_{B,1}}\right|\right|_{B,\infty} \ll_\varepsilon p^{1/8+\varepsilon} |d_K|^{-1/16+\varepsilon}, \end{align} $$

where again, $|\!| \cdot |\!|_{B,1}$ and $|\!| \cdot |\!|_{B,\infty }$ denote the pullback of, respectively, the $L^1$ -norm and sup norm under the isomorphism $V_{p,\infty }\cong \mathbb {R}^n$ defined by $B=\{e_1,\ldots , e_n\}$ . We note that the statement (2.5) is exactly parallel to Theorem 1.5. Translating back to the language of elliptic curves, (2.5) implies the following analogue of Linnik’s Theorem on the smallest prime in arithmetic progressions.

We think of Corollary 1.7 (and more generally Corollary 8.1 below) as a real quadratic analogue of the above.

3.1 Real quadratic fields and closed geodesics

We will refer to [Reference PopaPop06] and [Reference DarmonDar94] for an in-depth account of the following material. Let K be a real quadratic extension of $\mathbb {Q}$ of discriminant $d_K>0$ . Let p be a prime which splits in K and fix throughout a residue $r \ \mathrm {mod}\ 2p$ such that $r^2\equiv d_K \ \mathrm {mod}\ 4p$ . Then we have the following equality of ideals:

$$ \begin{align*} p\mathcal{O}_K= \left[p,\frac{r-\sqrt{d_K}}{2}\right]\left[p,\frac{r+\sqrt{d_K}}{2}\right],\end{align*} $$

where we use the following notation for $\alpha ,\beta \in K$ :

(3.1) $$ \begin{align}[\alpha,\beta]:=\mathbb{Z}\alpha+\mathbb{Z}\beta \subset K.\end{align} $$

In other words, $[p,\frac {r\pm \sqrt {d_K}}{2}]$ are the two prime ideals of $\mathcal {O}_K$ lying over p.

We let $\mathrm {Cl}_K^+$ denote the narrow class group of K (i.e., fractional ideals modulo principal ideals generated by elements of positive norms). Below, when the discriminant $d_K $ is clear from context, we let $I\in \mathrm {Cl}_K^+$ denote the class containing the principal fractional ideal $(1)=\mathcal {O}_K\subset K $ and $J\in \mathrm {Cl}_K^+$ denote the class containing the different $(\sqrt {d_K})=\sqrt {d_K}\mathcal {O}_K\subset K $ . Observe that for fundamental discriminants divisible by a prime $q\equiv 3 \ \mathrm {mod}\ 4$ , there exists no unit with norm $-1$ (as $-1$ is not a quadratic residue modulo q) which implies that $J\neq I$ . In this case, a principal ideal belongs to the class J exactly if it has a generator with negative norm.

We will now show the following result, which is closely related to the conditions appearing in our main results as we will see below.

Proof. Pick an odd prime $q\equiv 3 \ \mathrm {mod}\ 4$ such that $-p$ is a quadratic residue mod q. Then by simple considerations about quadratic residues, there is a residue $t \ \mathrm {mod}\ 8 q$ such that

$$ \begin{align*} q| t^2+p,\, \, q^2\!\!\not| t^2+p \quad \text{and}\quad (t^2+p \ \mathrm {mod}\ 16)\in \{ 1, 5, 8,9,12,13\}, \end{align*} $$

meaning in particular that $t^2+p\equiv 0\text { or }1 \ \mathrm {mod}\ 4$ . Now for each $n\equiv t \ \mathrm {mod}\ 8q$ , there is a unique positive fundamental discriminant $d>0$ and integer $m>0$ such that $d m^2=n^2+p$ . By construction, we have $q| d$ , and thus $J\neq I$ inside $\mathrm {Cl}_K^+$ where $K=\mathbb {Q}(\sqrt {d})$ . Furthermore, we have the factorization

$$ \begin{align*} -p=(n-\sqrt{d}m)(n+\sqrt{d}m), \end{align*} $$

which means that $[p,\frac {r-\sqrt {d}}{2}]$ is a principal ideal generated by $ n\pm \sqrt {d}m$ (for some choice of sign $\pm $ ). By construction, the norm of $n\pm \sqrt {d}m$ is negative (and equal to $-p$ ). By the above, this implies $[p,\frac {r-\sqrt {d}}{2}]\in J$ in $\mathrm {Cl}_K^+$ as wanted.

Now it follows that if $J\neq I$ , then $J\notin (\mathrm {Cl}_K^+)^2$ . Thus, for $p,d$ as in Proposition 3.1 and each subgroup $H\leq (\mathrm {Cl}_K^+)^2$ , we have $[p,\frac {r-\sqrt {d}}{2}]\notin H$ . This gives plenty of examples for which Theorems 1.1 and 1.5 apply, recalling that $\{\mathfrak {p}_1, \mathfrak {p}_2\}=\{[p,\frac {r\pm \sqrt {d}}{2}]\} $ .

3.1.1 Closed geodesics and class groups

Let p be prime and K a real quadratic field of discriminant $d_K$ such that p splits in K. Let $r \ \mathrm {mod}\ 2p$ as above be such that $r^2\equiv d_K \ \mathrm {mod}\ 4p$ . Denote by $\mathcal {Q}_{K,p}$ (suppressing r in the notation) the set of all integral binary quadratic form

$$ \begin{align*}Q(x,y)=ax^2+bxy+cy^2\end{align*} $$

of discriminant $b^2-4ac=d_K$ and level p, meaning that $a\equiv 0 \ \mathrm {mod}\ p$ and $b\equiv r \ \mathrm {mod}\ 2p$ . The group $\Gamma _0(p)$ acts naturally on $\mathcal {Q}_{K,p}$ by coordinate transformation. It is a classical fact, essentially due to Gauß, that there is a natural bijection (depending on the choice of $r \ \mathrm {mod}\ p$ )

(3.2) $$ \begin{align} \mathrm{Cl}_K^+ \xrightarrow{\sim} \Gamma_0(p)\backslash \mathcal{Q}_{K,p}. \end{align} $$

When the level is trivial, the above bijection is induced by mapping $ax^2+bxy+cy^2$ to the narrow ideal class of the fractional ideal $[1,\frac {-\mathrm {sign}(a)b+\sqrt {d_K}}{2|a|}]$ (using the notation (3.1)).

Given an integral binary quadratic form $Q(x,y)=ax^2+bxy+cy^2$ of discriminant $d_K $ and level p, we associate the following matrix:

(3.3) $$ \begin{align}\gamma_Q:=\begin{pmatrix} u+bv & 2cv \\ -2av & u-bv\end{pmatrix}\in \Gamma_0(p),\end{align} $$

where $u,v$ are positive half-integers satisfying Pell’s equation $u^2-d_Kv^2=1$ and such that v is minimal among all such solutions (i.e., the fundamental positive unit of K is $\epsilon _K=u+v\sqrt {d_K}$ ).

For $A\in \mathrm {Cl}_K^+$ , we denote by $\mathcal {C}_{A}(p)$ the oriented closed geodesic on $Y_0(p)$ obtained by projecting the oriented geodesic connecting $z_Q$ and $\gamma _Q z_Q$ , where

$$ \begin{align*} z_Q:=\frac{-\mathrm{sign} (a)b+i \sqrt{d_K}}{2|a|} \end{align*} $$

and $Q\in \mathcal {Q}_{K,p}$ corresponds to A under the isomorphism (3.2) (the image in $Y_0(p)$ is independent of the choice of representative Q).

3.2 (Co)homology of modular curves

Here and throughout, we will consider matrices in $\mathrm {SL}_2(\mathbb {R})$ as elements of $\mathrm {PSL}_2(\mathbb {R})$ without further mentioning. Let p be prime and consider the Hecke congruence group (or more precisely, its projection to $\mathrm {PSL}_2(\mathbb {R})$ )

Let

$$ \begin{align*}Y_0(p):=\Gamma_0(p)\backslash \mathbb{H},\quad X_0(p):=\overline{Y_0(p)}=Y_0(p)\cup \Gamma_0(p)\backslash \mathbb{P}^1(\mathbb{Q}) \end{align*} $$

be resp., the modular curve of level p and its compactification which is a compact Riemann surface of genus $g=\frac {p}{12}+O(1)$ (see, for example, [Reference ShimuraShi94, Proposition 1.40]). We can consider the integral singular homology group [Reference HatcherHat02, Chapter 2]

$$ \begin{align*}H_1(Y_0(p),\mathbb{Z})\cong \mathbb{Z}^{2g+1}, \end{align*} $$

which sits as a lattice inside the real homology

$$ \begin{align*}H_1(Y_0(p),\mathbb{R})\cong \mathbb{R}^{2g+1}. \end{align*} $$

We will be interested in the distribution of oriented closed geodesics inside the lattice $H_1(Y_0(p),\mathbb {Z})$ .

We have the cap product pairing

(3.4) $$ \begin{align} \langle\cdot, \cdot \rangle: H_1(Y_0(p),\mathbb{R})\times H^1(Y_0(p),\mathbb{R})\rightarrow \mathbb{R}\end{align} $$

between real homology and cohomology which identifies $H^1(Y_0(p),\mathbb {R})$ with the linear dual $H_1(Y_0(p),\mathbb {R})^\ast $ . Given a basis B of $H_1(Y_0(p), \mathbb {R})$ , we denote by $ B^{\ast }\subset H^1(Y_0(p), \mathbb {R})$ the dual basis of B with respect to the cap product pairing as in (3.4).

Recall that the de Rham isomorphism gives a description of $H^1(Y_0(p),\mathbb {R})$ in terms of real valued harmonic 1-forms on $Y_0(p)$ . Any such 1-form is a linear combination of forms of the type

$$ \begin{align*} \omega_f=2\pi i f(z)dz ,\quad \overline{\omega_f}=\overline{2\pi i f(z)dz}, \end{align*} $$

where $f\in \mathcal {M}_2(p)$ is a weight 2 and level p holomorphic form (not necessarily cuspidal). We also have a surjective map

(3.5) $$ \begin{align} \Gamma_0(p)\twoheadrightarrow H_1(Y_0(p), \mathbb{Z}),\quad \gamma\mapsto \{ z, \gamma z\},\end{align} $$

where $z\in \mathbb {H}$ (the class does not depend on the choice of z), and we are using the following notation for $z_1,z_2\in \mathbb {H}$ :

(3.6) $$ \begin{align}\{z_1, z_2\}:= [\text{class of the oriented geodesic connecting} z_1 \text{ and } z_2]\in H_1(Y_0(p), \mathbb{R}), \end{align} $$

which defines an element of the real homology $H_1(Y_0(p), \mathbb {R})$ via integration against $1$ -forms. The map (3.5) induces an isomorphism

(3.7) $$ \begin{align}\Gamma_0(p)^{\mathrm{ab}}/(\Gamma_0(p)^{\mathrm{ab}})_{\mathrm{tor}}\cong H_1(Y_0(p),\mathbb{Z}).\end{align} $$

We note that for a closed oriented geodesic $\mathcal {C}_{A}(p)$ as in the previous section, the homology class $[\mathcal {C}_{A}(p)]\in H_1(Y_0(p),\mathbb {Z})$ corresponds exactly to the image of (any) $\gamma _Q$ as in (3.3) under the map (3.5). Using the above identifications, the cap product pairing is induced by the map

$$ \begin{align*} \Gamma_0(p)\times \mathcal{M}_2(p)\ni (\gamma, f)\mapsto \int^{\gamma z}_z \omega_f, \end{align*} $$

for any $z\in \mathbb {H}$ .

The natural pullback map induced by inclusion fits into a short exact sequence of $\mathbb {R}$ -vector spaces

(3.8) $$ \begin{align} 0\rightarrow H^1(X_0(p),\mathbb{R})\rightarrow H^1(Y_0(p),\mathbb{R})\rightarrow \mathbb{R} \rightarrow 0, \end{align} $$

using here that we have two cusps (which, as we will see, implies that the Eisenstein space of weight $2$ is one-dimensional). This identifies $H^1(X_0(p),\mathbb {R})$ with the parabolic classes in $H^1(Y_0(p),\mathbb {R})$ (i.e., classes which vanish on all parabolic elements of $\Gamma _0(p)$ using the above pairing). Under the de Rham isomorphism, the parabolic classes correspond to $1$ -forms obtained from holomorphic cusp(!) forms of weight $2$ and level p (see, for example, [Reference ShimuraShi94, Section 5]). Similarly, we have the pushforward map $H_1(Y_0(p),\mathbb {R})\rightarrow H_1(X_0(p),\mathbb {R})$ whose kernel is exactly given by the image of the parabolic elements of $\Gamma _0(p)\otimes \mathbb {R}$ inside $H_1(Y_0(p),\mathbb {R})$ under the map (3.5).

3.2.1 Hecke operators

Due to the arithmetic nature of $\Gamma _0(p)$ , we have a family of commuting linear operators acting on all of the above-mentioned (co)homology groups, namely the Hecke operators. The Hecke action is induced by the following: on the space of holomorphic forms $\mathcal {M}_2(p)$ of weight 2 and level p, the n-th Hecke operator $T_n$ acts by (see, for example, [Reference Iwaniec and KowalskiIK04, (14.46)])

(3.9) $$ \begin{align} T_n f(z):= \frac{1}{n}\sum_{\substack{ad=n,\\ (a,p)=1}} a^2 \sum_{0\leq b<d} f\left( \frac{az+b}{d} \right), \end{align} $$

the Fricke involution $W_p$ acts as

(3.10) $$ \begin{align} W_p f(z):=p^{-1}z^{-2} f(-1/(pz))\end{align} $$

and we also have the involution $\iota $ given by

(3.11) $$ \begin{align}\iota f(z):= f(-\overline{z}),\end{align} $$

defined for $f\in \mathcal {M}_2(p)\oplus \overline {\mathcal {M}_2(p)}$ . Notice that, with this normalization, the Ramanujan conjecture amounts to the bound $\leq d(n)n^{1/2}$ for the Hecke eigenvalues. See also [Reference ShimuraShi94, Section 5] for an intrinsic definition in terms of group cohomology using double cosets. Similarly, we can define an action on the homology groups by (using the notation (3.6))

(3.12) $$ \begin{align} T_n \{z_1, z_2\}:= \sum_{\substack{ad=n,\\ (a,p)=1}} \,\,\sum_{0\leq b<d} \left\{\frac{az_1 +b}{d}, \frac{a z_2 +b}{d}\right\}, \end{align} $$

(3.13) $$ \begin{align}W_p\{z_1, z_2\}:=\{-1/(pz_1), -1/(pz_2)\} \end{align} $$

and

(3.14) $$ \begin{align} \iota \{z_1, z_2\}:= \{-\overline{z_1}, -\overline{z_2}\}. \end{align} $$

One can now check that all of these operators are pairwise adjoint with respect to the cap product pairing as above. Furthermore, it can be shown that all of these linear operators commute, and thus we can find a common eigen-basis. Explicitly, such a Hecke eigen-basis for $H^1(Y_0(p),\mathbb {R})$ is given by

(3.15) $$ \begin{align} B_{\mathrm{Hecke}}(p):=\{\omega_f^{\epsilon}: f\in \mathcal{B}_p,\epsilon\in\{\pm\}\}\cup\{ \omega_{E}(p) \}, \end{align} $$

where

$$ \begin{align*} \omega_f^{\pm}:= \frac{2\pi i f(z)dz\pm \overline{2\pi i f(z)dz}}{1+i \pm (1-i)}\in H^1(Y_0(p), \mathbb{R}), \end{align*} $$

with $\mathcal {B}_p:=\{f_1,\ldots , f_g\}\subset \mathcal {S}_2(p)$ a basis of Hecke normalized (i.e., the first Fourier coefficient is $1$ ) holomorphic cuspidal eigen forms of weight $2$ and level p. And

(3.16) $$ \begin{align} \omega_{E}(p):=E_{2,p}(z) dz\in H^1(Y_0(p), \mathbb{R})\end{align} $$

is the normalized Eisenstein class defined from the weight $2$ Eisenstein series of level p:

$$ \begin{align*} E_{2,p}(z):=\frac{pE_2(pz)-E_2(z)}{p-1}=\frac{pE^\ast_2(pz)-E^\ast_2(z)}{p-1}=1+\frac{24}{p-1}e^{2\pi i z}+\ldots,\end{align*} $$

where $E_2$ denotes the the weight $2$ Eisenstein series of level $1$ and $E^\ast _2$ the modified Eisenstein series given by

(3.17) $$ \begin{align}E_2(z):=1-24\sum_{n\geq 1} \sigma_1(n)e^{2\pi i n z}=E_2^\ast(z)+\frac{3}{\pi y}.\end{align} $$

Here, $\sigma _1(n)=\sum _{d|n}d$ is the sum of divisors function. One obtains a Hecke eigen basis for the homology as the dual basis of $B_{\mathrm {Hecke}}(p)$ with respect to the cap product pairing (3.4) which we denote by

(3.18) $$ \begin{align} B_{\mathrm{Hecke}}(p)^\ast:=\{v_f^{\pm}: f\in \mathcal{B}_p,\pm\}\cup\{ v_{E}(p) \}\subset H_1(Y_0(p),\mathbb{R}) , \end{align} $$

where $\langle v_f^\pm ,\omega _f^\pm \rangle =1$ for $f\in \mathcal {B}_p$ and $\langle v_E(p),\omega _E(p)\rangle =1$ . Since the constant Fourier coefficient of $E_{2,p}$ is $1$ , one sees that, in fact,

(3.19) $$ \begin{align}v_{E}(p)=\{i,i+1\}\in H_1(Y_0(p),\mathbb{Z}), \end{align} $$

with notation as in (3.6). In other words, $v_{E}(p)$ is the (normalized) Eisenstein class in homology appearing in Theorem 1.1 which geometrically is the class of a simple loop going around the cusp at $\infty $ .

We have the following classical formula for $\gamma \in \mathrm {PSL}_2(\mathbb {Z})$ (see, for example, [Reference Duke, Imamoḡlu and TóthDIT18, (55)]):

$$ \begin{align*} \int_{z}^{\gamma z} E_2^\ast(z) dz= \Psi(\gamma), \end{align*} $$

where

is the Rademacher symbol with

$$ \begin{align*}s(a,c):=\sum_{n=1}^{ c } (\!(\tfrac{n}{c})\!)(\!(\tfrac{na}{c})\!),\end{align*} $$

the classical Dedekind sum with

$$ \begin{align*}(\!(x)\!)=\begin{cases}x-\lfloor x\rfloor-1/2,& x\notin \mathbb{Z}\\ 0,& x\in \mathbb{Z}\end{cases}\end{align*} $$

the sawtooth function. Notice that $\Psi (\gamma )$ does not depend on the representative of $\gamma \in \mathrm {PSL}_2(\mathbb {Z})$ as should be the case. This implies the following key formula for $\gamma \in \Gamma _0(p)$ :

(3.20) $$ \begin{align}\langle \{z,\gamma z\}, \omega_{E}(p) \rangle= \frac{\Psi\left(\gamma'\right)- \Psi(\gamma)}{p-1},\end{align} $$

where

with

. Using the trivial bound $\Psi (\gamma )\ll \frac {|a+d|}{ c }+ c $ (for $c> 0$ ), we get the following useful estimate:

(3.21)

Recall the short exact sequence (3.8) above. A Hecke-equivariant splitting is given by mapping $1\in \mathbb {R}$ to the Eisenstein class $\omega _{E}(p)=E_{2,p}(z)dz$ (this fits into a general framework due to Franke [Reference FrankeFra98]). This defines an isomorphism

(3.22) $$ \begin{align}H^1(Y_0(p),\mathbb{R})\cong H^1(X_0(p),\mathbb{R})\oplus \mathbb{R} \omega_{E}(p),\end{align} $$

and we will denote the projection to the ‘cuspidal subspace’ (which we will identify with $H^1(X_0(p),\mathbb {R})$ ) with respect to this splitting by

(3.23) $$ \begin{align} \mathbb{P}_{\mathrm{cusp}}:H^1(Y_0(p),\mathbb{R})\rightarrow H^1(X_0(p),\mathbb{R})\subset H^1(Y_0(p),\mathbb{R}), \end{align} $$

which is explicitly given by

$$ \begin{align*} \mathbb{P}_{\mathrm{cusp}} \omega= \omega- \langle v_{E}(p),\omega\rangle\omega_{E}(p), \end{align*} $$

where $v_{E}(p), \omega _{E}(p)$ are the Eisenstein classes defined above.

4.1 Fundamental polygons

Let $\Gamma \subset \mathrm {PSL}_2(\mathbb {R})$ be a Fuchsian subgroup of the first kind (i.e., a discrete and cofinite subgroup of $\mathrm {PSL}_2(\mathbb {R})$ ). A fundamental domain for $\Gamma $ is a locally finite domain $\mathcal {F}\subset \mathbb {H}$ such that

1. 1. For any $z\in \mathbb {H}$ , we have $\gamma z\in \overline {\mathcal {F}}$ for some $\gamma \in \Gamma $ .

2. 2. If $z_1,z_2\in \overline {\mathcal {F}}$ are $\Gamma $ -equivalent, then $z_1=z_2$ or $z_1,z_2\in \partial \mathcal {F}$ .

We say that $\mathcal {F}$ is a fundamental polygon for $\Gamma $ if $\mathcal {F}$ is furthermore (hyperbolically) convex with a piecewise geodesic boundary. We define a side of $\mathcal {F}$ as a nonempty subset of the shape $\gamma \overline {\mathcal {F}}\cap \overline {\mathcal {F}}$ with $ \mathrm {Id}\neq \gamma \in \Gamma $ . We define a vertex of $\mathcal {F}$ as a non-empty subset of the shape $\gamma _1 \overline {\mathcal {F}}\cap \gamma _2 \overline {\mathcal {F}} \cap \overline {\mathcal {F}}$ with $ \mathrm {Id},\gamma _1,\gamma _2\in \Gamma $ pairwise distinct. It can be shown that a fundamental polygon $\mathcal {F}$ has an even number of sides which are pairwise $\Gamma $ -equivalent, and $\Gamma \mathcal {F}$ gives a tessellation of $\mathbb {H}$ . The set of elements identifying sides of $\mathcal {F}$ are called the side pairing transformations associated to $\mathcal {F}$ which we denote by $\mathcal {S}(\mathcal {F})\subset \Gamma $ (which formally is a multiset if there are order two elements in $\Gamma $ ). Given a side L of $\mathcal {F}$ we refer to the side pairing transformation associated to the side L as the element $\sigma \in \mathcal {S}(\mathcal {F})$ such that $\sigma ^{-1}L$ is also a side of $\mathcal {F}$ . Similarly we say that $\sigma $ is the side pairing transformation associated to the side $\gamma L$ of the translate $\gamma \mathcal {F}$ for each $\gamma \in \Gamma $ .

It is known that $\mathcal {S}(\mathcal {F})$ generates $\Gamma $ for any fundamental polygon $\mathcal {F}$ , and it is a fundamental fact that one can understand the relation between the elements of $\mathcal {S}(\mathcal {F})$ from the geometry of a fundamental polygon and vice versa (see Lemma 4.3). A simple but key incarnation is the following.

Proposition 4.1. Let $\mathcal {F}$ be a fundamental polygon for a discrete and co-finite subgroup $\Gamma \subset \mathrm {PSL}_2(\mathbb {R})$ . Consider a sequence of consecutive $\Gamma $ -translates of $\mathcal {F}$ :

$$ \begin{align*} \mathcal{F},\gamma_1 \mathcal{F},\ldots, \gamma_n \mathcal{F}. \end{align*} $$

Then

$$ \begin{align*}\gamma_n=\sigma_1\sigma_2\cdots \sigma_n, \end{align*} $$

where $\sigma _i\in \mathcal {S}(\mathcal {F})$ denotes the side pairing transformation associated to the side shared between $\gamma _{i-1} \mathcal {F}$ and $\gamma _{i} \mathcal {F}$ (here, we put $\gamma _0=\mathrm {Id}$ ).

Proof. Note that the side of $\mathcal {F}$ shared with $\gamma _1 \mathcal {F}$ will exactly have associated $\sigma _1=\gamma _1\in \mathcal {S}(\mathcal {F})$ . Now if the side shared between $\gamma _1 \mathcal {F}$ and $\gamma _2 \mathcal {F}$ has associated $\sigma _2\in \mathcal {S}(\mathcal {F})$ , then we have

$$ \begin{align*}\gamma_2\mathcal{F}=(\sigma_1\sigma_2\sigma_1^{-1})(\sigma_1 \mathcal{F})=\sigma_1 \sigma_2 \mathcal{F}. \end{align*} $$

Continuing like this, we get

$$ \begin{align*} \gamma_n\mathcal{F}=\sigma_1 \cdots \sigma_n \mathcal{F}, \end{align*} $$

and since $\gamma \mathcal {F}=\gamma ' \mathcal {F}$ implies $\gamma =\gamma '\in \Gamma $ by condition (2), we conclude the wanted equality.

A first application of Proposition 4.1 is the following slight reformulation.

Corollary 4.2. Let $c:[0,1]\rightarrow \mathbb {H}$ be a continuous, injective curve with $c(0)=z\in \mathcal {F}$ and $c(1)=\gamma z$ for some $\gamma \in \Gamma $ . Assume that $c([0,1])$ does not intersect the set of $\Gamma $ -translates of the vertices of $\mathcal {F}$ . Then one can code the element $\gamma $ in the following way; let $\sigma _1,\ldots , \sigma _n\in \mathcal {S}(\mathcal {F})$ be the (ordered) sequence of side pairing transformations associated to the intersection between the curve c and $\Gamma $ -translates of the sides of $\mathcal {F}$ . Then one has

$$ \begin{align*} \gamma=\sigma_1\cdots\sigma_n. \end{align*} $$

When $c([0,1])$ is a closed geodesic (when projected to $\Gamma \backslash \mathbb {H}$ ), this is known as geometric coding of geodesics (see [Reference KatokKat96] for a nice treatment).

As a second consequence, we note that if the sequence of translates ‘loops around’, meaning that ${\gamma _n=1}$ , then one obtains a relation between the side pairing transformations $S(\Gamma )$ . This yields immediately that if two sides of $\mathcal {F}$ are paired, then the associated elements in $\mathcal {S}(\mathcal {F})$ are inverses. We call these the inverse relations of $\mathcal {F}$ . We also get relations by looping around the vertices of $\mathcal {F}$ which we will now make precise. Observe that the embedding $\mathcal {F}\subset \mathbb {H}$ defines an orientation on the boundary $\partial \mathcal {F}$ . Let $L_1$ be a side of $\mathcal {F}$ with leftmost (wrt. the orientation) vertex $v_1$ and let $\sigma _1\in \mathcal {S}(\mathcal {F})$ be the associated side pairing transformation. Let $L_2$ be the side of $\mathcal {F}$ different from $\sigma _1^{-1} L_1$ containing the vertex $v_2=\sigma _1^{-1} v$ and let $\sigma _2\in \mathcal {S}(\mathcal {F})$ be the associated side pairing transformation. Continuing like this yields a periodic sequence of pairs

$$ \begin{align*} (L_1,v_1), (L_2,v_2),\ldots, \end{align*} $$

with minimal period $m\geq 1$ , say, which we call a cycle of $\mathcal {F}$ . It is now clear that $\sigma _1\sigma _2\ldots \sigma _m$ fixes $v_1$ . Thus, if we put $\nu =|\Gamma _{v_1}|$ (i.e., the size of the stabilizer of $v_1$ inside $\Gamma $ ), then we get the following relation on the side pairing transformations:

$$ \begin{align*} (\sigma_1\sigma_2\ldots \sigma_m)^\nu=1, \end{align*} $$

where if $\nu =\infty $ (i.e., $v_1$ is a boundary vertex), this is understood as the empty relation. Notice that the relation does not depend on the choice of starting point $(L_1,v_1)$ . We call these the cycle relations of $\mathcal {F}$ , and we have the following key theorem of Poincaré; see [Reference MaskitMas71] for a proof.

Secondly, it is clear that each side appears in exactly one cycle. Thus, we obtain the following useful fact.

4.1.1 Bounding coordinates of side pairing transformations

Now we consider the image of the inverse and cycle relations when mapped to $V_p$ under the composition of maps:

$$ \begin{align*}\Gamma_0(p)\twoheadrightarrow \Gamma_0(p)^{\mathrm{ab}}\twoheadrightarrow H_1(Y_0(p),\mathbb{Z})\hookrightarrow V_p=H_1(Y_0(p),\mathbb{R})\cong \mathbb{R}^{2g+1}.\end{align*} $$

Notice that since $V_p$ is torsion-free, we can divide the relations by $\nu $ so that all coefficients in the relations are contained in $\{-1,0,1\}$ . By killing the inverse relations, this yields a system of linear relations with a variable for each pair $\{\sigma ,\sigma ^{-1}\}\subset \mathcal {S}(\mathcal {F})$ of side pairing transformations so that each such variable appears once with positive and once with negative sign (which follows from Lemma 4.4).

Let $B=\{v_0,\ldots , v_{2g}\}\subset V_p$ be a basis of the real homology corresponding to side pairing transformations $\sigma \in \mathcal {S}(\mathcal {F})$ under the surjection (3.5). Combining the cycle relations with the choice of B gives rise to a system of linear equations in $V_p$ :

$$ \begin{align*}\mathcal{L}_{\mathcal{F},B}(x_0,\ldots, x_n)= \left\{ \sum_{0\leq i\leq n} a_{ik}x_i+\sum_{0\leq j\leq 2g} b_{jk}v_j=0 \right\}_{0\leq k \leq K},\end{align*} $$

where $x_0,\ldots , x_n$ are variables, one for each pair $\{\sigma ,\sigma ^{-1}\}\subset \mathcal {S}(\mathcal {F})$ not corresponding to an element of B. Lemmata 4.3 and 4.4 translate to two key properties:

1. 1. $\mathcal {L}_{\mathcal {F},B}$ has exactly one solution $(x_0,\ldots , x_n)\in (V_p)^{n+1}$ .

2. 2. For any subset $A\subset \{0,\ldots , K\}$ , and any indices $0\leq i\leq n,0\leq j\leq 2g$ , we have

   $$ \begin{align*} \sum_{k\in A}a_{ik},\, \sum_{k\in A}b_{jk} \in \{-1,0,1\}. \end{align*} $$

We have the following general result about such systems of linear equations.

Lemma 4.5. Let

$$ \begin{align*}\mathcal{L}(x_0,\ldots, x_n)= \left\{ \sum_{0\leq i\leq n} a_{ik}x_i+\sum_{0\leq j\leq 2g} b_{jk}v_j=0 \right\}_{0\leq k \leq K}\end{align*} $$

be a system of linear equations in $V_p$ satisfying (1) and (2).

Then the unique solution $(x_0,\ldots , x_n)\in (V_p)^{n+1}$ satisfies

(4.1) $$ \begin{align}x_i=\sum_{0\leq j\leq 2g} c_{ij} v_j,\quad c_{ij}\in \{-1,0,1\},\end{align} $$

for all $0\leq i\leq n$ .

Proof. We proceed by induction on $K+1\geq 0$ (i.e., the number of equations). If $K+1=0$ , then there is nothing to prove. Now assume the claim is known for systems of $K'<K+1$ equations. We start by making the following reductions. We may assume that all equations contain a variable (i.e., $\forall k \exists i:a_{ik}\neq 0$ ), since otherwise we can remove such an equation. Furthermore, we may assume that there are some variables appearing exactly once (i.e., $\exists i : |\{k: a_{ik}\neq 0\}|=1$ ). If this is not the case, then by (1) and (2) every variable appears exactly twice with coefficients $\pm 1$ , respectively. This means that the equation obtained by adding all of the equations in $\mathcal {L}(x_0,\ldots , x_n)$ has to be trivial. Thus, by removing any equation, we obtain an equivalent system with K equations, and we are done by the induction hypothesis.

The key observation is that there is always an equation with exactly one variable appearing (i.e., $\exists k : |\{i: a_{ik}\neq 0\}|=1$ ). If not, then since every variable appears at most twice and there is one variable appearing once (by the above reductions) we have a system of linear equations with more variables than equations. Thus, the number of solutions $(x_0,\ldots , x_n)\in (V_p)^{n+1}$ is either $0$ or $\infty $ , which contradicts (1). Let $x_i$ be the only variable appearing in some equation. Then this equation determines $x_i$ , which by (2) satisfies (4.1). Now if $-x_i$ does not appear, we are done by the induction hypothesis. Otherwise, we add the equation containing $x_i$ to the one containing $-x_i$ which gives a system of K linear equations again satisfying (1) and (2). Thus, the claim follows from the induction hypothesis.

Corollary 4.6. Let $\Gamma $ be a Fuchsian group with a fundamental polygon $\mathcal {F}$ , and let B be a basis for $H_1(\Gamma \backslash \mathbb {H}, \mathbb {R})$ consisting of classes of the form $\{z,\sigma z\}$ with $\sigma \in \mathcal {S}(\mathcal {F})$ and $z\in \mathbb {H}$ .

Then we have for any $\sigma \in \mathcal {S}(\mathcal {F})$ , $z\in \mathbb {H}$ and $\omega \in B^{\ast }$ that

$$ \begin{align*} |\langle \{z, \sigma z\}, \omega \rangle|\leq 1.\end{align*} $$

Proof. Observe that if the variable $x_i$ corresponds to $\{\sigma ,\sigma ^{-1}\}\in \mathcal {S}(\mathcal {F})$ , then the numbers

$$ \begin{align*} (\langle \{ z, \sigma z\}, \omega \rangle)_{\omega\in B^{\ast}} \end{align*} $$

are exactly the coordinates of either $x_i$ or $-x_i$ in the basis B where $(x_0,\ldots , x_n)\in (V_p)^{n+1}$ is the unique solution to $\mathcal {L}_{\mathcal {F},B}$ . Now the result follows directly from Lemma 4.5.

4.2 Zagier’s fundamental polygon

We will now consider a fundamental polygon for $\Gamma _0(p)$ introduced by Zagier [Reference ZagierZag85, Section 3] which will give rise to a set of natural bases for the homology. The following is a fundamental polygon for $\Gamma _0(p)$ :

$$ \begin{align*} \tilde{\mathcal{F}}_{\mathrm{Zag}}(p)=\cup_{i=0}^{p} \sigma_i \mathcal{F}_{\mathrm{std}},\end{align*} $$

where

for $0\leq i <p$ and

, and

(4.2) $$ \begin{align}\mathcal{F}_{\mathrm{std}}:=\{z\in \mathbb{H}: |\operatorname{Re} z|<1/2, |z|>1\}\end{align} $$

is the standard fundamental polygon for $\mathrm {PSL}_2(\mathbb {Z})$ . This gives rise to the side pairing transformation set

where $0<a^*<p$ is such that $aa^*\equiv -1 \ \mathrm {mod}\ p$ . Recall that $\Gamma _0(p)$ is normalized by the matrix

, which implies that also $\mathcal {F}_{\mathrm {Zag}}(p):=W_{p} \tilde {\mathcal {F}}_{\mathrm {Zag}}(p)$ is a fundamental polygon for $\Gamma _0(p)$ with side pairing transformations given by

which, by Lemma 4.3, generate $\Gamma _0(p)$ (as claimed in the introduction). Both of these fundamental polygons have the nice property that all cuspidal sides (i.e., sides containing a cusp) are paired by a parabolic element of $\Gamma _0(p)$ (which is not the case for all fundamental polygons). The elements of $\mathcal {S}(\mathcal {F}_{\mathrm {Zag}}(p))$ are minimal in the sense that the archimedean sizes of the entries are as small as one can hope for ( $\leq p$ ). Explicitly, the fundamental polygon $\mathcal {F}_{\mathrm {Zag}}(p)$ has $p+3$ vertices: the two cusps $0$ and $\infty $ as well as $\frac {2j-1+i\sqrt {3}}{2p}$ for $0\leq j\leq p$ . We see that the matrix $W_{p}$ takes the hyperbolic triangle with vertices $ \{\tfrac {-1+i\sqrt {3}}{2p},0,\tfrac {1+i\sqrt {3}}{2p}\}$ to the hyperbolic triangle with vertices $\{ \tfrac {-1+i\sqrt {3}}{2},\infty ,\tfrac {1+i\sqrt {3}}{2}\}$ . In particular, the subgroup $\langle W_{p}, \Gamma _0(p)\rangle \leq \mathrm {PSL}_2(\mathbb {R})$ has a fundamental domain contained in

(4.3) $$ \begin{align} \{ z\in \mathbb{H} : |\operatorname{Re} z|\leq 1/2, \operatorname{Im} z\geq \sqrt{3}/(2p) \},\end{align} $$

which will be useful later on.

We define the following compatible family of bases of the homology groups.

4.3 Special fundamental polygons

By the Kurosh subgroup theorem, we know that any subgroup of $\mathrm {PSL}_2(\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\ast \mathbb {Z}/3\mathbb {Z}$ is isomorphic to a free product of a number of copies of $\mathbb {Z}/2\mathbb {Z},\mathbb {Z}/3\mathbb {Z}$ and $\mathbb {Z}$ . In particular, a torsion-free Hecke congruence subgroup $\Gamma _0(p)$ is a free group on $k=\mathrm {rank}_{\mathbb {Z}}\, \Gamma _0(p)^{\mathrm {ab}}$ generators. We will now describe an explicit geometric way due to Kulkarni [Reference KulkarniKul91] for constructing a set of independent generators of $\Gamma _0(p)$ . The starting points are so-called special fundamental polygons of $\Gamma _0(p)$ .

Let g be the genus of $Y_0(p)$ , and $e_2,e_3$ the number of conjugacy classes of subgroups in $\Gamma _0(p)$ of order, respectively, $2$ and $3$ . Following [Reference KulkarniKul91], we define a Farey symbol of level p as a sequence of reduced fractions

$$ \begin{align*} \frac{0}{1}=\frac{a_{0}}{b_{0}}<\frac{a_1}{b_1}<\ldots<\frac{a_{n-1}}{b_{n-1}}< \frac{a_{n}}{b_{n}}=\frac{1}{1} , \end{align*} $$

with $n=4g+e_2+e_3$ such that $a_{i+1}b_{i}-a_{i}b_{i+1}=1$ for all $1\leq i<n$ . Furthermore (considering below the indices modulo $n+2$ ),

• ○ there are $e_2$ even indices i such that

  $$ \begin{align*} b_i^2+b_{i+1}^2 \equiv 0\ \mathrm{mod}\ p, \end{align*} $$

• ○ there are $e_3$ odd indices i such that

  $$ \begin{align*} b_i^2+b_ib_{i+1}+b_{i+1}^2 \equiv 0\ \mathrm{mod}\ p, \end{align*} $$

• ○ for the remaining $4g$ free indices, there is a pairing $i\leftrightarrow i^\ast $ satisfying

  $$ \begin{align*} b_ib_{i^\ast}+b_{i+1}b_{i^\ast+1}\equiv 0\ \mathrm{mod}\ p. \end{align*} $$

Such a symbol always exists, and one can even find one which is symmetric around $1/2$ [Reference KulkarniKul91, Section 13]. Dooms–Jesper–Konolalov [Reference Dooms, Jespers and KonovalovDJK10] have described an algorithm for determining Farey symbols of a general level.

Consider the polygon $\mathcal {P}(p)$ with vertices at $\infty $ , at the fractions of the Farey symbol, at the midpoint of the geodesic circle connecting $a_i/b_i$ and $a_{i+1}/b_{i+1}$ for i an even index, and for an odd index i at the $\mathrm {PGL}_2(\mathbb {Z})$ -translate of $\frac {1+i\sqrt {3}}{2}$ lying between $a_i/b_i$ and $a_{i+1}/b_{i+1}$ (for details see [Reference KulkarniKul91, Section 2]). Note that $\mathcal {P}(p)$ consists of $\mathrm {PGL}_2(\mathbb {Z})$ -translates of

(4.4) $$ \begin{align}\mathcal{F}^+:=\{z\in \mathbb{H}: 0<\operatorname{Re} z<1/2, |z|>1\}. \end{align} $$

The map $\ast $ defines a side pairing transformation on this polygon by identifying the half circle connecting $\frac {a_{i}}{b_{i}},\frac {a_{i+1}}{b_{i+1}}$ and the one connecting $\frac {a_{i^\ast +1}}{b_{i^\ast +1}},\frac {a_{i^\ast }}{b_{i^\ast }}$ , as well as identifying the vertical sides of $\mathcal {P}(p)$ and the elliptic sides. By Poincaré’s theorem, $\mathcal {P}(p)$ together with this pairing defines a subgroup of $\mathrm {PSL}_2(\mathbb {R})$ which can be shown to be equal to $\Gamma _0(p)$ . Furthermore, since $\mathcal {P}(p)$ has a minimal number of sides, it follows that an independent set of generators of $\Gamma _0(p)$ is given by the matrices which map between the sides identified by the pairing induced from $\ast $ . These matrices are explicitly given by (see [Reference KulkarniKul91, Theorem 6.1])

(4.5)

and the $e_2$ matrices of order $2$ and $e_3$ matrices of order $3$

(4.6) $$ \begin{align} &\begin{pmatrix} a_{i+1}b_{i+1}+a_{i}b_i & -a_i^2-a_{i+1}^2 \\ b_i^2+b_{i+1}^2 & -a_{i+1}b_{i+1}-a_{i}b_{i}\end{pmatrix},\qquad\qquad\quad\qquad \end{align} $$

(4.7) $$ \begin{align} &\begin{pmatrix} a_{i+1}b_{i+1}+a_{i}b_{i+1}+a_{i}b_i & -a_i^2-a_ia_{i+1}-a_{i+1}^2 \\ b_i^2+b_ib_{i+1}+b_{i+1}^2 & -a_{i+1}b_{i+1}-a_{i+1}b_i-a_{i}b_{i}\end{pmatrix}, \end{align} $$

together with the $2g$ hyperbolic matrices

(4.8) $$ \begin{align} \begin{pmatrix} a_{i^\ast+1}b_{i+1}+a_{i^\ast}b_i & -a_ia_{i^\ast}-a_{i+1}a_{i^\ast+1} \\ b_ib_{i^\ast}+b_{i+1}b_{i^\ast+1} & -a_{i+1}b_{i^\ast+1}-a_{i}b_{i^\ast}\end{pmatrix} \text{ for } i<i^\ast \text{ a pair}. \end{align} $$

Observe that the $2g$ hyperbolic matrices above, together with T, define a basis for $H_1(Y_0(p),\mathbb {R})$ under the map (3.5). Recently, Doan–Kim–Lang–Tan [Reference Doan, Kim, Lang and TanDKLP22] have shown that one can find minimal special fundamental polygons $\mathcal {P}_{\mathrm {min}}(p)$ , meaning that we have

$$ \begin{align*} b_ib_{i^\ast}+b_{i+1}b_{i^\ast+1}=p, \quad 0\leq i\leq n, \end{align*} $$

which implies that, in fact, $\mathcal {S}(\mathcal {P}_{\mathrm {min}}(p))\subset \mathcal {S}(\mathcal {F}_{\mathrm {Zag}}(p))$ .

5.1 The Eisenstein contribution

In order to estimate the Eisenstein contribution, we will rely on the following classical formula of Hecke (see [Reference Duke, Imamoḡlu and TóthDIT18, (68)]):

$$ \begin{align*} \sum_{A\in \mathrm{ Cl}_K^+} \int_{\mathcal{C}_{A}(p)} E^*_2(z)dz\, \chi(A)= 6(1-\chi(J))L(\chi,0) ,\end{align*} $$

where $E^*_2(z)=E_2(z)-\frac {3}{\pi \operatorname {Im} z}$ (with $E_2(z)$ defined in (3.17)), $J=(\sqrt {D})\in \mathrm {Cl}_K^+$ is the different of K and $L(\chi ,s)$ denotes the (finite part) of the Hecke L-function associated to the narrow class group character $\chi $ . Thus, by a change of variable, we get

$$ \begin{align*} \sum_{A\in \mathrm{Cl}_K^+} \int_{\mathcal{C}_{A}(p)} pE^*_2(pz)dz \, \chi(A)=\sum_{A\in \mathrm{Cl}_K^+} \int_{p\,\mathcal{C}_{A}(p)} E^*_2(z)dz \, \chi(A). \end{align*} $$

We notice by direct computation that $p\,\mathcal {C}_{A}(p)$ (i.e., the dilation of $\mathcal {C}_{A}(p)$ by the factor p) is again a closed geodesic as follows. If $\mathcal {C}_{A}(p)$ corresponds to the quadratic form $ax^2+bxy+cy^2$ of discriminant $d_K $ and level p, then $p\,\mathcal {C}_{A}(p)$ corresponds to $\tfrac {a}{p}x^2+bxy+cpy^2$ (which now might not be of level p). Recall that in Section 3.1 we fixed a residue $r \ \mathrm {mod}\ 2p$ such that $r^2\equiv d_K \ \mathrm {mod}\ 4p$ . One can now check by direct computation on ideals that

$$ \begin{align*} \left(\mathbb{Z} \frac{a}{p} + \mathbb{Z} \frac{b - \sqrt{d_K}}{2}\right) \left(\mathbb{Z} p + \mathbb{Z} \frac{r - \sqrt{d_K}}{2}\right) = \mathbb{Z} a + \mathbb{Z} \frac{b - \sqrt{d_K}}{2} \end{align*} $$

if $a> 0$ , and similarly,

$$ \begin{align*} \left(\mathbb{Z} \left(-\frac{a}{p}\sqrt{d_K}\right) + \mathbb{Z} \frac{d - b\sqrt{d_K}}{2}\right) \left(\mathbb{Z} p + \mathbb{Z} \frac{r - \sqrt{d_K}}{2}\right) = \mathbb{Z} (-a\sqrt{d_K}) + \mathbb{Z} \frac{d - b\sqrt{d_K}}{2} \end{align*} $$

if $a < 0$ . This implies that when projected to the modular curve $\mathrm {PSL}_2(\mathbb {Z})\backslash \mathbb {H}$ of level $1$ , we have

$$ \begin{align*} p\, \mathcal{C}_{A}(p)=\mathcal{C}_{AA_p}(1),\end{align*} $$

where $A_p=[p, \frac {r-\sqrt {d_K}}{2}]\in \mathrm {Cl}_K^+$ (using the notation (3.1)). This implies that

(5.7) $$ \begin{align}\nonumber \sum_{A\in \mathrm{Cl}_K^+} \langle [\mathcal{C}_{A}(p)],\omega_{E}(p)\rangle \chi(A)&=\sum_{A\in \mathrm{Cl}_K^+} \int_{\mathcal{C}_{A}(p)} E_{2,p}(z)dz\,\, \chi(A)\\ &= \frac{6}{p-1}(1-\chi(J))(\overline{\chi(A_p)}-1)L(\chi,0), \end{align} $$

using that $(p-1)E_{2,p}(z)=pE_{2}^*(pz)-E_{2}^*(z)$ . Note that the above vanishes when $\chi (J)=1$ or $\chi (A_p)=1$ .

By the functional equation for odd class group L-functions [Reference Duke, Imamoḡlu and TóthDIT18, p. 13] and the fact that class group characters are self-dual (i.e., $\chi \circ c=\overline {\chi }$ where c denotes complex conjugation), we conclude for $\chi \in \widehat {\mathrm {Cl}_K^+}$ with $\chi (J)=-1$ that

(5.8) $$ \begin{align} \overline{L(\chi,0)}=L(\overline{\chi},0)=L(\chi\circ c,0)=L(\chi,0)=\pi^{-2} d_K^{1/2} L(\chi,1)>0.\end{align} $$

By standard estimates for L-functions on the critical line for all $\varepsilon>0$ , there is some uniform (but ineffective) constant $c_\varepsilon>0$ such that

(5.9) $$ \begin{align} L(\chi,0)\geq c_\varepsilon d_K^{1/2-\varepsilon}.\end{align} $$

Proposition 5.2. Let $H\leq \mathrm {Cl}_K^+$ be a subgroup such that $A_p\notin H$ and $J\notin H$ . Then for each $\varepsilon>0$ , there exists a constant $c_\varepsilon>0$ such that

(5.10) $$ \begin{align} \sum_{A\in H} \langle [\mathcal{C}_{A}(p)], \omega_{E}(p) \rangle \leq -c_\varepsilon \frac{d_K^{1/2-\varepsilon}}{p}. \end{align} $$

Proof. By orthogonality, the Hecke formula (5.7), and equations (5.8) and (5.9), we conclude

(5.11) $$ \begin{align} \sum_{A\in H} \langle [\mathcal{C}_{A}(p)], \omega_{E}(p) \rangle&=\frac{|H|}{|\mathrm{Cl}_K^+|}\sum_{\chi\in \widehat{\mathrm{Cl}_K^+}: \chi_{|H}=1}\sum_{A\in \mathrm{Cl}_K^+} \langle [\mathcal{C}_{A}(p)], \omega_{E}(p) \rangle\chi(A)\qquad\qquad\qquad \end{align} $$

(5.12) $$ \begin{align} &\qquad\quad\qquad=-\frac{12|H|}{(p-1)|\mathrm{Cl}_K^+|}\sum_{\substack{\chi\in \widehat{\mathrm{Cl}_K^+}:\\ \chi_{|H}=1,\chi(J)=-1}}\sum_{A\in \mathrm{Cl}_K^+} (1-\operatorname{Re} \chi(A_p))L(\chi,0) \end{align} $$

(5.13) $$ \begin{align} &\qquad\!\!\quad\leq -c_\varepsilon \frac{d_K^{1/2-\varepsilon}}{p}\frac{|H|}{|\mathrm{Cl}_K^+|}\sum_{\substack{\chi\in \widehat{\mathrm{Cl}_K^+}:\\ \chi_{|H}=1,\chi(J)=-1}} (1-\operatorname{Re} \chi(A_p)),\qquad \end{align} $$

where we are using that $(1-\operatorname {Re} \chi (A_p))\geq 0$ . Now we observe that if $H':=\langle H,J\rangle \leq \mathrm {Cl}_K^+$ denotes the group generated by $H\leq \mathrm {Cl}_K^+$ and $J=(\sqrt {d_K})$ , then we can write

(5.14) $$ \begin{align} \sum_{\substack{\chi\in \widehat{\mathrm{Cl}_K^+}:\\ \chi_{|H}=1,\chi(J)=-1}} (1-\operatorname{Re} \chi(A_p)) &= \frac{|\mathrm{Cl}_K^+|}{|H'|} -\operatorname{Re} \left( \sum_{\substack{\chi\in \widehat{\mathrm{Cl}_K^+/H}}} \chi(A_p H)-\sum_{\substack{\chi\in \widehat{\mathrm{Cl}_K^+/H'}}}\chi(A_p H') \right) \end{align} $$

(5.15) $$ \begin{align} &\,\kern-1.3pt= \begin{cases} \tfrac{|\mathrm{Cl}_K^+|}{2|H|},& A_p\notin H'\\ \frac{|\mathrm{Cl}_K^+|}{|H|},& A_p\in H'\end{cases}.\quad\quad\qquad \end{align} $$

Here, we are using that $J\notin H$ , which implies that $|H'|=2|H|$ , and $A_p\notin H$ . Inserting this into the above yields the wanted estimate.

5.2 The cuspidal contribution

In the case of cuspidal Weyl sums, we will use the following formula due to Popa [Reference PopaPop06, Theorem 6.3.1] (see also the first formula in the introduction of [Reference PopaPop06]):

(5.16) $$ \begin{align} \left|\sum_{A\in \mathrm{Cl}_K^+} \langle [\mathcal{C}_{A}(p)], \omega_f \rangle \overline{\chi}(A)\right|^2=d_K^{1/2}L(\pi_f\otimes \pi_\chi,1/2), \end{align} $$

where $\pi _f$ denotes the $\mathrm {GL}_2$ -automorphic representation associated to f, $\pi _\chi $ denotes the $\mathrm {GL}_2$ -automorphic representation associated to $\chi $ via automorphic induction and $L(\pi _f\otimes \pi _\chi ,1/2)$ denotes the central value of the (finite part of the) Rankin–Selberg L-function of $\pi _f$ and $\pi _\chi $ . This implies

(5.17) $$ \begin{align} \sum_{A\in \mathrm{Cl}_K^+} \langle [\mathcal{C}_{A}(p)], \omega_f^{\pm} \rangle \overline{\chi}(A) = \frac{d_K^{1/4}}{2}\left( \epsilon_{f,\chi} |L(\pi_f\otimes \pi_\chi,1/2)|^{1/2}\pm \epsilon_{f,\overline{\chi}} |L(\pi_f\otimes \pi_{\overline{\chi}},1/2)|^{1/2} \right), \end{align} $$

with $|\epsilon _{f,\chi }|=|\epsilon _{f,\overline {\chi }}|=1$ . By the subconvexity bound

$$ \begin{align*} L(\pi_f\otimes \pi_{\chi},1/2)\ll_f d_K^{1/2-1/1057}\end{align*} $$

due to Michel [Reference MichelMic04, Theorem 2], we conclude the following bound on the twisted Weyl sums:

(5.18) $$ \begin{align}\sum_{A\in \mathrm{Cl}_K^+} \langle [\mathcal{C}_{A}(p)], \omega_f^{\pm} \rangle \overline{\chi}(A)\ll_f d_K^{1/2-1/2114}.\end{align} $$

Combining this with the lower bound for the Eisenstein contribution yields the proof of our main result.

Proof of Theorem 5.1

Let $H\leq \mathrm {Cl}_K^+$ be a subgroup as in the statement. Then by Proposition 5.2, we have for all $\varepsilon>0$ that there exists an absolute constant $c_\varepsilon>0$ such that

(5.19) $$ \begin{align}\left|\sum_{A\in H} \langle [\mathcal{C}_{A}(p)], \omega_{E}(p) \rangle\right|=-\sum_{A\in H} \langle [\mathcal{C}_{A}(p)], \omega_{E}(p) \rangle\geq c_\varepsilon d_K^{1/2-\varepsilon}p^{-1}.\end{align} $$

Now the result follows from (5.6) (with $C=1$ ) by dividing through by (5.19) and bounding the cuspidal contribution using (5.18). Here, we are using that the number of narrow class group characters $\chi $ such that $\chi _{|H}=1$ is exactly $\tfrac {|\mathrm {Cl}_K^+|}{|H|}$ , and the remaining terms in (5.6) do not depend on $d_K$ .

6.1 The method of proof

The first natural approach to bounding the left-hand side of (6.1) would be to use a version of the amplified pre-trace formula approach to the sup norm problem in arithmetic quantum chaos (see, for example, [Reference Blomer and HolowinskyBH10]). This is, however, not very effective due to the possible non-positivity of (6.3) (i.e., we are working in a Banach space rather than a Hilbert space). However, it has become apparent from the work of Steiner [Reference SteinerSte20], Khayutin–Steiner [Reference Khayutin and SteinerKS20] and Khayutin–Nelson–Steiner [Reference Khayutin, Nelson and SteinerKNS22] that one in many cases can obtain very strong sup norm bounds by using the theta correspondence. We will employ a version of this argument in our setting. The basic fact is that for $v_0\in H_1(X_0(p),\mathbb {R})$ and $\omega \in H^1(X_0(p),\mathbb {R})$ ,

(6.4) $$ \begin{align}\sum_{n\geq 1} \langle T_n v_0, \omega\rangle e^{2\pi i nz}\end{align} $$

defines a cusp form of level p. This follows easily by the fact that the $\pm 1$ isotypic components of $\iota $ acting on $H_1(X_0(p), \mathbb {R})$ are isomorphic as Hecke modules to the space of holomorphic cusp forms $\mathcal {S}_2(p)$ of weight 2 and level p (which is an incarnation of the Eichler–Shimura isomorphism [Reference ShimuraShi94, Section 8.2]). This can be seen as an instance of the theta correspondence (see [Reference Darmon, Harris, Rotger and VenkateshDHRV21, Section 5.3]). We will give a simple proof below of the exact statement that we need. Now the key fact is that if $\iota v_0=\pm v_0$ , then the $L^2$ -norm of (6.4) is (by spectrally expanding) equal to

$$ \begin{align*} \sum_{f\in \mathcal{B}_p} |\langle v_0 , \omega_f^{\pm}\rangle|^ 2 |\langle v_f^{\pm}, \omega \rangle| ^2.\end{align*} $$

Note that now we have obtained positivity for free! Thus, we are reduced to, one the one hand, a lower bound for $|\langle v_0 , \omega _f^{\pm }\rangle |^ 2$ which we resolve in Corollary 6.5, and on the other hand, a bound for the $L^2$ -norm of (6.4) which is essentially equal to

(6.5) $$ \begin{align}\sum_{1\leq n\leq p} \frac{|\langle T_n v_0, \mathbb{P}_{\mathrm{cusp}}\,\omega\rangle|^2}{n}.\end{align} $$

The above is an analogue of the second moment matrix count that appears in, for example, [Reference Khayutin and SteinerKS20, Section 9]. We will bound (6.5) using geometric coding of geodesics.

Remark 6.2. Let B be a basic basis of level p and let $v\in B$ and $v^\ast \in B^{\ast }$ such that $\langle v, v^\ast \rangle =1$ . Then we get by expanding in the Hecke basis

$$ \begin{align*} 1= \sum_{f\in \mathcal{B}_p,\pm } \langle v_f^\pm, v^\ast \rangle \langle v, \omega_f^\pm \rangle.\end{align*} $$

As we will see in Corollary 6.5, we have on average that $\langle v, \omega _f^\pm \rangle \asymp 1$ . Thus, if the classes $v_f^\pm $ are perfectly distributed (relative to B), then we would have $\langle v_f^\pm , v^\ast \rangle \asymp 1/p$ . Thus, the strongest sup norm conjecture one can hope for is

$$ \begin{align*} \langle v_f^\pm, \omega \rangle \ll_\varepsilon p^{-1+\varepsilon}, \end{align*} $$

for $\omega \in B^{\ast }$ . This might, however, be too much to hope for.

Remark 6.3. The dual sup norm problem corresponds to obtaining bounds for the modular symbols

$$ \begin{align*} \sup_{v\in B}\, |\langle v, \omega_f \rangle|, \end{align*} $$

for $v\in B$ as $p\rightarrow \infty $ , where $\omega _f=f(z)dz$ with $f\in \mathcal {B}_p$ a holomorphic Hecke eigenform of weight $2$ and level p. Let $v=\{\gamma z,z\}\in H_1(Y_0(p),\mathbb {Z})$ with $\gamma \in \mathcal {S}(\mathcal {F}_{\mathrm {Zag}}(p))$ , which we may assume is hyperbolic. Picking z on the fixed circle of $\gamma $ and using Hölder’s inequality, we see that

$$ \begin{align*}\left|\langle v, \omega_f \rangle\right| \leq 2\pi \int_{\mathcal{C}_\gamma} |\operatorname{Im} (z)f(z)|\frac{dz}{\operatorname{Im} (z)}\ll_\varepsilon p^{1/4+\varepsilon} , \end{align*} $$

using the sup norm estimate coming from [Reference Khayutin, Nelson and SteinerKNS22, Theorem 1.6]. Here, $\mathcal {C}_\gamma $ denotes the closed geodesic on $X_0(p)$ associated to $\gamma $ with hyperbolic length $\ell (\mathcal {C}_\gamma )\ll \log p$ . Notice that we also have the trivial bound

$$ \begin{align*} \left( \sum_{f\in \mathcal{B}_p}\left|\langle v, \omega_f \rangle\right|{}^2\right)^{1/2} \leq 2\pi \int_{\mathcal{C}_\gamma} \left(\sum_{f\in \mathcal{B}_p} |\operatorname{Im} (z)f(z)|^2\right)^{1/2}\frac{dz}{\operatorname{Im} (z)}\ll_\varepsilon p^{1/2+\varepsilon} \end{align*} $$

by Minkowski’s integral inequality and the pre-trace formula (thanks to the referee for pointing this out).

6.2 A lower bound for modular symbols

In this section, we will construct the homology class $v_0\in H_1(X_0(p), \mathbb {R})$ mentioned above such that $\langle v_0, \omega _f^{\pm }\rangle $ is ‘not too small’ in terms of the level. We will be constructing $v_0$ in terms of classes of the shape $\{\tfrac {x}{p},\infty \}$ for $1\leq x< p$ (using the notation (3.6)). This reduces the problem to a lower bound for the additive twist L-functions of f. One could naively try to calculate the average of all additive twists with $1\leq x<p$ . But this yields very poor results as there is a lot of cancellation in this sum. Instead, we use that the second moment coincides with the second moment of Dirichlet twists of the L-function of f (by the Birch–Stevens formula). By Cauchy–Schwarz, it suffices to calculate the first moment of these L-functions instead. Such a calculation is quite standard using an approximate functional equation. The exact case we need, however, does not seem to have been considered before as we are in the case of joint ramification. For the sake of completeness, we have provided a detailed argument below.

To be more precise, let $f\in \mathcal {B}_p$ be a Hecke normalized eigenform and let $\chi $ be a primitive Dirichlet character modulo q with $p|q$ . Then we define the twisted L-function as the analytic continuation of

$$ \begin{align*} L(f,\chi,s):= \sum_{n\geq 1}\frac{\lambda_f(n)\chi(n)}{n^{s+1/2}} \end{align*} $$

(here the Ramanujan conjecture is $|\lambda _f(n)|\leq d(n)n^{1/2}$ ). By the Birch–Stevens formula (see, for example, [Reference NordentoftNor21, Proposition 6.1]), we have for a primitive $\chi $ with $\chi (-1)=\pm 1$

(6.6) $$ \begin{align} i^{(\chi(-1)-1)/2}\tau(\overline{\chi})L(f,\chi,s)= \sideset{}{^\ast}\sum_{a \ \mathrm {mod}\ q} L^{\pm}(f, a/q,s)\overline{\chi(a)},\end{align} $$

where

$$ \begin{align*} L^\pm(f, a/q,s):=\frac{L(f, a/q,s)\pm L(f, -a/q,s)}{1+i\pm (1-i)}, \quad L(f, a/q,s):=\sum_{n\geq 1} \frac{\lambda_f(n)e(na/q)}{n^{s+1/2}},\quad \operatorname{Re} s>1 \end{align*} $$

are the additive twist L-series satisfying analytic continuation and the functional equation

$$ \begin{align*} \gamma_f(s)q^s L(f,a/q,s)= -\gamma_f(1-s)q^{1-s} L(f,-\overline{a}/q,1-s), \end{align*} $$

where $a\overline{a}\equiv 1\ \mathrm{mod}\ q$ and $\gamma_f(s)=\Gamma(s+1/2)(2\pi)^{-s}$ (see e.g. Reference NordentoftNor21, Proposition 3.3). This implies the functional equation

(6.7) $$ \begin{align}\gamma_f(s)q^{s} L(f,\chi,s)=\varepsilon_\chi \gamma_f(1-s)q^{1-s} L(f,\overline{\chi},1-s),\end{align} $$

where $\varepsilon _\chi = \frac {-\tau (\chi )^2}{q}$ . Note that by inserting the Fourier expansion of f and interchanging sum and integral, we arrive at the identity

(6.8) $$ \begin{align}\langle \{\tfrac{a}{q},\infty\}, \omega^\pm_f\rangle=-L^\pm(f,\tfrac{a}{q}, 1/2 ).\end{align} $$

Lemma 6.4. Let $f\in \mathcal {B}_p$ and let q be an integer such that $p|q$ . Then we have (uniformly in p and f) that

$$ \begin{align*} \frac{1}{\varphi^\ast_\pm(q)}\,\,\,\sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} L(f,\chi,1/2)=1+O_\varepsilon(q^{-1/4+\varepsilon}),\end{align*} $$

where the sum is restricted to primitive characters $\chi $ modulo q with $\chi (-1)=\pm 1$ , and $\varphi ^*_\pm (q)$ denotes the total number of such characters.

Proof. By the approximate functional equation [Reference Iwaniec and KowalskiIK04, Theorem 5.3] using (6.7), we can write the moment in question as

(6.9) $$ \begin{align} \frac{2}{\varphi^*(q)}\,\,\,\sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} \sum_{n\geq 1} \frac{\lambda_f(n)}{n}\left(\chi(n) V\left(\frac{n}{q^\lambda}\right)+ \varepsilon_\chi \overline{\chi}(n) V\left(\frac{n}{q^{2-\lambda}}\right)\right), \end{align} $$

for some $0<\lambda <2$ to be chosen and $V: \mathbb {R}_{>0}\rightarrow \mathbb {R}$ a smooth, rapidly decaying function satisfying $V(y)=1+O(y^{1/3})$ as $y\rightarrow 0$ (see [Reference Iwaniec and KowalskiIK04, Proposition 5.4]). Now by a simple application of Möbius inversion, we get that for $(n,q)=1$ ,

(6.10) $$ \begin{align} \sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} \chi(n) &=\sum_{d|q} \mu(q/d)\varphi(d)(\delta_{n\equiv 1 \, (d)}\pm \delta_{n\equiv -1 \, (d)})/2 , \end{align} $$

(6.11) $$ \begin{align} \sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} \varepsilon_\chi \overline{\chi}(n) &=-\frac{1}{q} \sum_{d|q} \mu(q/d)\varphi(d)\frac{K_2(n;d)\pm K_2(-n;d)}{2}, \end{align} $$

where $K_2(n;d)= \sum _{xy\equiv n\, (d)} e((x+y)/d)$ denotes the usual $2$ -dimensional Kloosterman sum. By Weil’s bound and standard estimates, we conclude that for $(n,q)=1$ ,

(6.12) $$ \begin{align} \sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} \varepsilon_\chi \overline{\chi}(n) \ll_\varepsilon q^{1/2+\varepsilon}. \end{align} $$

Now for the primary part of the sum (6.9) we obtain using the analytic properties of V mentioned above as well as (6.10), we get

(6.13) $$ \begin{align} &\frac{1}{\varphi^*_\pm(q)}\,\,\,\sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ q} \sum_{n\geq 1} \frac{\lambda_f(n)}{n} \chi(n) V\left(\frac{n}{q^\lambda}\right) \quad\qquad\qquad\end{align} $$

(6.14) $$ \begin{align} &\quad\qquad=1+O_{\varepsilon}\left(\frac{q^{1-\lambda/3}}{\varphi^*_\pm(q)}+\frac{1}{\varphi^*_\pm(q)}\sum_{d|q} \varphi(d) \sum_{\substack{n\equiv \pm 1\, (d)\\ 1<n\ll q^{\lambda+\varepsilon}}} \frac{|\lambda_f(n)|}{n} \right), \end{align} $$

with the main term corresponding to $n=1$ . Now using the Ramanujan bound $|\lambda _f(n)|\leq d(n)n^{1/2}\ll _\varepsilon n^{1/2+\varepsilon }$ , we get

(6.15) $$ \begin{align} \sum_{\substack{n\equiv \pm 1\, (d)\\ 1<n\ll c^{\lambda+\varepsilon}}} \frac{|\lambda_f(n)|}{n}\ll_\varepsilon \sum_{ 1\leq m \ll q^{\lambda+\varepsilon}/d}\sum_\pm (md\pm 1)^{-1/2+\varepsilon}\ll q^{\lambda/2+\varepsilon}d^{-1}, \end{align} $$

which bounds the error-term in (6.13) by $O_\varepsilon ((q^{1-\lambda /3}+q^{\lambda /2+\varepsilon })/\varphi ^\ast (q))$ . Similarly, by using (6.11), we can bound the dual sum by $O_\varepsilon (q^{3/2-\lambda /2+\varepsilon }/\varphi ^\ast _\pm (q))$ . Choosing $\lambda =3/2$ and recalling that $\varphi ^\ast _\pm (q)\gg _\varepsilon q^{1-\varepsilon }$ , we get the wanted asymptotic formula.

Corollary 6.5. There exists an absolute constant $c_0>0$ such that for any prime p, $f\in \mathcal {B}_p$ and $\epsilon \in \{\pm \}$ , we have

(6.16) $$ \begin{align} \frac{1}{p-1}\sum_{0<a <p} |\langle\{\tfrac{a}{p},\infty\},\omega_f^{\epsilon} \rangle|^2\geq c_0. \end{align} $$

Proof. First of all, recall that the following matrices generate $\Gamma _0(p)$ :

where $0<a^\ast <p$ are such that $a\overline {a}\equiv -1 \ \mathrm {mod}\ p$ (as is proved in Section 4.2). Since $\omega _f^{\pm }$ are nonzero cohomology classes vanishing on parabolic elements, we conclude that the left-hand side of (6.16) is always nonzero.

We now define the Fourier transform of

$$ \begin{align*} (\mathbb{Z}/p\mathbb{Z})^\times \ni a\mapsto \langle\{\tfrac{a}{p},\infty\},\omega_f^{\pm} \rangle \end{align*} $$

as follows for a Dirichlet character $\chi \ \mathrm {mod}\ p$ :

$$ \begin{align*} \widehat{L}_f^{\pm}(\chi):= \sideset{}{^\ast}\sum_{a \ \mathrm {mod}\ p} \langle\{\tfrac{a}{p},\infty\},\omega_f^{\pm} \rangle \overline{\chi(a)}.\end{align*} $$

For $\chi $ primitive with $\chi (-1)=\pm 1$ , we have by the equality (6.8) and the Birch–Stevens formula (6.6) that

$$ \begin{align*} \widehat{L}_f^{\pm}(\chi)= -i^{(\chi(-1)-1)/2}\tau(\overline{\chi})L(f,\chi,1/2). \end{align*} $$

Now by Parseval, we have

$$ \begin{align*} \sum_{0<a <p} |\langle\{\tfrac{a}{p},\infty\},\omega_f^{\pm} \rangle|^2=\frac{1}{p-1}\sum_{\chi \ \mathrm {mod}\ p} |\widehat{L}_f^{\pm}(\chi)|^2 \geq \frac{p}{p-1}\,\,\,\sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ p} |L(f,\chi,1/2)|^2. \end{align*} $$

Using the previous lemma, we conclude by Cauchy–Schwarz that

$$ \begin{align*} \frac{1}{\varphi^\ast_\pm(p)}\,\,\,\sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ p} |L(f,\chi,1/2)|^2\geq \left( \frac{1}{\varphi^\ast_\pm(p)}\,\,\, \sideset{}{^{\ast,\pm}}\sum_{\chi \ \mathrm {mod}\ p} L(f,\chi,1/2)\right)^2\geq 1/2, \end{align*} $$

for p large enough. This yields the wanted lower bound since $\varphi _\pm ^\ast (p)\asymp p-1$ .

6.3 Reduction to a counting problem

We will now use the theta correspondence in a simple form to reduce the sup norm problem to a certain counting problem.

Lemma 6.7. Let $\omega \in H^1(X_0(p),\mathbb {R})$ be a cuspidal cohomology class. Then we have

$$ \begin{align*} \sum_{\epsilon\in \{\pm \}}\sum_{f\in \mathcal{B}_p} | \langle v_f^{\epsilon}, \omega\rangle|^2 \ll_\varepsilon p^{-1+\varepsilon}\sum_{n\geq 1} \frac{\frac{1}{p}\sum_{0< x<p}(|\langle T_n \{\tfrac{x}{p}, \infty\}, \omega \rangle|^2+|\langle T_n \{\tfrac{1}{x}, 0\}, \omega \rangle|^2)}{n} e^{-8 n/p} \end{align*} $$

uniformly in p, where $\langle \cdot , \cdot \rangle $ denotes the cap product pairing (3.4) between homology and cohomology and $T_n$ denotes the n-th Hecke operator.

Proof. For $0<x<p$ and ${\boldsymbol {\epsilon }}=(\epsilon _1,\epsilon _2)$ with $\epsilon _i\in \{\pm 1\}$ , we consider the following class in the compactly supported homology

(6.17) $$ \begin{align} &v_x^{\boldsymbol{\epsilon}}:=(1+\epsilon_1 \iota)(1+\epsilon_2 W_p)\{\tfrac{x}{p} , \infty \}\qquad\qquad\qquad\qquad\quad \end{align} $$

(6.18) $$ \begin{align} &\quad \qquad\qquad\qquad\qquad= \{\tfrac{x}{p} , \infty \}+\epsilon_1 \{-\tfrac{x}{p} , \infty \} +\epsilon_2\{-\tfrac{1}{x},0\}+\epsilon_1\epsilon_2 \{\tfrac{1}{x},0\}\in H_1(X_0(p),\mathbb{Z}). \end{align} $$

By construction, $v_x^{\boldsymbol {\epsilon }}$ is contained in the $\epsilon _1$ eigenspace of the involution $\iota $ as in (3.14), and in the $\epsilon _2$ eigenspace of the Fricke involution $W_p$ as in (3.13). Associated to $0<x<p$ , ${\boldsymbol {\epsilon }}\in \{(\pm 1,\pm 1)\}$ and $\omega \in H^1(X_0(p),\mathbb {R})$ (suppressed in the notation), we define $g:\mathbb {H}\rightarrow \mathbb {C}$ by

(6.19) $$ \begin{align} g(z):=\sum_{n\geq 1} \langle T_n v_x^{\boldsymbol{\epsilon}}, \omega \rangle e^{2\pi i nz}.\end{align} $$

By expanding in the Hecke basis of cohomology, we obtain

$$ \begin{align*} \langle T_n v_x^{\boldsymbol{\epsilon}}, \omega \rangle =\sum_{\substack{f\in \mathcal{B}_p:\\ W_p f=\epsilon_2 f}} \lambda_f(n) \langle v_x^{\boldsymbol{\epsilon}}, \omega_f^{\epsilon_1}\rangle \langle v_f^{\epsilon_1}, \omega\rangle. \end{align*} $$

By newform theory, we know that for $f\in \mathcal {B}_p$ , we have $\lambda _f(n)=a_f(n)$ , where $a_f(n)$ denotes the Fourier coefficients (at $\infty $ ) of f. Thus, we conclude that

$$ \begin{align*} g(z)= \sum_{\substack{f\in \mathcal{B}_p:\\ W_p f=\epsilon_2 f}} \langle v_x^{\boldsymbol{\epsilon}}, \omega_f^{\epsilon_1}\rangle \langle v_f^{\epsilon_1}, \omega\rangle f(z), \end{align*} $$

and, in particular, $g\in S_2(p)$ is a holomorphic cusp form of weight $2$ for $\Gamma _0(p)$ contained in the $\epsilon _2$ eigenspace of $W_p$ . This can be seen as an instance of the theta correspondence as explained above. By orthogonality of Hecke eigenforms, this implies the key identity

(6.20) $$ \begin{align} \langle g, g \rangle_{\mathrm{Pet}} = \sum_{\substack{f\in \mathcal{B}_p:\\ W_p f=\epsilon_2 f}} \langle f, f \rangle_{\mathrm{Pet}} |\langle v_x^{{\boldsymbol{\epsilon}}}, \omega_f^{\epsilon_1}\rangle \langle v_f^{\epsilon_1}, \omega\rangle|^2 , \end{align} $$

where $\langle f, g \rangle _{\mathrm {Pet}}= \int _{Y_0(p)} f(z)\overline {g(z)} dxdy$ denotes the Petersson inner-product on $\mathcal {S}_2(p)$ . Recall from (4.3) that the subgroup

$$ \begin{align*} \Gamma_0^\ast(p):=\langle W_{p}, \Gamma_0(p)\rangle\subset \mathrm{PSL}_2(\mathbb{R}) \end{align*} $$

has $\Gamma _0(p)$ as an index two subgroup and has a fundamental domain contained in

$$ \begin{align*} \{z\in \mathbb{H}: |\operatorname{Re} z| \leq 1/2, \operatorname{Im} z \geq \sqrt{3}/(2p) \}.\end{align*} $$

By unfolding and using that $|g|^2$ is invariant under the Fricke involution $W_p$ , by construction we arrive at the following bound:

(6.21) $$ \begin{align}\nonumber\sum_{\substack{f\in \mathcal{B}_p:\\ W_p f=\epsilon_2 f}} \langle f, f \rangle_{\mathrm{Pet}}|\langle v_x^{{\boldsymbol{\epsilon}}}, \omega_f^{\epsilon_1}\rangle \langle v_f^{\epsilon_1}, \omega\rangle|^2=2\int_{\Gamma_0^\ast(p)\backslash \mathbb{H}} |g(z)|^2 dxdy &\ll \int_{\sqrt{3}/(2p)}^\infty \int_{-1/2}^{1/2} |g(z)|^2 dxdy\\ \nonumber&\ll \sum_{n\geq 1} |\langle T_n v_x^{{\boldsymbol{\epsilon}}}, \omega \rangle|^2 \int_{\sqrt{3}/(2p)}^\infty e^{-4\pi n y} dy\\ &\ll \sum_{n\geq 1} \frac{|\langle T_n v_x^{{\boldsymbol{\epsilon}}}, \omega \rangle|^2}{n} e^{-2 \sqrt{3}\pi n/p}. \end{align} $$

Notice that for $f\in \mathcal {B}_p$ with $W_p f=\epsilon _2 f$ , we have by self-adjointness of $\iota $ and $W_p$ with respect to the cap product pairing that

$$ \begin{align*} \langle v_x^{{\boldsymbol{\epsilon}}}, \omega_f^{\epsilon_1}\rangle=4 \langle \{\tfrac{x}{p},\infty\}, \omega_f^{\epsilon_1}\rangle.\end{align*} $$

Now we sum over $0<x<p$ , $\boldsymbol {\epsilon }\in \{(\pm 1,\pm 1)\}$ and apply Corollary 6.5 to the left-hand side of (6.21). Finally, by expressing the Petersson inner product in terms of a special value of an adjoint L-function (see, for example, [Reference Petridis and RisagerPR18, (8.6)]) and using the lower bounds of Hoffstein–Lockhart [Reference Hoffstein and LockhartHL94], we arrive at

$$ \begin{align*} \langle f, f \rangle_{\mathrm{Pet}}=\frac{pL(\mathrm{sym}^2 f, 1)}{8\pi^3}\gg_\varepsilon p^{1-\varepsilon}.\end{align*} $$

Finally, we have by linearity of $T_n$ that

$$ \begin{align*}|\langle T_n v_x^{{\boldsymbol{\epsilon}}}, \omega \rangle|^2 \ll |\langle T_n \{\tfrac{x}{p} , \infty \}, \omega \rangle|^2+|\langle T_n \{-\tfrac{x}{p} , \infty \}, \omega \rangle|^2+|\langle T_n \{\tfrac{1}{x} , \infty \}, \omega \rangle|^2+|\langle T_n \{-\tfrac{1}{x} , \infty \}, \omega \rangle|^2, \end{align*} $$

from which we conclude the wanted inequality since $2 \sqrt {3}\pi>8$ .

6.4 The counting argument

By Lemma 6.7, we are reduced to a certain second moment count. We think of this as an analogue of the matrix counts that show up in most approaches to the (arithmetic) sup norm problem (see, for example, [Reference TemplierTem15]). Recall the Ramanujan bound $|\lambda _f(n)|\leq n^{1/2}d(n)$ for Hecke eigenforms $f\in \mathcal {B}_p$ . This implies for any class $v\in H_1(X_0(p),\mathbb {R})$ and $\omega \in H^1(X_0(p),\mathbb {R})$ , we have

$$ \begin{align*} \langle T_n v, \omega \rangle\ll_{v,\omega} n^{1/2} d(n),\quad \text{as }n\rightarrow \infty. \end{align*} $$

This is, however, not useful as for us the main point is exactly the dependence on v and $\omega $ . Our approach is to use the explicit description of the Hecke operators and then lift the counting from $\Gamma _0(p)^{\mathrm {ab}}$ to $\Gamma _0(p)$ . As a first step, we use geometric coding as in Proposition 4.1 to obtain the following.

Lemma 6.8. Let $\omega \in H^1(X_0(p),\mathbb {R})$ be a cuspidal cohomology class and with $c> 0$ . Then we have

(6.22) $$ \begin{align} | \langle \{\gamma \infty, \infty \}, \omega \rangle| \ll \left(\max_{0<x<p} |\langle \{\tfrac{x}{p} ,\infty\}, \omega\rangle|\right)\log c,\end{align} $$

uniformly in p.

Proof. Observe that $\{\gamma \infty , \infty \}$ only depends on $a/c \ \mathrm {mod}\ 1$ . This means that without changing the homology class, we may assume that $a,d$ are integers satisfying $2c\leq |a+d|\leq 4 c $ as well as $-c<a-d<c$ . This implies that all entries of $\gamma $ are $O( c )$ , and that the half circle $S_\gamma $ fixed by $\gamma $ intersects the standard fundamental domain $\mathcal {F}_{\mathrm {std}}$ for $\mathrm {PSL}_2(\mathbb {Z})$ . Recall Zagier’s fundamental polygon $\mathcal {F}_{\mathrm {Zag}}'(p)$ for $\Gamma _0(p)$ defined in Section 4.2 which consists of $\mathrm {PSL}_2(\mathbb {Z})$ -translates of $\mathcal {F}_{\mathrm {std}}$ , as well as the fundamental polygon $\mathcal {F}_{\mathrm {Zag}}(p):=W_{p} \mathcal {F}_{\mathrm {Zag}}'(p)$ . This latter polygon has the advantage that all entries of the associated side pairing transformations are of size $\leq p$ , whereas for $\mathcal {F}_{\mathrm {Zag}}'(p)$ , the lower left entry can be of magnitude $ p^2$ . Notice also that the assumptions on $\gamma $ above ensure that the half circle $S_\gamma $ fixed by $\gamma $ intersects the Siegel domain $\{z\in \mathbb {H}: |\operatorname {Re} z|\leq 1/2, \operatorname {Im} z\geq 1\}$ which is contained in $\mathcal {F}_{\mathrm {Zag}}(p)$ .

We now consider the geometric coding of $\gamma $ with respect to $ \mathcal {F}_{\mathrm {Zag}}(p)$ as in Corollary 4.2. This yields an expression

$$ \begin{align*} \gamma= \sigma_1\cdots \sigma_{N}, \end{align*} $$

where $\sigma _i\in \mathcal {S}(\mathcal {F}_{\mathrm {Zag}}(p))$ are side pairing transformations. When considering the class $\{\gamma \infty , \infty \}\in H_1(X_0(p), \mathbb {Z})$ in the compact homology, we can ignore the parabolic elements among the $\sigma _i$ ’s. More precisely, if $\sigma ^{\prime }_1,\ldots , \sigma ^{\prime }_\ell $ is the subsequence of $\sigma _1,\ldots ,\sigma _{N}$ consisting of non-parabolic elements, then we have

$$ \begin{align*} \{\gamma \infty, \infty \} =\sum_{i=1}^\ell \{ \sigma^{\prime}_i\infty , \infty \}\in H_1(X_0(p), \mathbb{Z}). \end{align*} $$

Now $\ell $ is exactly the number of intersections between the geodesic from w to $\gamma w$ (for any $w\in \mathcal {F}_{\mathrm {Zag}}(p)\cap S_\gamma $ ) and $\Gamma _0(p)$ -translates of the sides of $\mathcal {F}_{\mathrm {Zag}}(p)$ being paired by non-parabolic elements. This is equal to the number of intersections between the geodesic from w to $\gamma ' w$ (where $\gamma '=W_{p}\gamma W_{p}^{-1}$ and $w\in \mathcal {F}_{\mathrm {Zag}}'(p)\cap S_{\gamma '}$ ) and $\Gamma _0(p)$ -translates of the non-parabolic sides of $\mathcal {F}_{\mathrm {Zag}}'(p)$ (this follows by conjugating everything which preserves parabolicity). Recall that for $\mathcal {F}_{\mathrm {Zag}}'(p)$ , all sides containing a cusp are paired by parabolic elements. Since $\mathcal {F}_{\mathrm {Zag}}'(p)$ consists of $\mathrm {PSL}_2(\mathbb {Z})$ -translates of $\mathcal {F}_{\mathrm {std}}$ , we can bound $\ell $ by the number of intersections between the geodesic from w to $\gamma ' w$ (for $w\in \mathcal {F}_{\mathrm {std}}\cap S_{\gamma '}$ ) and $\mathrm {PSL}_2(\mathbb {Z})$ -translates of the non-parabolic side of $\mathcal {F}_{\mathrm {std}}$ – that is, the arc

$$ \begin{align*} \{z\in \mathbb{H}: |z|=1, -1/2<\operatorname{Re} z< 1/2\}. \end{align*} $$

It now follows from a result of Eichler [Reference EichlerEic65, Satz 1] that

$$ \begin{align*} \ell \ll \log ((a')^2+(b')^2+(c')^2+(d')^2) \ll \log c , \end{align*} $$

where $a',b',c',d'$ are the entries of $\gamma '$ , which by the assumptions on the entries of $\gamma $ are all $O( p c)= O(c ^2)$ . By definition of $\mathcal {F}_{\mathrm {Zag}}(p)$ , we have for all $1\leq i\leq \ell $ that $\sigma ^{\prime }_i\infty $ is of the form $\tfrac {x}{p}$ with $0<x<p$ . Thus, we conclude

$$ \begin{align*} |\langle \{\gamma \infty, \infty \},\omega \rangle| \ll \ell \left(\max_{0<x<p} |\langle \{\tfrac{x}{p} ,\infty\}, \omega\rangle|\right)\ll \log c \left(\max_{0<x<p}|\langle \{\tfrac{x}{p} ,\infty\}, \omega\rangle|\right), \end{align*} $$

as wanted.

This implies the following result for basic bases. Recall the definition of the cuspidal projection operator $\mathbb {P}_{\mathrm {cusp}}$ in (3.23).

Corollary 6.9. Let ${B}$ be a basic basis of level p. Then for with $c>0$ and $\omega \in B^{\ast }$ , we have

$$ \begin{align*} |\langle \{\gamma \infty,\infty\}, \mathbb{P}_{\mathrm{cusp}} \, \omega\rangle|\ll \log c, \end{align*} $$

uniformly in p.

Proof. By Proposition 6.8, we are reduced to proving that

$$ \begin{align*} \max_{0<x<p}|\langle \{\tfrac{x}{p} ,\infty\}, \mathbb{P}_{\mathrm{cusp}}\omega\rangle|\ll 1 \end{align*} $$

for $\omega \in B^{\ast }$ . Recall that we have

$$ \begin{align*} \langle \{\tfrac{x}{p} ,\infty\}, \mathbb{P}_{\mathrm{cusp}}\omega\rangle= \langle \{\gamma_x z ,z\}, \omega\rangle-\langle v_{E}(p), \omega\rangle\langle \{\gamma_x z ,z\}, \omega_{E}(p)\rangle, \end{align*} $$

where with $0<x^\ast <p$ such that $xx^\ast \equiv -1 \ \mathrm {mod}\ p$ and $z\in \mathbb {H} $ is arbitrary. Now by Corollary 4.6, we have

$$ \begin{align*} \langle v_{E}(p), \omega\rangle\ll 1,\quad \langle \{\gamma_x z ,z\}, \omega\rangle \ll 1,\end{align*} $$

and by (3.21), we have

$$ \begin{align*}\langle \{\gamma_x z ,z\}, \omega_{E}(p)\rangle\ll \frac{p|x-x^\ast|+p^2}{p^2}\ll 1,\end{align*} $$

which yields the wanted bound.

6.5 Proof of Theorem 6.1

Combining all of the above, we are now ready to prove our sup norm bounds.

Proof of Theorem 6.1

Let B be a basic basis of level p. By the definition of the Hecke operators acting on homology (3.12), we have for $\omega \in B^{\ast }$ by Corollary 6.9

(6.23) $$ \begin{align} \nonumber |\langle T_n \{\tfrac{x}{p} , \infty \}, \mathbb{P}_{\mathrm{cusp}}\, \omega \rangle| \nonumber &\leq \sum_{\substack{ad=n,\\ (a,p)=1}}\sum_{0\leq b <d} \left|\left\langle \left\{ \frac{ax+bp}{pd}, \infty \right\}, \mathbb{P}_{\mathrm{cusp}}\,\omega \right\rangle\right|\\ & \ll \sum_{\substack{ad=n,\\ (a,p)=1}}\sum_{0\leq b <d} \log pd \ll_\varepsilon p^{\varepsilon}n^{1+\varepsilon}. \end{align} $$

Here, we are using that $(ax+bp,p)=1$ , meaning that $\frac { ax+bp}{pd}$ is of the shape $\gamma \infty $ for $\gamma \in \Gamma _0(p)$ with left lower entry of size $O(pd)$ . By changing contours, we have the following equality of compactly supported homology classes:

A similar argument as above, again using Corollary 6.9, yields

$$ \begin{align*} |\langle T_n \{\tfrac{1}{x} , \infty \}, \mathbb{P}_{\mathrm{cusp}}\, \omega \rangle|\ll_\varepsilon p^{\varepsilon}n^{1+\varepsilon}, \quad \omega\in B^\ast.\end{align*} $$

Thus, by Lemma 6.7, we conclude that

$$ \begin{align*} \sum_{f\in \mathcal{B}_p} | \langle v_f^{\pm}, \omega\rangle|^2 \ll_\varepsilon p^{-1+\varepsilon}\sum_{n\geq 1} \frac{n^{2+\varepsilon}}{n} e^{-8 n/p} \ll_\varepsilon p^{1+\varepsilon} ,\quad \omega\in B^{\ast},\end{align*} $$

as wanted.

8.1 A group theoretic application

Recall the independent generators of $\Gamma _0(p)$ described in Section 4.3 coming from special fundamental polygons. Given a real quadratic field K of discriminant $d_K $ such that p splits in K, the oriented closed geodesics $\mathcal {C}_{A}(p)$ associated to $A\in \mathrm {Cl}_K^+$ corresponds to a conjugacy class of matrices inside $\Gamma _0(p)$ (given by the matrices $\gamma _Q$ defined in (3.3) where $Q\in \mathcal {Q}_{K,p}$ runs through integral binary quadratic form of discriminant $d_K $ and level p corresponding to A under the identification (3.2)). We will apply our results to understand the representation of the matrices $\gamma _Q$ in terms of the independent generators (4.5) (4.6), (4.7) and (4.8) coming from a special fundamental domain of $\Gamma _0(p)$ .

Proof. Let $B_{\mathrm {sp}}\subset H_1(Y_0(p),\mathbb {R})$ be a basis obtained from the side pairing transformations of $\mathcal {P}(p)$ . Note that $v_{E}(p)\in B_{\mathrm {sp}}$ and let $\omega _0\in B_{\mathrm {sp}}^{\ast }$ be characterized by $\langle v_{E}(p),\omega _0\rangle =1$ . Let ${B}$ be a basic basis of level p containing $v_{E}(p)$ . Then by expanding in this basis, we get by Theorem 7.2 that

(8.1) $$ \begin{align} \nonumber\left\langle \frac{\sum_{A\in H}[\mathcal{C}_{A}(p)]}{|\!|\sum_{A\in H}[\mathcal{C}_{A}(p)]|\!|} , \omega_0 \right\rangle&=-\left\langle \frac{v_{E}(p)}{|\!|v_{E}(p)|\!|}, \omega_0 \right\rangle+ \left\langle \frac{\sum_{A\in H}[\mathcal{C}_{A}(p)]}{|\!|\sum_{A\in H}[\mathcal{C}_{A}(p)]|\!|}+\frac{v_{E}(p)}{|\!|v_{E}(p)|\!|}, \omega_0 \right\rangle \\ \nonumber &=-1+\sum_{v\in B}\left\langle v, \omega_0 \right\rangle \left\langle \frac{\sum_{A\in H}[\mathcal{C}_{A}(p)]}{|\!|\sum_{A\in H}[\mathcal{C}_{A}(p)]|\!|}+\frac{v_{E}(p)}{|\!|v_{E}(p)|\!|}, v^\ast \right\rangle \\ &= -1+ O_\varepsilon\left( d_K^{-1/12+\varepsilon}p^{2+\varepsilon}\sum_{v\in B}|\langle v, \omega_0 \rangle |\right), \end{align} $$

where $|\!|\cdot |\!|=|\!|\cdot |\!|_{B,\infty }$ and $v^\ast \in B^\ast $ is characterized by $\langle v, v^\ast \rangle =1$ . For $v\in {B}-\{v_{E}(p)\}$ , let $\gamma _v\in \mathcal {S}(\mathcal {F}_{\mathrm {Zag}}(p))$ be such that

$$ \begin{align*} v=\{z,\gamma_v z\}\in H_1(Y_0(p),\mathbb{R}),\end{align*} $$

using the notation (3.6) (note that the lower left entry of $\gamma _v$ is equal to p). Now for $Y>0$ , consider a curve $c_Y: [0,1] \rightarrow \mathbb {H}$ connecting the three points

$$ \begin{align*} \gamma_v^{-1} \infty +iY ,\quad \gamma_v \infty +iY,\quad \gamma_v(\gamma_v^{-1} \infty +iY)=\gamma_v\infty+ip^{-2}Y^{-1}\end{align*} $$

by a (Euclidean) straight line. We assume that Y is large enough so that

$$ \begin{align*} \{x +iY:\min (\gamma_v\infty,\gamma_v^{-1} \infty) < x < \max(\gamma_v\infty,\gamma_v^{-1} \infty)\}\subset \mathcal{P}(p). \end{align*} $$

Observe that by Corollary 4.2, the quantity $|\langle v, \omega _0 \rangle |$ is bounded by the number of intersections between $c_Y$ and $\Gamma _0(p)$ -translates of $\{iy:y>0\}$ (which is a side of the special fundamental polygon $\mathcal {P}(p)$ ). Since $0<\gamma _v\infty , \gamma _v^{-1}\infty <1$ , there is no such intersection for the horizontal segment of $c_Y$ . Intersections with the vertical segment correspond to integers $a,b,c,d$ such that

(8.2) $$ \begin{align}p|c,\qquad ad-bc=1,\qquad\text{and}\quad \frac{a}{c}<\gamma_v\infty <\frac{b}{d}\text{ or }\frac{b}{d}<\gamma_v\infty <\frac{a}{c}.\end{align} $$

In particular, we have $\gamma _v\neq \frac {a}{c}$ , which implies

$$ \begin{align*} \left|\gamma_v\infty-\frac{a}{c}\right|\geq \frac{1}{|c|},\qquad \left|\frac{b}{d}-\frac{a}{c}\right|=\frac{1}{|cd|},\end{align*} $$

but this contradicts (8.2). This implies that $\langle v, \omega _0 \rangle =0$ for $v\in {B}-\{v_{E}(p)\}$ which by (8.1) yields the wanted error-term.

From the above, Corollary 1.7 follows in the case where K has wide class number one and narrow class number two by writing out explicitly the matrices $\gamma _Q$ associated to $Q\in \mathcal {Q}_{K,p}$ as in (3.3) and recalling that since K does not have a unit of norm $-1$ , the condition $\mathfrak {p}_1 \notin (\mathrm {Cl}_K^+)^2$ is equivalent to $\mathfrak {p}_1$ not having a generator of positive norm.

8.2 An application to modular forms

Recall the following definition mentioned in the introduction for a modular form $f\in \mathcal {M}_2(p)$ :

$$ \begin{align*} M_{f}:=\inf \{c\geq 0: |a_f(n)|\leq c \sigma_1(n) ,\, n\geq 1\}<\infty, \end{align*} $$

where $a_f(n)$ denotes the Fourier coefficients of f (at $\infty $ ) and $\sigma _1(n)=\sum _{d|n}d$ . We have the following nonvanishing result for cycle integrals of modular forms.

Corollary 8.2. Let p be prime and let $f\in \mathcal {M}_2(p)$ be a holomorphic modular form of weight $2$ and level p with constant Fourier coefficient equal to $a_f(0)=1$ . Consider a real quadratic field $K$ of discriminant $d_K$ with no unit of norm $-1$ such that $p$ splits in $K$ with $p\mathcal{O}_K=\mathfrak{p}_1\mathfrak{p}_2$ and $\mathfrak{p}_1 \notin (\mathrm{Cl}_K^+)^2$ . Then for $d_K \gg_\varepsilon (M_{f})^{12+\varepsilon} p^{48+\varepsilon}$ there is some $A\in (\mathrm{Cl}_K^+)^2$ such that

$$ \begin{align*}\langle [\mathcal{C}_{A}(p)], f(z)dz\rangle= \int_{\mathcal{C}_{A}(p)} f(z)dz\neq 0.\end{align*} $$

Proof. Let

$$ \begin{align*} f(z)dz= \omega_+ +i\omega_-\in H^1(Y_0(p),\mathbb{R})\oplus i H^1(X_0(p),\mathbb{R}) \end{align*} $$

be the cohomology class (with complex coefficients) associated to f. By expanding in a basic basis $B\subset H_1(Y_0(p),\mathbb {R})$ as in (8.1) and using Cauchy–Schwarz followed up by (7.4), we conclude that

$$ \begin{align*} \left|\left\langle \frac{\sum_{A\in H}[\mathcal{C}_{A}(p)]}{|\!|\sum_{A\in H}[\mathcal{C}_{A}(p)]|\!|_{B,\infty}}-v_{E}(p) , \omega_\pm \right\rangle\right|\ll_\varepsilon p^{3+\varepsilon} d_K^{-1/12+\varepsilon}\sup_{v\in {B}}|\langle v, \omega_\pm \rangle |. \end{align*} $$

Now by the assumptions on the Fourier expansion of f, we get

$$ \begin{align*} f(x+iy)&\ll 1+ M_{f}\sum_{n\geq 1}\sigma_1(n) e^{-2\pi yn}\ll_\varepsilon 1+M_{f}\int_1^\infty t^{1+\varepsilon} e^{-2\pi ty} \frac{dt}{t}\ll_\varepsilon \mathrm{max} (1, M_{f} y^{-1-\varepsilon}). \end{align*} $$

In order to bound the quantities $\langle v, \omega _\pm \rangle $ for $v\in {B}$ , we recall that we can write $v=\{\gamma z, z\}$ for $z\in \mathbb {H}$ (which we may choose freely) and some matrix as in Section 4.2. In particular, we observe that by construction, we have $a,b,c,d\ll p$ and $ c\in \{ 0,p\}$ . If $c=0$ , then clearly $ \langle v, \omega _\pm \rangle \in \{0,1\}$ . If $c=p$ , we make the following convenient choice $z=\frac {-d}{p}+\frac {i}{p}$ satisfying $\gamma z=\frac {a}{p}+\frac {i}{p}$ . This gives

$$ \begin{align*} \langle v, \omega_\pm \rangle= \int_{\tfrac{a}{p}}^{-d/p} \frac{f(x+ip^{-1})\pm \overline{f(x+ip^{-1})}}{2i^{(1\mp 1)/2}} dx\ll_\varepsilon M_{f} \frac{|a+d|}{p}p^{1+\varepsilon}\ll_\varepsilon M_{f} p^{1+\varepsilon},\end{align*} $$

which we plug into the above. Since $\langle v_{E}(p),\omega _+ \rangle =1$ and $\langle v_{E}(p),\omega _- \rangle =0$ , we conclude that $\langle \sum _{A\in H}[\mathcal {C}_{A}(p)] , f(z)dz \rangle $ is indeed nonvanishing for $d_K \gg _\varepsilon (M_{f})^{12+\varepsilon } p^{48+\varepsilon }$ , as wanted.

Now Corollary 1.8 follows from the above in the case of wide class number one and narrow class number two.