Let f(x) and g(x) be polynomials in $\mathbb F_{2}[x]$ with ${\rm deg}\text{ } f=n$. It is shown that for $n\gg 1$, there is an $g_{1}(x)\in \mathbb F_{2}[x]$ with ${\rm deg}\text{ } g_{1}\leqslant \max\{{\rm deg}\text{ } g, 6.7\log n\}$ and $g(x)-g_{1}(x)$ having $ \lt 6.7\log n$ terms such that $\gcd(f(x), g_{1}(x))=1$. As an application, it is established using a result of Dubickas and Sha that given $f(x)\in \mathbb F_{2}[x]$ of degree $n\geqslant 1$, there is a separable $g(x)\in 2[x]$ with ${\rm deg}\text{ } g= {\rm deg}\text{ } f$ and satisfying that $f(x)-g(x)$ has $\leqslant 6.7\log n$ terms. As a simple consequence, the latter result holds in $\mathbb Z[x]$ after replacing ‘number of terms’ by the L1-norm of a polynomial and $6.7\log n$ by $6.8\log n$. This improves the bound $(\log n)^{\log 4 +\operatorname{\varepsilon}}$ obtained by Filaseta and Moy.

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Kobayashi–Ochiai proved that the set of dominant maps from a fixed variety to a fixed variety of general type is finite. We prove the natural extension of their finiteness theorem to Campana’s orbifold pairs.

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.

In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$ ) polynomials $F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$ , we show that the number of $f\in \mathbf{F}_{q}[t]$ of degree $n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$ such that all $F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$ , are irreducible is

$$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$

$N_{i}=n\deg _{x}F_{i}$

$F_{i}(t,f)$

$\deg f=n$

$\unicode[STIX]{x1D707}_{i}$

$F_{i}$

$\overline{\mathbf{F}}_{q}$

$\mathbf{F}_{q}(t)$

$F_{1},\ldots ,F_{m}$

$x$

$n$

$C$

$$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{m}F_{i}(t,x)=0 & & \displaystyle \nonumber\end{eqnarray}$$

$(\sum _{i=1}^{m}\deg F_{i})^{2}/2$

$\deg$

$t,x$

$$\begin{eqnarray}\displaystyle p=\text{char}\,\mathbf{F}_{q}>\max _{1\leqslant i\leqslant m}N_{i}, & & \displaystyle \nonumber\end{eqnarray}$$

$N_{i}$

$F_{i}(t,f)$

$\deg f=n$

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.

If is an integral model of a smooth curve X over a global field k, there is a localization sequence comparing the K-theory of and X. We show that K1 () injects into K1(X) rationally, by showing that the previous boundary map in the localization sequence is rationally a surjection, for X of “GL2 type” and k of positive characteristic not 2. Examples are given to show that the relative G1 term can have large rank. Examples of such curves include non-isotrivial elliptic curves, Drinfeld modular curves, and the moduli of -elliptic sheaves of rank 2.

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.

Let E/k be a function field over an infinite field of constants. Assume that E/k(x) is a separable extension of degree greater than one such that there exists a place of degree one of k(x) ramified in E. Let K/k be a function field. We prove that there exist infinitely many nonisomorphic separable extensions L/K such that [L:K]=[E:k(x)] and AutkL=AutKL≅Autk(x)E.

Silverman has discussed the problem of bounding the Mordell–Weil ranks of elliptic curves over towers of function fields (J. Algebraic Geom. 9 (2000), 301–308; J. reine. angew. Math. 577 (2004), 153–169). We first prove generalizations of the theorems of Silverman by a different method, allowing non-abelian Galois groups and removing the dependence on Tate's conjectures. We then prove some theorems about the growth of Mordell–Weil ranks in towers of function fields whose Galois groups are $p$-adic Lie groups; a natural question is whether the Mordell–Weil rank is bounded in such a tower. We give some Galois-theoretic criteria which guarantee that certain curves ${\mathcal{E}}/{\mathbb{Q}}(t)$ have finite Mordell–Weil rank over ${\mathbb{C}}(t^{p^{-\infty}})$, and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.

We study deformation rings of an n-dimensional representation $\overline{\rho}$, defined over a finite field of characteristic $\ell$, of the arithmetic fundamental group $\pi_1(X)$, where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p ($ \neq \ell$). When $\overline{\rho}$ has large image, we are able to show that the resulting rings are finite flat over $\mathbf{Z}_\ell$. The proof principally uses a Galois-theoretic lifting result of the authors in Part I of this two-part work, a lifting result for cuspidal mod $\ell$ forms of Ogilvie, Taylor–Wiles systems and the result of Lafforgue. This implies a conjecture of de Jong for representations of $\pi_1(X)$ with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation $\overline{\rho}$ as their reduction. A proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory. The methods are different.

We formulate and prove a Sato–Tate equidistribution law for Drinfeld modules.

The Hermitian function field H= K(x,y) is defined by the equation yq+y=xq +1 (q being a power of the characteristic of K). Over K=${\mirrored F}$q2 it is a maximal function field; i.e. the number N(H) of ${\mirrored F}$q2-rational places attains the Hasse–Weil upper boundN(H)=q2+1+2g(H)·q. All subfields K[subnE ] E⊆H are also maximal. In this paper we construct a large number of nonrational subfields E⊆H, by considering the fixed fields H$^{\mathcal G}$ under certain groups ${\mathcal G}$ of automorphisms of H/K. Thus we obtain many integers g[ges ]0 that occur as the genus of some maximal function field over ${\mirrored F}$q2.

We extend a result of Vojta on height inequalities for algebraic points on curves over function fields to include the case of positive characteristic. The main tool used is the Kodaira—Spencer map and destabilizing flags for vector bundles on curves.

Let k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.