The pro-Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched ' -covers.' In many cases, the -power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for -covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.

Abstract. Let K be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over K whose ℓ-power torsion fields are arithmetically constrained for some rational prime ℓ. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-ℓ fundamental group of P 1 − {0, 1, ∞}.Under GRH, we demonstrate the set of classes is finite for any fixed K and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of K/Q and the dimension of abelian varieties are not too large, through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of ℓ) are uniform in the degree of the extension K/Q.

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.

Let K be a number field, and S a finite set of places in K containing all infinite places. We present an implementation for solving the S-unit equation x + y = 1, x, y ∈ O × K,S in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets S. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant where 2 is totally ramified. In addition, we improve bounds for S-unit equation solutions that would allow enumeration of a certain class of genus 2 curves, and use the implementation to find all solutions to some cubic Ramanujan-Nagell equations.1 See also the recent translation [16] by Fuchs.

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.We consider the following point of view for Dirichlet's result. The F p -rational mor-p into two sets, those that are the image of an F p -rational point of the domain (that is, the quadratic residues) and those that are not (the non-residues). The character χ φ := (·/p) is now precisely the natural identification of the cokernel of φ with {±1} which makes the following sequence exact:Note that it is the properties of φ, not the underlying algebraic group G m , which allow this construction. This paper demonstrates that an analogous procedure, arising from a different morphism of algebraic groups, yields new character sums with similar arithmetic properties.