Explicit root numbers of abelian varieties

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist.Comment: Corrected formulation of Theorem 7.4; to appear in Transactions of the AM

Frobenius elements in Galois representations with SL_n image

Suppose we have a elliptic curve over a number field whose mod $l$ representation has image isomorphic to $SL_2(\mathbb{F}_l)$. We present a method to determine Frobenius elements of the associated Galois group which incorporates the linear structure available. We are able to distinguish $SL_n(\mathbb{F}_l)$-conjugacy from $GL_n(\mathbb{F}_l)$-conjugacy; this can be thought of as being analogous to a result which distinguishes $A_n$-conjugacy from $S_n$-conjugacy when the Galois group is considered as a permutation group.Comment: 5 page

Tame torsion, the tame inverse Galois problem, and endomorphisms

Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$.Comment: v2: Expanded to include application to tame inverse Galois problem. To appear in Manuscripta Mathematic

Conductors of twisted Weil--Deligne representations

We study the behaviour of conductors of L-functions associated to certain Weil--Deligne representations under twisting. For each global field K we prove: a sharp upper bound for the conductor of the Rankin--Selberg L-function associated to a pair of abelian varieties; and a formula for the conductor of the twisted L-function associate to an abelian variety A/K and sufficiently ramified Artin character over K. Our methods apply to more general Weil--Deligne representations than just those associated to abelian varieties, and we give similar results in this setting.Comment: 12 pages, comments welcome

Tame torsion and the tame inverse Galois problem

Fix a positive integer $g$ and a squarefree integer $m$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $m$ representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields

A user's guide to the local arithmetic of hyperelliptic curves

A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.Comment: Minor changes. To appear in the Bulletin of the London Mathematical Societ

Root number of the Jacobian of y2=xp+ay^2=x^p+a

Let $C/\mathbb{Q}$ be a hyperelliptic curve with an affine model of the form $y^2=x^p+a$. We explicitly determine the root number of the Jacobian of $C$, with particular focus on the local root number at $p$ where $C$ has wild ramification.Comment: 7 pages. Comments welcome