arXiv subjects

Matthew Bisatt

Publications and source records attributed to Matthew Bisatt.

11 recordsLinked to original sources

arXiv2026

Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.

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arXiv2023

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.

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arXiv2021

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.

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arXiv2021

Clusters, inertia, and root numbers

In a recent paper of Dokchitser--Dokchitser--Maistret--Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first étale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.

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arXiv2021

Root number of the Jacobian of $y^2=x^p+a$

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.

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arXiv2020

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 $\mathbb{Q}$.

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arXiv2019

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.

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arXiv2019

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.

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arXiv2017

On the Birch--Swinnerton-Dyer conjecture and Schur indices

For every odd prime $p$, we exhibit families of irreducible Artin representations $τ$ with the property that for every elliptic curve $E$ the order of the zero of the twisted $L$-function $L(E,τ,s)$ at $s\!=\!1$ must be a multiple~of~$p$. Analogously, the multiplicity of $τ$ in the Selmer group of $E$ must also be divisible by $p$. We give further examples where $τ$ can moreover be twisted by any character that factors through the $p$-cyclotomic extension, and examples where the $L$-functions are those of twists of certain Hilbert modular forms by Dirichlet charaters. These results are conjectural, and rely on a standard generalisation of the Birch--Swinnerton-Dyer conjecture. Our main tool is the theory of Schur indices from representation theory.

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arXiv2017

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.

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arXiv2015

Explicit open image theorems for abelian varieties with trivial endomorphism ring

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all sufficiently large primes $\ell$, we provide a semi-effective bound $\ell_0(A/K)$ such that $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all primes $\ell > \ell_0(A/K)$. The bound is given in terms of the Faltings height of $A$ and of the cardinality of the residue field at a suitably generic place of $K$. We also describe an algorithmic approach to obtain better bounds for abelian threefolds over $\mathbb{Q}$.

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