13 research outputs found
Frobenius elements in Galois representations with SL image
No full textSuppose we have a elliptic curve over a number field whose mod representation has image isomorphic to . Wepresent a method to determine Frobenius elements of the associated Galoisgroup which incorporates the linear structure available. We are able todistinguish -conjugacy from -conjugacy; this can be thought of as being analogous to a result whichdistinguishes -conjugacy from -conjugacy when the Galois groupis considered as a permutation group
Tame torsion, the tame inverse Galois problem, and endomorphisms
No full textFix a positive integer g and rational prime p. We prove the existence of a genus g curve C/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 Q
Explicit root numbers of abelian varieties
No full textThe 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.</p
Clusters, inertia, and root numbers
No full textIn 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
Conductors of twisted Weil--Deligne representations
No full textWe 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.Comment: Major restructuring: strengthened main result on Rankin--Selberg
L-functions and removed results on character twists. Comments welcome
Conductors of twisted Weil–Deligne representations
No full textWe study the behavior 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 L(A⊠B,s) where A,B/K are abelian varieties
Tame torsion and the tame inverse Galois problem
No full textFix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve C/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
On the Birch-Swinnerton-Dyer conjecture and Schur indices
No full textFor 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
