71 citations · 395 across the 32 of their papers we have counts for
33 papers · 1 filter
Reduction Types of Genus 2 Curves
Edwina Aylward, Lilybelle Cowland Kellock, Vladimir Dokchitser +1
Tate produced a table for elliptic curves over local fields that beautifully summarises their arithmetic invariants in terms of the Kodaira type. We present an analogous set of tab…
A note on the parity conjecture and base change
Vladimir Dokchitser
The parity conjecture predicts that the parity of the rank of an abelian variety is determined by its global root number, that is by the sign in the conjectural functional equation…
Reduction of Plane Quartics and Cayley Octads
Raymond van Bommel, Jordan Docking, Vladimir Dokchitser +2
We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently…
Root numbers and parity phenomena
Lilybelle Cowland Kellock, Vladimir Dokchitser
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct th…
Parity of ranks of Jacobians of curves
Vladimir Dokchitser, Holly Green, Alexandros Konstantinou +1
We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks.…
A note on hyperelliptic curves with ordinary reduction over 2-adic fields
Vladimir Dokchitser, Adam Morgan
We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled us…
Character formula for Weil representations in terms of Frobenius traces
Tim Dokchitser, Vladimir Dokchitser
It is known that the etale cohomology of a potentially good abelian variety over a local field K is determined by its Euler factors over the extensions of K. We extend this to all…
A user's guide to the local arithmetic of hyperelliptic curves
Alex J. Best, L. Alexander Betts, Matthew Bisatt +8
A new approach has been recently developed to study the arithmetic of hyperelliptic curves over local fields of odd residue characteristic via combinatorial data associa…
Parity conjecture for abelian surfaces
Vladimir Dokchitser, Celine Maistret
Assuming finiteness of the Tate--Shafarevich group, we prove that the Birch--Swinnerton-Dyer conjecture correctly predicts the parity of the rank of semistable principally polarise…
On a BSD-type formula for L-values of Artin twists of elliptic curves
Vladimir Dokchitser, Robert Evans, Hanneke Wiersema
This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. W…
Tate module and bad reduction
Tim Dokchitser, Vladimir Dokchitser, Adam Morgan
Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable. This allows us…
Arithmetic of hyperelliptic curves over local fields
Tim Dokchitser, Vladimir Dokchitser, Céline Maistret +1
We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p…
On the Birch--Swinnerton-Dyer conjecture and Schur indices
Matthew Bisatt, Vladimir Dokchitser
For every odd prime , we exhibit families of irreducible Artin representations with the property that for every elliptic curve the order of the zero of the twisted -f…
Constructing hyperelliptic curves with surjective Galois representations
Samuele Anni, Vladimir Dokchitser
In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the gener…
Quotients of hyperelliptic curves and etale cohomology
Tim Dokchitser, Vladimir Dokchitser
We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first etale cohom…
l-Adic Representations and their Associated Invariants
Vladimir Dokchitser, Samuele Anni
These are notes from a 3-lecture course given by V. Dokchitser at the ICTP in Trieste, Italy, 1st--5th of September 2014, as part of a graduate summer school on "L-functions and mo…
Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
L. Alexander Betts, Vladimir Dokchitser
We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this t…
Growth of Sha in towers for isogenous curves
Tim Dokchitser, Vladimir Dokchitser
We study the growth of the Tate-Shafarevich and p-infinity Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth…
A remark on Tate's algorithm and Kodaira types
Tim Dokchitser, Vladimir Dokchitser
We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v(a_i)/i. As an appl…
Local invariants of isogenous elliptic curves
Tim Dokchitser, Vladimir Dokchitser
We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime deg…
Euler factors determine local Weil representations
Tim Dokchitser, Vladimir Dokchitser
We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illus…
Surjectivity of mod 2^n representations of elliptic curves
Tim Dokchitser, Vladimir Dokchitser
For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional pri…
Identifying Frobenius elements in Galois groups
Tim Dokchitser, Vladimir Dokchitser
We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler's criterion. It is a part of the…
A note on the Mordell-Weil rank modulo n
Tim Dokchitser, Vladimir Dokchitser
Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers,…