Vladimir Dokchitser
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View article: Parity of ranks of Jacobians of curves
Parity of ranks of Jacobians of curves Open
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. Under the Shafarevich–Tate conjecture, we give an expressio…
View article: Reduction of Plane Quartics and Cayley Octads
Reduction of Plane Quartics and Cayley Octads Open
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 give the stable reduction type amongst the 42 possible ty…
View article: A note on the parity conjecture and base change
A note on the parity conjecture and base change Open
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 of its L-function. Assuming the Shafarevich-Tate conjectur…
View article: Root numbers and parity phenomena
Root numbers and parity phenomena Open
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 them, and is responsible for a wide range of unexplained arit…
View article: Reduction of Plane Quartics and Cayley Octads
Reduction of Plane Quartics and Cayley Octads Open
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 give the stable reduction type amongst the 42 possible typ…
View article: On the parity conjecture for abelian surfaces
On the parity conjecture for abelian surfaces Open
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 polarised abelian surfaces. If the surface in question is the Jacobia…
View article: Root numbers and parity phenomena
Root numbers and parity phenomena Open
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 them, and is responsible for a wide range of unexplained arit…
View article: Parity of ranks of Jacobians of curves
Parity of ranks of Jacobians of curves Open
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. Under the Shafarevich--Tate conjecture, we give an expressi…
View article: A user's guide to the local arithmetic of hyperelliptic curves
A user's guide to the local arithmetic of hyperelliptic curves Open
A new approach has been recently developed to study the arithmetic of hyperelliptic curves y 2 = f ( x ) $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of f $f$ . Since its introdu…
View article: A note on hyperelliptic curves with ordinary reduction over 2-adic fields
A note on hyperelliptic curves with ordinary reduction over 2-adic fields Open
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 using `cluster pictures', similarly to the case of odd residu…
View article: Arithmetic of hyperelliptic curves over local fields
Arithmetic of hyperelliptic curves over local fields Open
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 -adic distances between the roots of f ( x ), and s…
View article: Character formula for Weil representations in terms of Frobenius traces
Character formula for Weil representations in terms of Frobenius traces Open
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 abelian varieties, show that it is enough to take extension…
View article: Character formula for conjugacy classes in a coset
Character formula for conjugacy classes in a coset Open
Let $G$ be a finite group and $N
View article: On a BSD-type formula for <i>L</i>-values of Artin twists of elliptic curves
On a BSD-type formula for <i>L</i>-values of Artin twists of elliptic curves Open
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. We translate expected properties of L -functions into purel…
View article: A user's guide to the local arithmetic of hyperelliptic curves
A user's guide to the local arithmetic of hyperelliptic curves Open
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 p…
View article: Parity conjecture for abelian surfaces
Parity conjecture for abelian surfaces Open
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 polarised abelian surfaces. If the surface in question is the Jacob…
View article: Constructing hyperelliptic curves with surjective Galois representations
Constructing hyperelliptic curves with surjective Galois representations Open
In this paper we show how to explicitly write down equations of hyperelliptic curves over such that for all odd primes the image of the mod Galois representation is the general symplectic group. The proof relies on understanding the act…
View article: On a BSD-type formula for L-values of Artin twists of elliptic curves
On a BSD-type formula for L-values of Artin twists of elliptic curves Open
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. We translate expected properties of L-functions into purely …
View article: Semistable types of hyperelliptic curves
Semistable types of hyperelliptic curves Open
In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining eq…
View article: Issue Information
Issue Information Open
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View article: Arithmetic of hyperelliptic curves over local fields
Arithmetic of hyperelliptic curves over local fields Open
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-adic distances between the roots of f(x), and show that th…
View article: Variation of Tamagawa numbers of semistable abelian varieties in field extensions
Variation of Tamagawa numbers of semistable abelian varieties in field extensions Open
We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions…
View article: Arithmetic of hyperelliptic curves over local fields
Arithmetic of hyperelliptic curves over local fields Open
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-adic distances between the roots of f(x), and show that this…