Related papers: A note on local formulae for the parity of Selmer …

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…

We derive a local formula for the parity of the $2^{\infty}$-Selmer rank of Jacobians of curves of genus $2$ or $3$ with a $K$-rational $2$-torsion point. We give an explicit example to show how this local formula gives rank parity…

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This…

In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus \eta \oplus \tau $: W(E/K,1\oplus \eta \oplus \tau)=(-1)^{< 1\oplus\eta \oplus \tau, X_{p}(E/L)>}, where $E$ is…

For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group…

We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the…

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a…

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…

The present paper illustrates the utility of Brauer relations, Galois covers of curves and the theory of regulator constants in the context of studying isogenies between Jacobians and their relevance to the parity conjecture. This framework…

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is…

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rank of…

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness…

The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the…

We study the variation of Selmer ranks of Jacobians of twists of hyperelliptic curves and superelliptic curves. We find sufficient conditions for such curves to have infinitely many twists whose Jacobians have Selmer ranks equal to $r$, for…

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is…

Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians…

Let $A$ be an abelian variety defined over a number field $K$ and let $A^{\vee}$ be the dual abelian variety. For an odd prime $p$, we consider two Selmer groups attached to $A[p]$ and relate the orders of these groups along with those of…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra…

For an elliptic curve with complex multiplication over a number field, the $p^{\infty}$--Selmer rank is even for all $p$. \v{C}esnavi\v{c}ius proved this using the fact that $E$ admits a $p$-isogeny whenever $p$ splits in the complex…