Vladimir Dokchitser

On the Birch-Swinnerton-Dyer quotients modulo squares

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 of LII, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of III(E/K(E[2]))[6 ∞ ] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p ∞ -Selmer rank and the analytic rank agree.

Regulator constants and the parity conjecture

AbstractThe p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of

Root numbers of non-abelian twists of elliptic curves

\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})

Parity of ranks for elliptic curves with a cyclic isogeny

. We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

Elliptic curves with all quadratic twists of positive rank

We study the global root number of the complex

Computations in non-commutative Iwasawa theory

L

Root numbers and parity of ranks of elliptic curves

-function of twists of elliptic curves over

Self-duality of Selmer groups

\mathbb{Q}

Local invariants of isogenous elliptic curves

by real Artin representations. We obtain examples of elliptic curves over

Root numbers of elliptic curves in residue characteristic 2

\mathbb{Q}