Tim 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.

Computing Special Values of Motivic L-Functions

We present an algorithm to compute values L(s) and derivatives L (k) (S) of L-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any L-serieswhose Γ-factor is of the form AS with d arbitrary and complex λ j , not necessarily distinct. The algorithm relies on the known (or conjectural) functional equation for L(s).

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

Parity of ranks for elliptic curves with a cyclic isogeny

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

Elliptic curves with all quadratic twists of positive rank

. 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.

Computations in non-commutative Iwasawa theory

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p⩾3 and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.

Root numbers and parity of ranks of elliptic curves

Imagine you had an elliptic curve E/K with everywhere good reduction, defined over a number field K that has no real and an odd number r of complex places. Then the global root number w(E/K) is (−1)r = −1, and it becomes (−1)2r = +1 over every quadratic extension of K. As the root number is the sign in the (conjectural) functional equation for the L-function of E, the Birch–Swinnerton-Dyer conjecture predicts that the Mordell–Weil rank of E goes up in every quadratic extension of K. Equivalently, every quadratic twist of E/K has positive rank, a behaviour that does not occur over Q (and would contradict Goldfeld’s “1/2 average rank” conjecture). These curves do exist (1). For example, the elliptic curve over Q E : y = x + 5 4x 2 − 2x− 7 (121C1) has discriminant −114 and acquires everywhere good reduction over any cubic extension of Q which is totally ramified at 11. So one may take K = Q(ζ3, 3 √ 11m) or K = Q( 6 √ −11m) for any positive m, coprime to 11. (Those who prefer abelian extensions can take E = 1849C1 and K to be the degree 6 field inside Q(ζ43).) It is even easier to construct curves all of whose quadratic twists have root number +1. For example,

Self-duality of Selmer groups

We study special values of

Surjectivity of mod 2 n representations of elliptic curves

L

Numerical verification of Beilinson's conjecture for K 2 of hyperelliptic curves

-functions of elliptic curves over