Hershy Kisilevsky

On the Vanishing of Twisted L-Functions of Elliptic Curves

Let E be an elliptic curve over Q with L-function LE (S). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted Lfunctions LE (l, X), as X runs over the Dirichlet characters of order 3 (cubic twists). We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values LE (l, X) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of LE (S).

Vanishing and non-vanishing Dirichlet twists of L-functions of elliptic curves

Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,\chi) of the twisted L-function as \chi ranges over Dirichlet characters of given order.

On the minimal ramification problem for ℓ -groups

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups.

BIG BIASES AMONGST PRODUCTS OF TWO PRIMES

We show that substantially more than a quarter of the odd integers of the form

Ranks of Elliptic Curves and Random Matrix Theory: Vanishing of L -functions of elliptic curves over number fields

pq

Indépendance Linéaire sur Q de logarithmes P-adiques de nombres algébriques et rang P-adique du groupe des unités d'un corps de nombres

up to

Rank determines semi-stable conductor

x

Galois representations with non surjective traces.

, with

Ranks of elliptic curves in cubic extensions

p,q

Decomposition types in minimally tamely ramified extensions of \(\mathbb {Q}\)

both prime, satisfy