Non-commutative p-adic L-functions for supersingular primes

Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.Comment: 13 pages; some minor corrections; to appear in International Journal of Number Theor

Factoriality of Bozejko-Speicher von Neumann algebras

We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II_1 factor which does not have the property Gamma. Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for type II_1 and type III case.Comment: 8 pages. NEW IN VERSION 2: considered factors do not have the property Gamm

A Class of Incomplete Character Sums

Using $\ell$-adic cohomology of tensor inductions of lisse $\overline{\mathbb Q}_\ell$-sheaves, we study a class of incomplete character sums.Comment: Following the suggestion of the referee, we use tensor induction to study a class of incomplete character sums. Originally we use transfer, which is a special case of tensor induction, and which only works for rank one sheaves. The paper is to appear in Quarterly Journal of Mathematic

Asymptotic growth of the signed Tate-Shafarevich groups for supersingular abelian varieties

Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. In this paper, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Buyukboduk--Lei, we define the multi-signed Mordell-Weil groups for supersingular abelian varieties, provide an explicit structure of the dual of these groups as an Iwasawa module and prove a control theorem. Furthermore, we define the multi-signed Tate-Shafarevich groups and, by computing their Kobayashi rank, we provide an asymptotic growth formula along the cyclotomic tower of $\mathbb{Q}$