New results on Ramanujan's function
Non-abelian -adic -functions and Eisenstein series of unitary groups – The CM method
In this work we prove various cases of the so-called “torsion congruences” between abelian -adic -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of variables and we obtain more explicit results in the...
Nonanalytic automorphic integrals on the Hecke groups
1. Introduction. Since its genesis over a century ago in work of Jacobi, Riemann, Poincar ́e and Klein [Ja29, Ri53, Le64], the theory of automorphic forms has burgeoned from a branch of analytic number theory into an industry all its own. Natural extensions of the theory are to integrals [Ei57, Kn94a, KS96, Sh94], thereby encompassing Hurwitz’s prototype, the analytic weight 2 Eisenstein series [Hu81], and to nonanalytic forms [He59, Ma64, Sel56, ER74, Fr85]. A generalization in both directions...
Nonvanishing of automorphic L-functions at special points
At some special points, we establish a nonvanishing result for automorphic L-functions associated to the even Maass cusp forms in short intervals by using the mollification technique.
Non-vanishing of class group -functions at the central point
Let be an imaginary quadratic field, and denote by its class number. It is shown that there is an absolute constant such that for sufficiently large at least of the distinct -functions do not vanish at the central point .
Non-vanishing of L-functions attached to automorphic representations of GL (") over Q.
Nonvanishing of L-functions for GL(2).
Non-vanishing of L-functions of 2 x 2 unitary groups.
Non-vanishing of modular L-functions on a disc
Non-vanishing of -th derivatives of twisted elliptic -functions in the critical point
Let be a modular elliptic curve over
Nonvanishing theorems for L-functions of modular forms and their derivatives.
Numerical calculation of twisted adjoint -values attached to modular forms.