Non-vanishing of derivatives of $GL(3) ×GL(2)$ $L$-functions

SPECIAL VALUES OF L-FUNCTIONS ON GSp4×GL2 AND THE NON-VANISHING OF SELMER GROUPS

In this paper we show how one can use an inner product formula of Heim giving the inner product of a pullback Eisenstein series from Sp10 to Sp2×Sp4×Sp4 with a newform on GL2 and a SaitoKurokawa lift to produce congruences between Saito-Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp4×GL2. The congruence is then used to produce nontrivial torsion...

Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip

We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ(f, 1/2) 6= 0 is ≥ pk +o(1) with pk > 0 and pk = 1/2 + O(k−2), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result ...

Non-vanishing Results of Special Values of L-functions

Here are the notes I am taking for Eric Urban’s ongoing course on non-vanishing results of special values of L-functions offered at Columbia University in Spring 2015 (MATH G6675: Topics in Number Theory). As the course progresses, these notes will be revised. I recommend that you visit my website from time to time for the most updated version. Due to my own lack of understanding of the materia...

Non-vanishing of Quadratic Twists of Modular L-functions

On Non-vanishing of Symmetric Square L-functions

We find a lower bound in terms of N for the number of newforms of weight k and level N whose symmetric square L-functions are non-vanishing at a fixed point s0 with 1 2 < Re(s0) < 1 or s0 = 1 2 .