Non-vanishing of symmetric square $L$-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 .

Averages of symmetric square L-functions, and applications

We exhibit a spectral identity involving L(s,Symf) for f on SL2. Perhaps contrary to expectations, we do not treat L(s,Symf) directly as a GL3 object. Rather, we take advantage of the coincidence that the standard L-function for SL2 is the symmetric square for a cuspform on GL2 restricted to SL2. [1] As SL2 = Sp2, the integral identities obtained from Sp2n × Sp2n ⊂ Sp4n produce standard L-funct...

Symmetric Square L-Functions and Shafarevich-Tate Groups

CONTENTS We use Zagier's method to compute the critical values of the 1. Introduction symmetric square L-functions of six cuspidal eigenforms of level 2. Calculating the Critical Values one with rational coefficients. According to the Bloch-Kato 3. Tables of Results conjecture, certain large primes dividing these critical values 4. An Observation must be the orders of elements in generalised Sh...

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