Department of Mathematics,
University of California San Diego
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Math 209: Number Theory Seminar
A. Raghuram
Fordham University
Congruences and the special values of L-functions
Abstract:
There is an idea in number theory that if two objects are congruent modulo a prime p, then the congruence can also be seen for the special values of L functions attached to the objects. Here is a context explicating this idea: Suppose f and f' are holomorphic cuspidal eigenforms of weight k and level N, and suppose f is congruent to f' modulo p; suppose g is another cuspidal eigenform of weight l; if the difference k - l is large then the Rankin-Selberg L function L(s, f x g) has enough critical points; same for L(s, f' x g); one expects then that there is a congruence modulo p between the algebraic parts of L(m, f x g) and L(m, f' x g) for any critical point m. In this talk, after elaborating on this idea, I will describe the results of some computational experiments where one sees such congruences for ratios of critical values for Rankin-Selberg L-functions. Towards the end of my talk, time-permitting, I will sketch a framework involving Eisenstein cohomology for GL(4) over Q which will permit us to prove such congruences. This is joint work with my student P. Narayanan.
February 5, 2025
4:00 PM
APM 7321 and online (see https://www.math.ucsd.edu/~nts
Research Areas
Number Theory****************************