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Seminar

Analytic Number Theory Seminar: Trace Inequalities and Non-vanishing of L-functions April 27, 2017 (02:00 PM PDT - 03:00 PM PDT)
Parent Program:
Location: MSRI: Simons Auditorium
Speaker(s) Dinakar Ramakrishnan (California Institute of Technology)
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Let f be a holomorphic newform of prime level N, weight 2 and trivial character, for example one associated to a elliptic curve E over Q. For any imaginary quadratic field K of discriminant -D in which N is inert, and an ideal class character \chi of K, one is led to the ubiquitous Rankin-Selberg L-function L(s, f x g_\chi), where g_\chi is the modular form of level D associated to \chi by Hecke. It is well known that the central value L(1/2, f x g_\chi) is non-zero for "many" (D, \chi), which is a consequence of equidistribution of special points. The object of this talk is to indicate how to derive a strengthening of this, namely that if we fix an f as above together with a finite number of even Dirichlet characters \eta_1, ..., \eta_r, then one can find many (D,\chi) for which one has the simultaneous non-vanishing of L(1/2, (f.\eta_j) x g_\chi) for all j. The additional ingredient used here is an inequality of traces for tori relative to a non-abelian twist.

 

The talk will hopefully be accessible to a variety of mathematicians.

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