Fundamental Fourier coefficients of Siegel modular forms – Soumya Das

Soumya Das
Indian Institute of Science, Bangalore

Date(s) : 23/01/2020   iCal
14 h 00 min - 15 h 00 min

Understanding the Fourier coefficients of Siegel modular forms which are « fundamantal » i.e. indexed by matrices with fundamental discriminant has many applications. Generalising A. Saha’s work on this topic, we show that if F is a non-zero (possibly non-cuspidal) vector-valued level one Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. As an application of a variant of our result and utilising the work of A. Pollack, this implies an unconditional proof of Andrianov’s conjecture for genus 3, i.e., shows the expected standard analytic properties of the spinor L-function of a holomorphic cuspidal Siegel Hecke eigenform of degree 3 and level one. If time permits, we can discuss mod p versions of this, and related results.
These are joint works with Siegfried Boecherer.

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