Nonvanishing modulo ℓof Fourier coefficients of Jacobi forms

Markus Schwagenscheidt
2016 Research in Number Theory  
Let φ = r 2 ≤4mn c(n, r)q n ζ r be a Jacobi form of weight k (with k > 2 if φ is not a cusp form) and index m with integral algebraic coefficients which is an eigenfunction of all Hecke operators T p , (p, m) = 1, and which has at least one nonvanishing coefficient c(n * , r * ) with r * prime to m. We prove that for almost all primes there are infinitely many fundamental discriminants D = r 2 − 4mn < 0 prime to m with ν (c(n, r)) = 0, where ν denotes a continuation of the -adic valuation on Q
more » ... dic valuation on Q to an algebraic closure. As applications we show indivisibility results for special values of Dirichlet L-series and for the central critical values of twisted L-functions of even weight newforms.
doi:10.1007/s40993-015-0035-1 fatcat:biudds3sird63jx7ffcbulk6m4