Power series expansions of modular forms and their interpolation properties

Details Power series expansions of modular forms and their interpolation properties Power series expansions of modular forms and their interpolation properties

2004-06-19

arxiv.org/abs/math/0406388v2

Mathematics

Number Theory

45 pages. In this new version of the paper the restriction on the weight in the expansion principle in the quaternionic case h

Scientific paper

Let x be a CM point on a modular or Shimura curve and p a prime of good reduction, split in the CM field K. We define an expansion of an holomorphic modular form f in the p-adic neighborhood of x and show that the expansion coefficients give information on the p-adic ring of definition of f. Also, we show that letting x vary in its Galois orbit, the expansions coefficients allow to construct a p-adic measure whose moments squared are essentially the values at the centre of symmetry of L-functions of the automorphic representation attached to f based-changed to K and twisted by a suitable family of Grossencharakters for K.

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