Shimura correspondence for level $p^2$ and the central values of $L$-series II

Details Shimura correspondence for level $p^2$ and the central values of $L$-series II Shimura correspondence for level $p^2$ and the central values of $L$-series II

2010-05-27

arxiv.org/abs/1005.5135v1

Mathematics

Number Theory

45 pages

Scientific paper

Given a weight 2 and level $p^2$ modular form $f$,a method to construct two weight 3/2 modular forms of level $4p^2$ (with a character) mapping to $f$ under the Shimura correspondence was developed in [PT09b].By a Theorem of Waldspurger (see [Wal81]), the Fourier coefficients of the weight 3/2 modular forms are related to the central values of the twists of $f$ by imaginary quadratic fields. The main result of this work is a proof of the formula conjectured in [PT09b] Conjecture 2 for these central values. A generalized Gross method (as in [PT09b]) with orders $\Ot$ of reduced discriminant $p^2$ is not useful to construct weight 3/2 modular forms mapping to $f$ under the Shimura correspondence. Nevertheless, in this paper we show that if $D<0$ is the discriminant of the ring of integers of an imaginary quadratic field, then the set of special points of discriminant $D$ for $\Ot$ can be splitted in subsets, indexed by bilateral $\Ot$-ideals of norm $p$. The number of special points in each subset is a Fourier coefficient of a certain theta series for a suborder of discriminant $p^3$, which coincides with the definition given in [PT09b]}. This fact plays crucial in the proof of the Main Theorem.

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