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Fourier coefficients of half-integral weight modular forms modulo $$\ell$$. (English) Zbl 0907.11017
Ann. Math. (2) 147, No. 2, 453-470 (1998); corrigendum ibid. 148, No. 1, 361 (1998).
The authors study the indivisibility of the (algebraic) Fourier coefficients for a class of half-integral weight eigenforms. One of the motivations is provided by the connection between the Fourier coefficients of half-integral weight modular forms and class numbers of imaginary quadratic fields (Example 3). The theorem has a number of other arithmetic consequences. These concern the central critical $$L$$-values of quadratic twists of even-weight newforms and connections of such values to orders of the Tate-Shafarevich groups of elliptic curves (Corollaries 1-5). The main ingredients in the proof of the theorem are: the proper “bookkeeping” of newforms which are related via twisting and Galois conjugation, and the Chebotarev density theorem.

##### MSC:
 11F37 Forms of half-integer weight; nonholomorphic modular forms 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F30 Fourier coefficients of automorphic forms 11F33 Congruences for modular and $$p$$-adic modular forms
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