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Hecke operators in half-integral weight. (English. French summary) Zbl 1350.11060

The seminal work of Shimura in establishing the theory of modular forms of half integral weight, and in particular the (Hecke)-correspondence with modular forms of integral weight, is by now well known. In this paper, the author works out some aspects of the theory of Hecke operators for half-integral weight modular forms and the Shimura correspondence. More precisely the author determines:
the Hecke relations that determine \(T_{p^{2 \ell}}\) as a polynomial in \(T_{p^2}\) uniformly for all primes \(p\);
the explicit equivariance properties the \(t\)-th Shimura map with the Hecke operators on half-integral and integral weight respectively (extending W. Kohnen’s work [J. Reine Angew. Math. 333, 32–72 (1982; Zbl 0475.10025)]);
an upper bound on the number of generators of the algebra of Hecke operators as a subset of the endomorphisms of the space of half-integral weight cusp form in question, which is an explicit function of the index of the congruence subgroup and the weight. The result for integral weight can be found in the book by W. Stein [Modular forms, a computational approach. Providence, RI: AMS (2007; Zbl 1110.11015)].

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
11F25 Hecke-Petersson operators, differential operators (one variable)

References:

[1] F. Diamond and J. Shurman, A First Course in Modular Forms, GTM 228, Springer-Verlag, 2005. · Zbl 1062.11022
[2] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32-72. · Zbl 0475.10025
[3] T. Miyake, Modular Forms, Springer-Verlag, 1989. · Zbl 0701.11014
[4] S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Mathematical Journal 56 (1975), 147-161. · Zbl 0303.10027
[5] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and \(q\)-Series, CBMS 102, American Mathematical Society, 2004. · Zbl 1119.11026
[6] G. Shimura, On Modular Forms of Half Integral Weight, Annals of Mathematics, Second Series, Vol. 97, 3 (1973), pp. 440-481. · Zbl 0266.10022
[7] W. Stein, Modular forms, a computational approach, Graduate Studies in Mathematics 79, American Mathematical Society, 2007. · Zbl 1110.11015
[8] J. Sturm, On the Congruence of Modular Forms. Number theory (New York, 1984-1985), Lecture Notes in Math. 1240, Springer, Berlin, (1987), 275-280. · Zbl 0615.10035
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