p-adic congruences for cycle integrals associated to elliptic curves with complex multiplication
\textit{N. Koblitz} [Math. Ann. 274, 199-220 (1986; Zbl 0571.10030)] made several conjectures according to which congruences modulo certain rational primes p between cusp forms associated to powers of a Hecke character of an imaginary quadratic field should descend via the Shimura correspondence to congruences between cusp forms of half-integral weight. He was able to verify his conjecture numerically in several cases using the fact that according to \textit{J.-L. Waldspurger}'s theory [J. Math. Pures Appl., IX. Sér. 60, 375-484 (1981; Zbl 0431.10015)] the central values of the L-series of the powers of the Hecke character twisted with quadratic characters are essentially equal to the squares of the Fourier coefficients of forms of half-integral weight. In the present paper we give a conjecture which is similar to that of N. Koblitz (loc. cit.), but which is based on a generalization of Waldspurger's theory according to which the product of any two Fourier coefficients of a Jacobi-Hecke eigenform can be expressed in terms of certain cycle integrals of a corresponding form of integral weight [\textit{B. Gross}, the author and \textit{D. Zagier}, Heegner points and derivatives of L-series. II, Math. Ann. (to appear)]. We prove our conjecture in the case that the discriminant of the quadratic forms occurring in the cycle integrals is a non-zero square modulo p. In fact, in this case using results of \textit{N. M. Katz} [Ann. Math., II. Ser. 104, 459-571 (1976; Zbl 0354.14007)] we prove a p-adic congruence already for individual cycle integrals.
scientific article; zbMATH DE number 4000037
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