Fourier Coefficients of Half - Integral Weight Modular Forms modulo `
نویسندگان
چکیده
S. Chowla conjectured that every prime p has the property that there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. Gauss’ genus theory guarantees the existence of infinitely many such fields when p = 2, and the work of Davenport and Heilbronn [D-H] suffices for the prime p = 3. In addition, the DavenportHeilbronn result demonstrates that a positive proportion of such fields have this property. Using an elementary argument based on the Kronecker recurrence relations, Hartung [Ha] proved that for any odd prime p there are indeed infinitely many such fields whose class numbers are not multiples of p. His argument has been employed in other similar studies [Ho1,Ho2, Ho-On]. It is easy to capture the flavor of the Kronecker relations: if r(n) denotes the number of representations of a positive integer n as a sum of three squares, then
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