DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 163-172

W. Duke

Title: Bounds for Arithmetic Multiplicities

This paper will describe some recent applications of techniques giving non-trivial upper bounds for multiplicities in certain arithmetic instances. These applications include estimates for dimensions of spaces of cusp forms of weight one, multiplicities of number fields with a given degree and discriminant, and the number of elliptic curves over the rationals whose reductions have the same number of points for a few small primes. The techniques share a common strategy, which combines approximate orthogonality with rigidity properties of arithmetic Fourier coefficients.

1991 Mathematics Subject Classification: 11F,11N

Keywords and Phrases: modular forms of weight one, number fields, class groups, elliptic curves

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