Superlacunary cusp forms

Families of Cusp Forms

1. Families of cusp forms and the local equidistribution problem 2 2. Families of L-functions 6 3. Analogy with sieve 8 4. Examples of local equidistribution 8 5. Families of cusp forms according to Sarnak’s letter 11 6. Families of L-functions according to Conrey, Farmer, Keating, Rubinstein and Snaith 13 7. Direct consequences of strong local equidistribution 14 8. Quantitative local equidist...

Primitive Cusp Forms

For example, let f11A denote the (unique) level 11 weight 2 cusp form, then both f11A(z) and f11A(2z) are level 22 cusp forms. Similarly, both f14A(z) and f14A(2z) are level 28 cusp forms, and both f15A(z) and f15A(2z) are level 30 cusp forms. None of these are “new” at N = 22, 28 or 30 since they arise from lower levels. Define S k (N) to be the vector space of weight k primitive cusp forms (o...

Fourier Coefficients of Hilbert Cusp Forms Associated with Mixed Hilbert Cusp Forms

We express the Fourier coefficients of the Hilbert cusp form Lhf associated with mixed Hilbert cusp forms f and h in terms of the Fourier coefficients of a certain periodic function determined by f and h. We also obtain an expression of each Fourier coefficient of Lhf as an infinite series involving the Fourier coefficients of f and h.

Mellin Transforms of Mixed Cusp Forms

We define generalized Mellin transforms of mixed cusp forms, show its convergence, and prove that the function obtained by such a Mellin transform of a mixed cusp form satisfies a certain functional equation. We also prove that a mixed cusp form can be identified with a holomorphic form of the highest degree on an elliptic variety.

On the Coeecients of Cusp Forms