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Sign changes of Fourier coefficients of cusp forms of half-integral weight over split and inert primes in quadratic number fields. (English) Zbl 1460.11063

In the paper under review, the authors investigate sign changes of Fourier coefficients of half-integral weight cusp forms over some certain number fields. In a fixed square class \(t\mathbb{Z}^2\), they investigate the sign changes in the \(tp^2\)-th coefficient as \(p\) runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. They show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level. The main tool for the proofs is the Shimura lift, where they can “carry” some results from integral weight case as well as analytic number theory tools. They use some quadratic forms as a source of modular forms.

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
11F30 Fourier coefficients of automorphic forms
11N69 Distribution of integers in special residue classes
11R11 Quadratic extensions
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
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References:

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