Sign changes of coefficients of half integral weight modular forms

For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\neq 0$, we show that for all but finitely many primes $p$ the sequence $(a(tp^{2m}))_{m}$ has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms $f$ which are not necessarily Hecke eigenforms

On Fourier coefficients of modular forms of half-integral weight (Analytic, geometric and pp-adic aspects of automorphic forms and LL-functions)

In my talk, I reported about recent joint work with S. Gun in which a new proof was given that for any non-zero cusp form of half-integral weight in the plus space of level 4 (not necessarily a Hecke eigenform) there exist infinitely many fundamental discriminants D such that the Fourier coefficients evaluated at IDI are non-zero. The proof uses a new type of Dirichlet series built out of the squarefree coefficients of a form of half-integral weight. The result was first proved (usuing totally different methods) by Saha. By adapting the resonance method due to K. Soundararajan one can in fact demonstrate that such coefficients must take quite large values. The above mentioned results and their proofs can be found in [S. Gun, W. Kohnen and K. Soundararajan, Large Fourier coefficients of modular forms of half-integral weight, http://arxiv.org/abs/2994.14450]

Locally harmonic Maass forms and the kernel of the Shintani lift

In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and Zagier of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight cusp forms. Although our functions share many properties in common with harmonic weak Maass forms, they also have some properties which strikingly contrast those exhibited by harmonic weak Maass forms. As a first application of the new theory developed in this paper, one obtains a new perspective on the fact that the even periods of Zagier's cusp forms are rational as an easy corollary