This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about a problem on certain holomorphic differential operators on the space of Jacobi forms.

In the first part we will briefly discuss how the statistics of signs of newforms determine them (work of Matomaki-Soundararajan-Kowalski) and then introduce certain ‘Linnik-type’ problems (the original problem was concerning the size of the smallest prime in an arithmetic progression in terms of the modulus) which ask for the size of the first negative eigenvalue (in terms of the analytic conductor) of various types modular forms, which has seen a lot of recent interest. Also specifically we will discuss the problem in the context of Yoshida lifts (a certain subspace of the Siegel modular forms), where in the thesis, we have improved upon the previously known result on this topic significantly. We will prove that the smallest $n$ with $\lambda(n)<0$ satisfy $n < Q_{F}^{1/2-2\theta+\epsilon}$, where $Q_{F}$ is the analytic conductor of a Yoshida lift $F$ and $0<\theta <1/4$ is some constant. The crucial point is establishing a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level.

We will focus on a similar question concerning the first negative Fourier coefficient of a Hilbert newform. If ${C(\mathfrak{m})}_{\mathfrak{m}}$ denotes the Fourier coefficients of a Hilbert newform $f$, then we show that the smallest among the norms of ideals $\mathfrak{m}$ such that $ C(\mathfrak{m})<0$, is bounded by $Q_{f}^{9/20+\epsilon}$ when the weight vector of $f$ is even and $Q_{f}^{1/2+\epsilon}$ otherwise. This improves the previously known result on this problem significantly. Here we would show how to use certain ‘good’ Hecke relations among the eigenvalues and some standard tools from analytic number theory to achieve our goal.

Finally we would talk about the statistical distribution of the signs of the Fourier coefficients of a Hilbert newform and essentially prove that asymptotically, half of them are positive and half negative. This was a breakthrough result of Matomaki-Radziwill for elliptic modular forms, and our results are inspired by those. The proof hinges on establishing some of their machinery of averages multiplicative functions to the number field setting.

In the second part of the talk we will introduce Jacobi forms and certain differential operators indexed by $\{D_{v}\}_{0}^{2m}$ that maps the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the direct sum of the differential operators $D_{v}$ for $v={1,2,…,2m}$ maps $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. Inspired by certain conjectures of Hashimoto on theta series, S. Bocherer raised the question whether any of the differential operators be removed from that map while preserving the injectivity. In the case of even weights S. Das and B. Ramakrishnan show that it is possible to remove the last operator. In the talk we will discuss the case of the odd weights and prove a similar result. The crucial step (and the main difference from the even weight case) in the proof is to establish that a certain tuple of congruent theta series is a vector valued modular form and finding the automorphy of the Wronskian of this tuple of theta series.

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Last updated: 24 Mar 2023