Twist-minimal trace formula for holomorphic cusp forms

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg.

A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

Modular iterated integrals associated with cusp forms

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.

ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.