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#### Some analytic aspects of automorphic forms on GL(2) of minimal type

Let $\pi$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $\phi$ be an $L^2$-normalized automorphic form in the space of $\pi$. ...

#### Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve ...

#### Hybrid sup-norm bounds for Maass newforms of powerful level

(Mathematical Sciences Publishers, 2017-07-12)

Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$, character $\chi$ and Laplace eigenvalue $\lambda$. Let $N_1$ denote the smallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer ...

#### On the order of vanishing of newforms at cusps

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of $E$ by $X_0(N)$. Our ...

#### Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

(American Mathematical Society, 2014-01-01)

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the ...

#### Local and global Maass relations

(Springer Verlag, 2017-10-01)

We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special ...

#### A relation between multiplicity one and Bocherer's conjecture

We show that a weak form of the generalized Bocherer's conjecture implies multiplicity one for Siegel cusp forms of degree 2.

#### On sup-norms of cusp forms of powerful level

(European Mathematical Society, 2017-11-01)

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue \lambda. It is shown that |f|_\infty <<_{\lambda, \epsilon} N^{-1/12 + \epsilon} for any \epsilon>0. The exponent is further improved ...