Let f be an L
 2-normalized weight zero Hecke-Maas cusp form of square-free level N, character χ and Laplacian eigenvalue λ≥1/4. It is shown that ‖f‖∞≪
 λ
 N
 −1/37, from which the hybrid bound ‖f‖∞≪λ
 1/4(N
 λ)−δ
 (for some δ>0) is derived. The first bound holds also for f=y
 k/2
 F where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al. The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers.

/pdf/transcendental-values-of-certain-eichler-integrals-17w5893sxv.pdf

Transcendental values of certain Eichler integrals

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φλ
 ℂ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If $\{\phi_{j_{k}}\}$
 is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g$
 , where $[Z_{\phi_{j_k}^{\mathbb{C}}}]$
 is the current of integration over the complex zeros and where $\overline{\partial}$
 is with respect to the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel.

/pdf/complex-zeros-of-real-ergodic-eigenfunctions-64k8q07l3h.pdf

Complex zeros of real ergodic eigenfunctions

On a family of arithmetic hyperbolic 3-manifolds of square-free level, we prove an upper bound for the sup-norm of Hecke–Maas cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of Diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.

/pdf/bounds-for-eigenforms-on-arithmetic-hyperbolic-3-manifolds-1gew00cbfi.pdf

Bounds for eigenforms on arithmetic hyperbolic $3$-manifolds

We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.

On the asymptotic distribution of zeros of modular forms

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On the asymptotic distribution of zeros of modular forms