Related papers: Sign changes along geodesics of modular forms

We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of…

We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form $F$ of even integral weight on the Siegel modular group of arbitrary genus $ g\geq 2 $.

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute…

In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for $GL(2)$ over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maa{\ss} forms. Our results generalise…

We study expansions of Drinfeld modular forms of rank \(r \geq 2\) along the boundary of moduli varieties. Product formulas for the discriminant forms \(\Delta_{\mathfrak{n}}\) are developed, which are analogous with Jacobi's formula for…

We study generating series encoding linking numbers between geodesics in arithmetic hyperbolic $3$-folds. We show that the series converge to functions on genus $2$ Siegel space and that certain explicit modifications have the…

We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove…

We prove a quantitative lower bound on the number of nodal domains of the real-analytic Eisenstein series. The main tool in the proof is a quantitative restricted QUE theorem where the support of the test function is allowed to shrink with…

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and…

For every positive integral level $k$ we study arithmetic properties of certain holomorphic modular forms associated to modular invariant spaces spanned by graded dimensions of $L_{\hat{sl_2}}(k \Lambda_0)$-modules. We found a necessary and…

In [B-G1] and [B-G2], Borisov and Gunnells constructed for each level (N > 1) and for each weight (k > 1) a modular symbol with values in $Sk(\Gamma_1(N))$ using products of Eisenstein series. In this paper we generalize this result by…

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As…

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple…

We prove that the modular symbols appropriately normalized and ordered have a Gaussian distribution for all cofinite subgroups of SL_2(R). We use spectral deformations to study the poles and the residues of Eisenstein series twisted by…

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably…

Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with…

We define Eisenstein series twisted by modular symbols on the group SL(n), generalizing a construction of the first author. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points…

We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known…

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the…

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and…