Related papers: On Periods and $L$-functions for $\mathbf{GL}_4 \t…

Let $F$ be a totally real field and let $E/F$ be a CM quadratic extension. We construct a $p$-adic $L$-function attached to Hida families for the group ${\rm GL}_{2/F}\times {\rm Res}_{E/F}{\rm GL}_{1}$. It is characterised by an exact…

In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is \'etale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our…

Let $\mathbf{F}$ be a number field and $\mathfrak{q},\mathfrak{l}$ two coprime integral ideals with $\mathfrak{q}$ squarefree and $\pi_1,\pi_2$ two fixed unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional…

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and…

Let $\pi_1,\pi_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some finite extension $F/\mathbf{Q}_p$ of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on…

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll…

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,\pi,Sym^2\otimes\chi)$ of an irreducible cuspidal automorphic representation $\pi$ of $\GL_r(\A)$ twisted by a Hecke character $\chi$. In particular, we will show…

We prove the Bloch--Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in Hida families for…

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral…

These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to…

The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new…

We prove that one hundred percent of the closed geodesic periods of a Hecke--Maa{\ss} cusp form for the modular group are non-vanishing when ordered by length. We present applications to the non-vanishing of central values of…

We establish a connection between motivic cohomology classes over the Siegel threefold and special values of the degree four $L$-function of some cuspidal automorphic representations of $\mathrm{GSp}(4)$. Our computation relies on our…

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions…

We recall the relation between Zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of Zeta…

Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_N(\mathbb{A}_F)$ of symplectic type. When $\pi$ is spherical at all primes $\mathfrak{p}|p$, we construct a $p$-adic $L$-function…

The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal…

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $\Pi$ be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal…

This article is a companion to several works of the author and others on the arithmetic of automorphic forms for GSp(4), and their associated L-functions and Galois representations. These works require, at various points, an input from…