Related papers: On Rankin-Selberg integral structures and Euler sy…

In this paper, we study the special values of Rankin-Selberg L-functions as a continuation of [LLS24]. Utilizing the modular symbol approach, we prove the rationality and period relations for some critical values of Rankin-Selberg…

Recently we propose a class of infinite-dimensional integral representations of classical gl(n+1)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean…

We study arithmetic properties of certain quaternionic periods of Hilbert modular forms arising from base change of elliptic modular forms. These periods which we call the distinguished periods are closely related to the notion of…

We give a geometric derivation of Schottky's equation in genus four for the period matrices of Riemann surfaces among all period matrices. The equation arises naturally from the singularity theory of the Gauss map on the theta divisor, and…

We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the…

We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and…

We investigate properties of the Euler system associated to certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field $E$, constructed by Loeffler-Skinner-Zerbes. By adapting…

We propose a geometric interpretation of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program. We show that the geometric Langlands conjecture for an irreducible unramified local system $E$ of…

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell\neq p$ be a prime number, and $\mathrm{W}_F$ the Weil group of $F$. We classify the indecomposable $\mathrm{W}_F$-semisimple Deligne…

Let G be the unramified unitary group in three variables defined over a p-adic field of odd residual characteristic. In this paper, we establish a theory of newforms for the Rankin-Selberg integral for G introduced by Gelbart and…

We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindel\"of hypothesis on average, and recovers individually the…

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show that the invariant is rewritten as a sum…

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of…

We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit…

Form factor sequences of an integrable QFT can be defined axiomatically as solutions of a system of recursive functional equations, known as ``form factor equations''. We show that their solution can be replaced with the study of the…

Let $F$ be a $p$-adic field and $\pi$ be an irreducible smooth representation of $\textrm{Sp}_{2n}(F)$. In this paper, we show that if $\pi$ and $\pi^\kappa$ are both generic for a common generic character of the maximal unipotent of a…

We study integrable models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(2|1)$ or $\mathfrak{gl}(1|2)$ superalgebras. We obtain explicit determinant representations for form factors of the monodromy matrix…

This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is…

In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham…