Related papers: Periods and Reciprocity I

The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of…

In this paper, we establish some reciprocity formulas for certain generalized Hardy-Berndt sums by using the Fourier series technique and some properties of the periodic zeta function and the Lerch zeta function. It turns out that one of…

A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable…

We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences…

We propose a duality in the relative Langlands program. This duality pairs a Hamiltonian space for a group $G$ with a Hamiltonian space under its dual group $\check{G}$, and recovers at a numerical level the relationship between a period on…

In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical…

Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability measure preserving equivalence relations $\mathcal{R}$. We show that…

In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple…

Given a curve $C$ over a number field $K$ equipped with the action of a finite group $G$ by $K$-automorphisms, one obtains a factorisation of $L(C,s)$ into a product of $L$-functions of `motivic pieces of curves' associated to irreducible…

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem…

The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and \varpi=2.62205... is the real period of the elliptic…

On the example of the quantized spinor field, interacting with arbitrary external electromagnetic field, the commutation function is studied. It is shown that a proper time representation is available in any dimensions. Using it, all the…

Marvin Knopp developed the theory of automorphic integrals, which generalize automorphic forms; each automorphic integral has an additional period function in its automorphic relation. The period functions satisfy relations that arise from…

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…

We describe isomorphism patterns in the $p$-primary part of the Farrell cohomology ring ${\hat{H}}^*(Sp(p-1,Z[1/n]),Z)$ for any odd prime $p$ and suitable integers $0\neq n\in Z$, where $Sp(p-1,Z[1/n])$ denotes the group of symplectic…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

We give a new construction of a $p$-adic $L$-function $\mathcal{L}(f,\Xi)$, for $f$ a holomorphic newform and $\Xi$ an anticyclotomic family of Hecke characters of $\mathbb{Q}(\sqrt{-d})$. The construction uses Ichino's triple product…

The Lorentz reciprocity principle is a fundamental concept that governs light propagation in any optically linear medium in zero magnetic field. Here, we demonstrate experimentally a novel mechanism of reciprocity breaking in nonlinear…

We compute the first moment of cubic Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ evaluated at any $s$ inside the critical strip. The first moment for $s<\frac{1}{2}$ is particularly interesting, and we show there is a phase transition…

It is known that the algebraic \deRham cohomology group $\hDR{i}(X_0/\Q)$ of a nonsingular variety $X_0/\Q$ has the same rank as the rational singular cohomology group $\h^i\sing(\Xh;\Q)$ of the complex manifold $\Xh$ associated to the base…