Related papers: Elliptic Gauss Sums and Hecke L-values at s=1

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $\psi_{\D}$ of K with…

Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $\pi_2$. Let $H\subset[\![…

Let P and Q be relatively prime integers greater than 1, and f a real valued discretely supported function on a finite dimensional real vector space V. We prove that if f_{P}(x)=f(Px)-f(x) and f_{Q}(x)=f(Qx)-f(x) are both \Lambda-periodic…

Let $A_{\pi}(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\rm SL_3(\mathbb{Z})$ and $ \omega(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum…

In the study of the arithmetic structure of elliptic modular groups which are the fundamental groups of compactified modular curves, these truncated group algebras and their direct sums are considered to construct elliptic modular motives.…

We prove that the Galois pseudo-representation valued in the mod $p^n$ cuspidal Hecke algebra for GL(2) over a totally real number field $F$, of parallel weight $1$ and level prime to $p$, is unramified at any place above $p$. The same is…

This work builds on earlier results. We define universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable…

Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind…

For any elliptic curve $E$ over $k\subset \Bbb R$ with $E({\Bbb C})={\Bbb C}^\times/q^{\Bbb Z}$, $q=e^{2\pi iz}, \Im(z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\Bbb C})$, of the function $D_0(z) = \Im(z/(1-z))$. Let…

Consider an elliptic curve $\mathcal{C}$ with coefficients in $\mathbb{K}$ with $[\mathbb{K}:\mathbb{Q}]<\infty$ and $\delta \in \mathcal{C}(\mathbb{K})$ a non torsion point. We consider an elliptic difference equation $\sum_{i=0}^l a_i(p)…

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights…

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 present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the…

For any number field F, call a cusp form \pi on GL(2)/F {\it special icosahedral}, or just s-icosahedral for short, if \pi is not solvable polyhedral, and for a suitable "conjugate" cusp form \pi' on GL(2)/F, sym^3(\pi) is isomorphic to…

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo…

Let $E$ be an elliptic curve over $Q$, and $\tau$ an Artin representation over $Q$ that factors through the non-abelian extension $Q(\sqrt[p^n]{m},\mu_{p^n})/Q$, where $p$ is an odd prime and $n,m$ are positive integers. We show that…

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form, and let $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be prime. In this note we revisit the subconvexity problem addressed in `The circle method and bounds…

By telescoping method, Sun gave some hypergeometric series whose sums are related to $\pi$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper…