Related papers: On the order of vanishing of the cyclotomic p-adic…

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these…

Let $G$ be a countable group and $k$ a positive integer, we show that the $L^2$-Betti numbers of the group $G$ vanish up to degree $k$ provided that there is some infinite index subgroup $H$ with finite $k$th $L^2$-Betti number containing a…

Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this…

For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…

Let $f$ be a normalized, ordinary newform of weight $\ge 2$. For each prime $\mathfrak{p}$ of $F=\mathbb{Q}(a_n)_{n\in \mathbb{N}}$, there is an associated $\mathfrak{p}$-adic $L$-function $\mathcal{L}_\mathfrak{p}(f)\in \Lambda \otimes…

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption…

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were…

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…

We prove that the complete L-functions of classical holomorphic newforms have infinitely many simple zeros.

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each…

Fix a prime $p\geq5$, an integer $N\geq1$ relatively prime to $p$, and an irreducible residual global Galois representation $\bar{r}: Gal_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F}_p)$. In this paper, we utilize ghost series to study $p$-adic…

We prove that if a function $f \in L^p({\Bbb {R}} ^d)$ has vanishing periodizations then $\|f\|_{p'} \lesssim \|f\|_{p}$, provided $1 \le p < \frac {2d}{d + 2}$ and dimension $d \ge 3$.

We study, on average over f, zeros of the L-functions of primitive weight two forms of level q (fixed). We prove, on the one hand, density theorems for the zeros (similar to the results of Bombieri, Jutila, Motohashi, Selberg in the case of…

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 C be a non-empty finite set, and Gamma a subgroup of the symmetric group S(C). Given a bijection f:A cross C to B cross C, the problem of Gamma-equivariant division is to find a quotient bijection h:A to B respecting whatever symmetries…

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…

In this paper, we propose and explore a new connection in the study of $p$-adic $L$-functions and eigenvarieties. We use it to prove results on the geometry of the cuspidal eigenvariety for $\mathrm{GL}_{2n}$ over a totally real number…

We prove a new criterion for the solvability of the finite groups, depending on the function $\psi_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability…

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special…

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic…