Lefschetz formulae for p-adic groups

Hida families and p-adic triple product L-functions

We construct the three-variable p-adic triple product L-functions attached to Hida families of ellptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of p-adic L-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced p-adic triple product L-functions into a product of anticyclotomic p-adic L-functions for modular forms. By this factorization, we giv…

Hodge symmetry for rigid varieties via log hard Lefschetz

Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for lo…

p-adic l-functions and sums of powers

The purpose of this paper is to give the explicit formulae of p-adic l-functions and sums of powers which are related to Euler numbers.

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\…

Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

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 that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional z…

A p-adic analogue of a formula of Ramanujan

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.

Scaling group flow and Lefschetz trace formula for laminated spaces with $p-$adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function $\hatζ(s)$ (resp. the zeta function $ζ_Y(s)$ of a smooth projective curve $Y$ over a finite field $\mathbb{F}_q$, $q=p^f$)) one could possibly associate a foliated Riemannian laminated space $(S_{\mathbb{Q}}, \mathcal{F}, g, ϕ^t)$ (resp. $(S_Y, \mathcal{F}, g, ϕ^t)$) endowed with an action of a flow $ϕ^t$ whose primitive compact orbits should correspond to the primes of $\mathbb{Q}$ (resp. $Y$). The existence of such a foliated space and flow $ϕ^t$ is still unknown except when $Y$ is an ellipt…

具有指定判别式和Galois闭包群的强分歧四次扩张计数

Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group

本文针对2进数域上具有指定判别式赋值与Galois闭包群的完全分歧四次扩张建立计数公式，并据此改进Serre质量公式，为数域算术统计提供工具。

酉群上通过Bushnell-Kutzko类型的$p$-adic zeta积分

$p$-adic zeta integrals on unitary groups via Bushnell-Kutzko types

本文通过Bushnell-Kutzko类型计算酉群加倍方法中的$p$-adic zeta积分，给出显式公式，并扩展至非平凡超尖支撑情形。

Néron-Severi groups under specialization

André used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André's theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere p-adically dense. Our proof uses the ``p-adic Lefschetz (1,1) theorem'' of Berthelot and Ogus, combined with an anal…

通过 pro-p 对偶群理论的算术域理论

Arithmetic field theory via pro-p duality groups

本文通过pro-p对偶群理论定义适合算术拓扑的cobordism范畴，并利用Frobenius代数分类拓扑量子场论，推导出局部p进域上Galois扩展的计数公式。

超尖点表示的刻画与极正则元素

Characterization of supercuspidal representations and very regular elements

本文证明p进群正则超尖点表示可由极正则元素上的特征标值唯一刻画，并据此描述Kaletha的L-包，同时推广到一般及幂幺超尖点表示。

Noncommutative geometry and motives: the thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of $L$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative …