Related papers: Note on $p$-adic Local Functional Equation

Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over…

We define and study an $\ell$-adic Fourier transform for a relative version of Banach-Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic…

Let $F$ be a totally real field and let $E/F$ be a CM quadratic extension. We construct a $p$-adic $L$-function attached to Hida families for the group ${\rm GL}_{2/F}\times {\rm Res}_{E/F}{\rm GL}_{1}$. It is characterised by an exact…

The purpose of this article is to newly define the $p$-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special…

This work brings Mellin transforms into the realm of motivic integration. The new, larger class of motivic functions is stable under motivic Mellin and Fourier transforms, with general Fubini results and change of variables formulas. It…

This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any continuous representation of a…

What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much.…

Let $G$ be a locally compact group and $1\leq p<\infty$. Based on some important earlier works, in this paper the concept of $L_p^T-$function is introduced. Then the structure of the space $L^{T}_p(G)$, which is consisting of all…

In this paper we propose a construction of $p$-adic Euler $\ell$-function using Kubota-Leopoldt's approach and Washington's one. We also compute the derivative of $p$-adic Euler $\ell$-function at $s=0$ and the values of $p$-adic Euler…

We give a version of Ax-Katz's $p$-adic congruences and Moreno-Moreno's $p$-weight refinement that holds over any finite commutative ring of prime characteristic. We deduce this from a purely group-theoretic result that gives a lower bound…

We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the \'etale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy…

The spectral zeta functions have been found many application in several branches of modern physics, including the quantum field theory, the string theory and the cosmology. In this paper, we shall consider the spectral zeta functions and…

We develop a $p$-adic version of the so-called Grothendieck-Teichm\"uller theory (which studies $Gal(\bar{\bf Q}/{\bf Q})$ by means of its action on profinite braid groups or mapping class groups). For every place $v$ of $\bar{\bf Q}$, we…

Let $K$ be a $p$-adic local field. In this work we study a special kind of $p$-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms.…

In this, the eighth article in my Derived Langlands series, I describe the construction of a 2-variable L-function for two representations of general linear groups of a $p$-adic local field. Due to extenuating health circumstances, many of…

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in…

We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-etale site, which makes all constructions…

We axiomatize a class of existentially closed exponential fields equipped with an $E$-derivation. We apply our results to the field of real numbers endowed with $exp(x)$ the classical exponential function defined by its power series…

We build a one-variable $p$-adic $L$-function attached to two Hida families of ordinary $p$-stabilised newforms $\mathbf{f}$, $\mathbf{g}$, interpolating the algebraic part of the central values of the complex $L$-series $L(f \otimes…

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a…