A generalization of Serre’s Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over  $p$ . This characterization of the weights, which is formulated using $p$ -adic Hodge theory, is known under mild technical hypotheses if $p>2$ . In this paper we give, under the assumption that $p$ is unramified in $F$ , a conjectural alternative description for the set of weights. Our approach is to use the Artin–Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using $p$ -adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$-power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$, and define $G_{s}$ (resp. $G_{\infty }$) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp. $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$). In this paper, we study $G_{s}$-equivariantness properties of $G_{\infty }$-equivariant homomorphisms between torsion crystalline representations.

Let $F$ be a totally real number field, ${\wp}$ a place of $F$ above $p$ . Let ${\it\rho}$ be a $2$ -dimensional $p$ -adic representation of $\text{Gal}(\overline{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus ${\it\rho}$ is associated to Hilbert eigenforms). When the restriction ${\it\rho}_{{\wp}}:={\it\rho}|_{D_{{\wp}}}$ at the decomposition group of ${\wp}$ is semistable noncrystalline, one can associate to ${\it\rho}_{{\wp}}$ the so-called Fontaine–Mazur ${\mathcal{L}}$ -invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these ${\mathcal{L}}$ -invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil’s results [Breuil, Astérisque, 331 (2010), 65–115] in the $\text{GL}_{2}/\mathbb{Q}$ -case.

We propose an algorithm to verify the $p$-part of the class number for a number field $K$, provided $K$ is totally real and an abelian extension of the rational field $\mathbb{Q}$, and $p$ is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristics, including the wild case, was recently formulated by the author in the case where the given finite group acts linearly on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as nonlinear actions on possibly singular spaces and nonpermutation representations of nonabelian groups.

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$-adic hypergeometric function for any odd prime $d$.

We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$-function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$-adically analytic phases, which can be naturally seen as a $p$-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are ‘optimal’ in the following sense: they minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher-degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sen’s theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of the iterative residue.

Arithmetic duality theorems over a local field $k$ are delicate to prove if $\text{char}\,k>0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^{n}(k,G)$ for commutative finite type $k$-group schemes $G$. These ‘Čech topologies’, defined using Čech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^{n}(k,G)$: in the key case when $n=1$, identify $H^{1}(k,G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B}G$ and invoke Moret-Bailly’s general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these ‘classifying stack topologies’ enjoy the properties expected from the Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.

Given a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the p-adic analytic continuation around O(T) of functions defined by limits of sequences of restricted power series with p-adic integer coefficients. We also investigate applications to generating elements for or for some classes of closed subfields of .

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

Suppose that $G$ is a connected reductive algebraic group defined over $\mathbf{R}$, $G(\mathbf{R})$ is its group of real points, ${\it\theta}$ is an automorphism of $G$, and ${\it\omega}$ is a quasicharacter of $G(\mathbf{R})$. Kottwitz and Shelstad defined endoscopic data associated to $(G,{\it\theta},{\it\omega})$, and conjectured a matching of orbital integrals between functions on $G(\mathbf{R})$ and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on $G$ and ${\it\theta}$.

We construct $p$-adic families of Klingen–Eisenstein series and $L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$. When $r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$-adic $L$-function.

This paper proves two results on the field of rationality $\mathbb{Q}({\it\pi})$ for an automorphic representation ${\it\pi}$, which is the subfield of $\mathbb{C}$ fixed under the subgroup of $\text{Aut}(\mathbb{C})$ stabilizing the isomorphism class of the finite part of ${\it\pi}$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations ${\it\pi}$ such that ${\it\pi}$ is unramified away from a fixed finite set of places, ${\it\pi}_{\infty }$ has a fixed infinitesimal character, and $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ is bounded. The second main result is that for classical groups, $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions.

Let $\mathfrak{R}$ be a complete discrete valuation ring, $S=\mathfrak{R}[[u]]$ and $d$ a positive integer. The aim of this paper is to explain how to efficiently compute usual operations such as sum and intersection of sub-$S$-modules of $S^d$. As $S$ is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the $p$-adic and $u$-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with $S$-modules has applications in Iwasawa theory and $p$-adic Hodge theory. It is used in particular in Caruso and Lubicz (Preprint, 2013, arXiv:1309.4194) to compute the semi-simplified modulo $p$ of a semi-stable representation.

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

Let $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.