How far is an extension of $p$-adic fields from having a normal integral basis?

Galois cohomology of a $p$-adic field via $(Φ,Γ)$-modules in the imperfect residue field case

For a $p$-adic local field $K$ with perfect residue field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(Φ,Γ)$-modules. We shall generalize his work to the imperfect residue field (the residue field has a finite $p$-basis) case.

Rationality and p-adic properties of reduced forms of half-integral weight

In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.

On l-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations ρ_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (Γ) on a noncongruence subgroup Γ\subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, ρ_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where ρ_{\ell}^{\vee} is the dual of ρ_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^- \} of S_3 (Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer…

关于不完全剩余域情形下的p进Simpson与Riemann-Hilbert对应

On $p$-adic Simpson and Riemann-Hilbert correspondences in the imperfect residue field case

本文在剩余域具有有限p基的混合特征完全离散赋值域上，为Galois群的表示构造p进Simpson与Riemann-Hilbert对应，并由此证明Hodge-Tate与de Rham刚性定理，推广了Morita的结果。

p-adic boundary values

We study in detail certain natural continuous representations of G = GL(n,K) in locally convex vector spaces over a locally compact, non-archimedean field K of characteristic zero. We construct boundary value maps, or integral transforms, between subquotients of the dual of a ``holomorphic'' representation coming from a p-adic symmetric space, and ``principal series'' representations constructed from locally analytic functions on G. We characterize the image of each of our integral transforms as a space of functions on $G$ having certain transformation properties and satisfying a system of par…

Integral structures on $p$-adic Fourier theory

In this article, we give an explicit construction of the $p$-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of $K$-locally analytic functions on the ring of integers $\mathcal{O}_K$ for any finite extension $K$ of $\mathbb{Q}_p$, generalizing the basis constructed by Amice for locally analytic functions on $\mathbb{Z}_p$. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by K…

About the choice of a basis in Kedlaya's algorithm

Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of t…

Integral $p$-adic Hodge theory in the imperfect residue field case

Let $K$ be a mixed characteristic complete discrete valuation field with residue field admitting a finite $p$-basis, and let $G_K$ be the Galois group. We first classify semi-stable representations of $G_K$ by weakly admissible filtered $(φ,N)$-modules with connections. We then construct a fully faithful functor from the category of \emph{integral} semi-stable representations of $G_K$ to the category of Breuil-Kisin $G_K$-modules. Using the integral theory, we classify $p$-divisible groups over the ring of integers of $K$ by minuscule Breuil-Kisin modules with connections.

Probabilistic Galois Theory over $P$-adic Fields

We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$. In particular, we show that the Galois groups are cyclic with p…

The density of integral quadratic forms having a $k$-dimensional totally isotropic subspace

We investigate the probability that a random quadratic form in ${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result computes these local probabilities for quadratic forms over the $p$-adics. The formulae we obtain are rational functions in $p$ invariant upon substituting $p \mapsto 1/p$.

Affine quiver Schur algebras and $p$-adic $GL_n$

In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a bypr…

关于 Narasimhan 与 Seshadri 定理的 p 进版本

On a p-adic version of Narasimhan and Seshadri's theorem

本文研究 Faltings 关于 p 进 Simpson 对应像的提问，证明在好约化及 p 足够大的条件下，潜在强半稳定约化等价于强半稳定约化，从而给出否定回答。

Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.

Schneider--Teitelbaum 多项式在 $p$-adic Fourier 理论中的范数

Norms of Schneider--Teitelbaum Polynomials in $p$-adic Fourier Theory

本文在 $p$-adic Fourier 理论中，利用 Ardakov-Berger 赋值公式，明确计算了 Schneider-Teitelbaum 多项式在局部解析函数空间上的范数，并给出了 Bannai-Kobayashi 基的显式规范化。

Appearance of the Kashiwara-Saito singularity in the representation theory of $p$-adic $\mathop{GL}_{16}$

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packe…

一个 $p$-adic 第二基本定理

A $p$-adic Second Main Theorem

本文在非阿基米德完备代数闭域上，为解析映射与横截相交且非全为超平面的超曲面族建立了第二基本定理型估计，推广了 Levin 关于高次超曲面的结果。

Formal group exponentials and Galois modules in Lubin-Tate extensions

Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponent…

On the period-index problem in light of the section conjecture

Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $π_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points.

Generalized Kuga-Satake theory and good reduction properties of Galois representations

In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over…

Efficient computation of Cantor's division polynomials of hyperelliptic curves over finite fields

Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision. Consequently, after having possibly lifted the problem in the $p$-adics, we derive fast algorithms for computing explicitly Cantor's division polynomials of hyperelliptic curves defined over finite fields.

On the zeta function of a projective complete intersection

We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.

The $θ$-adics（θ进数）

The $θ$-adics

本文引入阿基米德型局部康托多元域$\u200b\mathcal{O}_θ$作为实二次域无穷位的$p$进数类似物，证明其为Marty多元域且与一维准晶体完备化相关，旨在发展更精细的类域论。

关于整形式值的商的p进密度

On $p$-adic denseness of quotients of values of integral forms

研究整形式值的商集在\(p\)进数中的密度问题，通过考虑由形式定义的簇并借助代数几何工具，证明某些条件下的结论，给出反例，猜想素数次形式的情况，还构造了特定的整形式。

特征\(p\)的整体函数域上椭圆曲线的\(p\)-adic \(L\)函数

$p$-adic $L$-functions for elliptic curves over global function fields

介绍与特征\(p\)的整体函数域\(K\)上的普通椭圆曲线\(A\)及\(\mathbb{Z}_{p}^{d}\)-扩张\(L/K\)相关的\(p\)-adic \(L\)函数\(\mathscr L_{A/L}\)，证明其满足函数方程等，在多种情形下证明了岩泽主猜想。

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5, \ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly $3$ (resp. $2$) monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ within deterministic time $\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of $f$ in $K$, and for each such root generates an approximation in $\mathbb{Q}$ with logarithmic height $O(\log^3(dH))$ that converges at a rate of $O\!\left((1…

带有Bianchi模形式的有符号p-adic L-函数

Signed $p$-adic $L$-functions of Bianchi modular forms

本文研究了Bianchi模形式的有符号p-adic L-函数，通过p-稳定化形式和局部解析分布，将双变量幂级数分解为有界系数的线性组合。

有限维p进赋范空间中算法问题的范数查询复杂度

Norm-Query Complexity of Algorithmic Problems in Finite-Dimensional p-adic Normed Spaces

该研究分析有限维p进赋范空间中正交化、最长向量问题（LVP）、最近向量问题（CVP）的确定性范数查询复杂度，证明正交化和CVP的复杂度无界，给出LVP的最优范数查询算法。

A complexity chasm for solving univariate sparse polynomial equations over $p$-adic fields

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $O(\log^{O(1)}(dH))$ in the classical Turing model. (The best previous algorithms were of complexity exponential in $\log d$, even for just counting roots in $\mathb…

Surfaces and p-adic fields I: Dehn twists

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus be applied to $G_{K}$, the absolute Galois groups of the $p$-adic field $K$. By moving both groups to the world of pro-$p$ groups (for $G_{K}$ we take its maximal pro-$p$ quotient, and for $π_{1}(S)$ we take its pro-$p$ completion), we see they both are pro-$p$ Poincare duality groups of dimension $2$, also known as Demuskin groups. Such groups have a very n…

An embedding problem of Noetherian rings into the Witt vectors

The aim of this article is to prove some results on the existence of an integral extension domain of a complete local Noetherian domain in mixed characteristic $p>0$ having certain distinguished properties with respect to the Frobenius map. We prove the main results by constructing required extension domains via Witt vectors and the method of maximal étale extensions. It is worth remarking that the resulting algebras have deep connections with the homological conjectures and the rings in $p$-adic Hodge theory.

Laurent级数域上的算术$\u0026#x1D49C;$-模：相对情形

Arithmetic $\mathcal{D}$-modules over Laurent series fields: the relative case

本研究在特征p>0的完全域k下，基于Berthelot的算术D-模理论，为Spec k[[t]]上的拟射影概形构建了Grothendieck六运算的p-进形式体系，推进了Laurent级数域上相对情形的算术D-模研究。