A matrix version of Dwork's congruences

通过 $q$-Lucas 定理的 Dwork 型 $q$-同余

Dwork-type $q$-congruences through the $q$-Lucas theorem

本文运用 $q$-Lucas 定理和已知 $q$-超同余，证明了若干 Dwork 型 $q$-同余，证实了三个先前提出的猜想，并验证了 Swisher (H.3) 猜想的较弱形式。

通过 q 形变得到的 Dwork 同余式

Dwork congruences via q-deformation

本文研究 Grassmannian 余切丛的 K 理论顶点函数截断多项式，证明其满足 Dwork 同余式的自然 q 形变，并在 q→1 极限下恢复已有结果。

Dwork-type supercongruences through a creative $q$-microscope

We develop an analytical method to prove congruences of the type $$ \sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv ω(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]} \quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers $m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic" hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in 1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of several Dwork-type congruences corresponding to $m\ge2$ (in other words, supercongruences) are based on constructing and proving t…

Truncated versions of Dwork's lemma for exponentials of power series and $p$-divisibility of arithmetic functiens

(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbi…

Congruence relations of GSpin Shimura varieties

In these notes I proved the Chai-Faltings version of Eichler-Shimura congruence relation for simple GSpin Shimura varieties. This extends the results by Bueltel, Wedhorn and Koskivirta.

($q$-)Supercongruences hit again

Using an intrinsic $q$-hypergeometric strategy, we generalise Dwork-type congruences $H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1})\pmod{p^3}$ for $s=1,2,\dots$ and $p$ a prime, when $H(N)$ are truncated hypergeometric sums corresponding to the periods of rigid Calabi--Yau threefolds.

超几何周期 $p^s$-近似的幽灵与同余

Ghosts and congruences for $p^s$-approximations of hypergeometric periods

本文证明洛朗多项式常数项的 Dwork 型同余，并借此研究超几何与 KZ 方程模 $p^s$ 多项式解的算术和 $p$-进解析性质，进而展示 $p$-进 KZ 联络存在不变线子丛而复数情形没有。

Quantitative Diophantine approximation with congruence conditions

In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.

关于西格尔问题与G-函数的德沃克猜想

On Siegel's problem and Dwork's conjecture for $G$-functions

本文否定回答了G-函数的西格尔问题，构造出无穷多个非超几何代数拉回的几何秩二局部系统，为德沃克猜想提供新反例并回答了Krammer的问题。

Some congruences on prime factors of class number of algebraic extensions K/Q

This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K. Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime. The paper is at elementary level and contains a lot of numerical examples.

Formal groups and congruences

We give a criterion of integrality of an one-dimensional formal group law in terms of congruences satisfied by the coefficients of the canonical invariant differential. For an integral formal group law a p-adic analytic formula for the local characteristic polynomial at p is given. We demonstrate applications of our results to formal group laws attached to L-functions, Artin--Mazur formal groups of algebraic varieties and hypergeometric formal group laws. (This is the final version submitted to a journal; new parts are added in the section on hypergeometric formal group laws.)

Hasse--Witt 矩阵的同余式与 $p$-adic KZ 方程的解

Congruences for Hasse--Witt matrices and solutions of $p$-adic KZ equations

本文证明 Laurent 多项式组 Hasse--Witt 矩阵的一般 Dwork 型同余式，并借此研究 KZ 方程模 $p^s$ 多项式解的算术与 $p$-adic 解析性质，进而证明超椭圆曲线族对应的 $p$-adic KZ 联络存在秩 $g$ 的不变子丛。

关于Hesse三次曲线的syntomic regulator与$p$-adic超几何函数

On the syntomic regulator of the Hesse cubic curves and $p$-adic hypergeometric functions

本文引入新型$p$-adic超几何函数，证明其满足Dwork型同余关系，并用于表达Hesse三次曲线的syntomic regulator，同时建立了与Asakura对数型$p$-adic超几何函数的变换公式。

Genus fields of abelian extensions of congruence rational function fields

In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5 and 5.2 We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions were obtained previously by G. Peng and Artin…

Metric Diophantine approximation with congruence conditions

We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.

Some supercongruences occurring in truncated hypergeometric series

For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are …

Lucas定理的一个推广

An extension of Lucas Theorem

本文给出计算模素数二项式系数的同余判据的初等证明，并据此推广Lucas定理，应用于推导二项式系数和的恒等式与同余式。