A quick introduction to Dwork's conjecture

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。

On Some Hypergeometric Supercongruence Conjectures of Long

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version…

Logarithmic growth filtrations for $(φ,\nabla)$-modules over the bounded Robba ring

In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(φ,\nabla)$-modules over the bounded Robba ring. The main result is a proof of a conjecture proposed by B. Chiarellotto and N. Tsuzuki on a comparison between the log-growth filtration and Frobenius slope filtration. One of the ingredients of the proof is a new criterion for pure of bounded quotient, which is a notion introduced by Chiarellotto and Tsuzuki to formulate their conjecture. We also give several applications to log-growth Newton polygons, including a conjecture of …

$L$-functions for families of generalized Kloosterman sums and $p$-adic differential equations

In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dwork's deformation equation for our family. Furthermore, with the help of Dwork's dual theory and deformation theory, the strong Frobenius structure of this equ…

Pure L-functions from algebraic geometry over finite fields

This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.

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。

通过同伦理论证明 Berthelot 猜想

Berthelot's conjecture via homotopy theory

本文利用 motive 方法快速证明了 Berthelot 猜想，指出结构层沿平滑固有映射在刚性上同调中的前推映射具有过度收敛 F-isocrystal 的典范结构。

On the parity of the number of partitions with odd multiplicities

Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of $a(2m)$ based solely on properties of $m.$ In this note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 …