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研究生： 王崇亘
Wang, Chung-Hsuan
論文名稱： p進超幾何函數的變換公式
On transformation formulas of p-adic hypergeometric functions
指導教授： 黃柏嶧
Huang, Po-Yi
學位類別： 博士
Doctor
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 43
中文關鍵詞： p進超幾何函數 、同餘關係式 、變換公式
外文關鍵詞： p-adic hypergeometric functions, congruence relations, transformation formulas
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    • 在這篇論文中，我們介紹三個 p進超幾何函數和它們的同餘關係式。接著我們會介紹關於它們的猜想: 變換公式。我們會證明其中一個變換公式能夠推導出另一個變換公式。最後，我們會討論這些變換公式在s=1和s=2時的結論。

    In this paper, we recall p-adic hypergeometric functions and their congruence relations. Then we introduce the conjecture of their transformation formulas and give a proof of one of transformation formulas implies another. Finally, we discuss transformation formulas when s=1 and s=2.

    Contents 摘要 iii Abstract iv 誌謝 v 1 Introduction 1 2 p-adic Hypergeometric Functions 8 2.1 p-adic Digamma functions and p-adic Euler Constant 8 2.2 Dwork’s p-adic Hypergeometric Functions 9 2.3 p-adic Hypergeometric Functions of Logarithmic Type 10 3 Congruence Relations 13 3.1 Congruence Relations of Bk/Ak and ̂ Bk/Ak 13 3.2 Congruence Relations for p-adic Hypergeometric Functions 20 3.3 Proof of Congruence Relations 26 4 Transformation Formulas 30 4.1 Involution of W 〈t, t−1, h(t)−1〉30 4.2 Transformation Formulas of F (σ)a (t) and ̂ F (̂ σ)a (t) 32 4.3 Transformation Formulas of Dwork’s p-adic Hypergeometric Func- tions 34 4.4 Transformation Formulas of F Dwa (t) Implies Transformation Formulas of F (σ)(t) and ̂ F (̂ σ)(t) 34 4.5 Case: s = 1 and Case: s = 2 40 Bibliography 43

    [1]Asakura, M.: New p-adic hypergeometric functions and syntomic regulators arXiv:1811.03770.
    [2]Dwork, B.: p-adic cycles. Publ. Math. IHES, tome 37 (1969), 27-115.
    [3]Hartshorne, R.: Algebraic geometry. Springer Science & Business Media, 2013.
    [4]Kedlaya, Kiran S.: p-adic Differential Equations. Cambridge University Press, 2010.
    [5]Poonen, B. Rational points on varieties. American Mathematical Soc., 2017.
    [6]Alain M. Robert.: A Course in p-adic Analysis. Vol. 198. Springer Science & Business Media, 2013.
    [7]W. H. Schikhof: Ultrametric Calculus: An Introduction to p-adic Analysis. Cambridge University Press
    [8]L. J. Slater: Generalized hypergeometric functions. Cambridge Univ. Press, Cambridge (1966).
    [9]Van Der Put, Marius.: The cohomology of Monsky and Washnitzer. Mém. Soc. Math. France (NS), 1986, 23.4: 33-59.
    [10]Wang Chung-Hsuan.: Congruence relations for p-adic hypergeometric functions $widehat{mathscr{F}}_{a,...,a}^{(sigma)}(t)$ and its transformation formula. manuscripta mathematica (to appear).

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