New $p$-adic hypergeometric functions and syntomic regulators

Masanori Asakura

doi:10.5802/jtnb.1250

New

p

-adic hypergeometric functions and syntomic regulators

Masanori Asakura ¹

¹ Department of Mathematics Hokkaido University Sapporo 060-0810, Japan

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 393-451.

Résumé
Abstract

Nous introduisons un nouveau type de fonctions hypergéométriques $p$ -adiques, que nous appelons fonctions hypergéométriques $p$ -adiques de type logarithmique. Le premier résultat principal de cet article est la preuve des relations de congruence similaires à celles de Dwork. Le deuxième résultat principal est que les valeurs spéciales de nos nouvelles fonctions apparaissent dans le calcul des régulateurs syntomiques pour les courbes hypergéométriques, courbes de Fermat et certaines courbes elliptiques. D’après la conjecture de Beilinson $p$ -adique de Perrin-Riou, on s’attend à ce qu’elles soient liées aux valeurs spéciales des fonctions $L$ $p$ -adiques. Nous en donnons un exemple.

We introduce a new type of $p$ -adic hypergeometric functions, which we call the $p$ -adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork’s. The second main result is that the special values of our new functions appear in the syntomic regulators for hypergeometric curves, Fermat curves and some elliptic curves. According to the $p$ -adic Beilinson conjecture by Perrin-Riou, they are expected to be related with the special values of $p$ -adic $L$ -functions. We provide one example for this.

Reçu le : 2021-09-29
Révisé le : 2023-01-23
Accepté le : 2023-04-05
Publié le : 2023-10-10

DOI : 10.5802/jtnb.1250

Classification : 14F30, 19F27, 11S80, 19F15
Mots clés : syntomic regulator, $p$-adic hypergeometric function, $p$-adic Beilinson conjecture

Affiliations des auteurs :

Masanori Asakura ¹

¹ Department of Mathematics Hokkaido University Sapporo 060-0810, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Masanori Asakura. New $p$-adic hypergeometric functions and syntomic regulators. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 393-451. doi : 10.5802/jtnb.1250. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1250/

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