New $p$-adic hypergeometric functions and syntomic regulators

Asakura, Masanori

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

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.

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.

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

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