Confluent hypergeometric functions and wild ramification
This paper provides `numerical' evidence to the fact that irregular singularities for differential equations and wild ramification for \(\ell\)-adic sheafs are similar phenomena, by computing the determinant of the period matrix attached to confluent hypergeometric functions on the one hand, and the determinant of the Frobenius action on the sheaves attached to the corresponding Kloosterman sums on the other hand. (See \textit{N. Katz} [Exponential sums and differential equations, Ann. Math. Stud. 124, Princeton Univ. Press (1990; Zbl 0731.14008)], or \textit{O. Gabber} and \textit{F. Loeser} [Duke Math. J. 83, 501-606 (1996)] for other aspects of this analogy). The explicit formulae established by the author for these determinants bear a strong similarity: products of gamma factors and of exponentials in the first case, products of Gauss sums and of values of the additive character in the second one. The proofs however are of different natures: the complex formula is deduced from the classical (i.e. non-confluent) case by a limiting process, while the \(\ell\)-adic one uses Laumon's theory of \(\varepsilon\)-factors. [Remark: for a related formula in the complex case, see \textit{Y. Haraoka}, Trans. Am. Math. Soc. 349, 675-712 (1997)].
irregular singularities for differential equations
scientific article; zbMATH DE number 941220
scientific article