Confluent hypergeometric functions and wild ramification
type: Item
instance of: scholarly article
title: Confluent hypergeometric functions and wild ramification
zbMATH Open document ID: 0873.12004
author: Tomohide Terasoma
published in: Journal of Algebra
publication date: 1997-10-30T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 12H99
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 33C70
zbMATH DE Number: 941220
zbMATH Keywords: irregular singularities for differential equations
zbMATH Keywords: wild ramification
zbMATH Keywords: confluent hypergometric functions
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: determinants
reviewed by: Daniel Bertrand
MaRDI profile type: Publication
full work available at URL: jabr.1996.0309
OpenAlex ID: W2032847066
DOI: 10.1006/JABR.1996.0309
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description: scientific article; zbMATH DE number 941220