\(p\)-adic \(L\)-functions for CM fields
type: Item
instance of: scholarly article
title: \(p\)-adic \(L\)-functions for CM fields
zbMATH Open document ID: 0417.12003
DOI: 10.1007/BF01390187
author: Nicholas M. Katz
published in: Inventiones Mathematicae
publication date: 1978-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/142602
review text: The values \(\zeta(-k)\) of the Riemann zeta-function \(\zeta\) at negative integers \(-k\) are known to be rational, from Euler. Furthermore, for any fixed prime number \(p\), the numbers \(\zeta_a^{(p)} (-k) = (1-a^{k+i})(1-p^k)\zeta(-k)\) are \(p\)-integral for all integers \(k\geq 0\) and integers \(a\) prime to \(p\) and satisfy the congruence conditions: \[ \sum c_r \zeta_a^{(p)}(-r)\equiv 0 \pmod{p^N}, \] whenever \(\sum c_rx^r \equiv 0\pmod{p^N}\), for every \(p\)-adic unit \(x\) and integers \(c_r\). The totality of these Kummer congruences (for \(m\geq 1\)gives rise to a measure \(\mu_a\), on the group \(\mathbb Z_p^\times\) of \(p\)-adic units (with values in the ring \(\mathbb Z_p\) of \(p\)-adic integers) such that for all integers \(k\geq 0\) \[ \int_{\mathbb Z_p^\times} x^k\,d\mu_a = (1-a^{k+i})(1-p^k) \zeta(-k). \] The corresponding rationality results for values of the Dedekind zeta-function and \(L\)-series for totally real algebraic number fields \(K\) were established by Siegel and Klingen. The analogues by way of the Kummer congruences and the measure \(\mu_a\) are due to Coates-Sinnott for quadratic \(K\) and to Deligne-Ribet for any (totally real) \(K\). If \(K\) is not totally real, the values of the \(L\)-series at negative integers are \(0\) (under the influence of \(\Gamma\)-factors in their functional equation!). One may, however, investigate the values at \(0\) of Hecke \(L\)-series for grössencharacters of type \(A_0\) (in the sense of Weil). If \(L\) is a totally complex quadratic extension of a totally real field \(K\) and \(\chi\), a grössencharacter of type \(A_0\) with a power of \(p\) as conductor (viewing \(\chi\) as a \(p\)-adic character of \(G=\text{Gal}(L(p^\infty)/L)\), where \(L(p^\infty)\) is maximal abelian and unramified outside \(p\) over \(L\)), the author constructs (under suitable conditions) a \(p\)-adic measure \(\mu\) on \(G\) such that the Mellin transform \(L_\mu(\chi)\) is closely related to the value \(L(0,\ch…
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 3648841
zbMATH Keywords: p-adic L-functions
zbMATH Keywords: CM fields
zbMATH Keywords: totally complex quadratic extension of a totally real field
zbMATH Keywords: Grössencharacter
zbMATH Keywords: p-adic measure
zbMATH Keywords: p-adic interpolation of Hecke L-function
zbMATH Keywords: functional equation
zbMATH Keywords: non-analytic Eisenstein series
zbMATH Keywords: Hilbert modular group
zbMATH Keywords: p-adic differential operators
zbMATH Keywords: p-adic Eisenstein series
MaRDI profile type: Publication
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OpenAlex ID: W2074722439
description: scientific article; zbMATH DE number 3648841