L-values at zero and multiplicative Galois module structure (also Galois Gauss sums and additive Galois module structure).

*(English)*Zbl 0693.12012Let N/K be a Galois extension of number fields, with Galois group Gal(N/K). Let \(K^ c\) denote the algebraic closure of K and \(\Omega_ K=Gal(K^ c/K)\). The ring of integers of N is \({\mathfrak o}_ N\). The field K is either a finite extension of \({\mathbb{Q}}\) or of \({\mathbb{Q}}_ p\); in the latter case \({\mathbb{Q}}^ c_ p\) is denoted by \({\mathbb{C}}_ p.\)

The ‘L-values at zero’ are the leading coefficients of the Taylor expansion at 0 of certain Artin L-functions (see below) and the basic theme of this paper is the interplay between those L-values and the multiplicative Galois module structure of units, especially in the case when Gal(N/K) is abelian and \(K={\mathbb{Q}}.\)

The author obtains novel and explicit results on units in a way which sheds light on why things are as they are. He also obtains integral variants of problems and theorems related to Stark’s conjecture; indeed ‘the general philosophy is that this conjecture has module theoretic consequences and... that the module theory yields stronger integral forms of the conjecture’.

The author develops new and powerful tools based on pairings b: \(X\times Y\to {\mathbb{A}}\), where X and Y are finite-dimensional vector spaces over a field k, the quotient field of some Dedekind domain \({\mathfrak o}\), and \({\mathbb{A}}\) is an algebraically closed field containing k. Those pairings enable one to introduce significant generalizations of the concepts of regulator (in the multiplicative case) and resolvent (in the additive case) and to present the analogy between L-values at 0 in the former case and Galois Gauss sums in the latter in a striking form, having deep connections with the functional equation of the L-functions.

The paper makes demands on the reader, who will be rewarded by rich ideas and fascinating results, some flavour of which is given by the following two theorems.

Let \(\Gamma\) be a group, K a number field and let q: \(\Omega_ K\to \Gamma\) be a continuous surjection of groups, that is an isomorphism \(\Gamma\) \(\cong Gal(N/K)\), where N is the fixed field of Ker q.

In the additive theory, K is a finite extension of the base field k (which is either \({\mathbb{Q}}\) or \({\mathbb{Q}}_ p)\). The set of field embeddings \(N\to {\mathbb{A}}\) (where \({\mathbb{A}}\) is \(k^ c)\) is denoted by S(N). The group \(\Gamma\) acts on N and on \({\mathfrak o}_ N=Int(N)\) on the right and hence on S(N) from the left. Let FS denote the free F-module on S (F is a finite extension of k in which certain representations are realizable). The pairing \(b=b_ N: N\otimes_{k\Gamma}kS\to {\mathbb{A}}\) is defined by \(b(x,ys)=x^ s\cdot y\), \(x\in N\), \(y\in k\), \(s\in S\) and in accordance with the general theory of pairings defines a lattice of the integral closure of \({\mathfrak o}_ N\). Such a lattice is called a generalized resolvent.

In the multiplicative theory, S is a finite set of places of K, including the infinite ones, and S(N) denotes the places of N over S. Let \({\mathfrak U}_{S,N}\) denote the group of S(N) units in N, modulo roots of unity, and let \({\mathfrak S}_{S,N}={\mathbb{Z}}S(N)/(\sum s)\) where \((\sum s)\) denotes the subgroup generated by the sums \(\sum s\), \(s\in S(N)\). The group \(\Gamma\) acts on the right on \({\mathfrak U}_{S,N}\) and on the left on \({\mathbb{Z}}S(N)\) and \({\mathfrak S}_{S,N}\), to yield a pairing \(b=b_{S,N}: {\mathfrak U}_{S,N}{\mathbb{Q}}\times {\mathfrak S}_{S,N}{\mathbb{Q}}\to {\mathbb{C}}\) which extends that given by \(b(u,s)=\log \| u\|_ s\), \(u\in {\mathfrak U}_{S,N}\), \(s\in S(N)\) and the corresponding lattices are the generalized regulators [cf. Tate’s formulation of Stark’s conjecture as formulated by the author in Sémin. Théor. Nombres, Univ. Bordeaux I 1986/87, Exp. No.32 (1987; Zbl 0642.12015)].

Suppose that \(K={\mathbb{Q}}\) and that N is real. The surjection q lifts each representation \(\rho\) of \(\Gamma\) over \({\mathbb{C}}\) to a continuous representation \(\rho\) \(\circ q\) of \(\Omega_ K\), which has Artin L- function \(L_ S(s,\rho \circ q)\), with the Euler factors of places in S removed. The leading non-zero coefficient of the Taylor expansion of \(L_ S\) at \(s=0\) is denoted by \(L_ S(\rho,N)\). If \({\mathfrak U}',{\mathfrak S}'\) are \({\mathbb{Z}}\Gamma\)-lattices spanning \({\mathfrak U}_{S,N}{\mathbb{Q}}\) and \({\mathfrak S}_{S,N}{\mathbb{Q}}\), respectively, and if \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\) with associated generalized regulator \({\mathfrak R}(\theta,{\mathfrak U}',{\mathfrak S}')\) then \[ {\mathfrak R}(\theta,{\mathfrak U}',{\mathfrak S}')^{-1}\cdot L_ S(\theta^{-1},N) \] is an ideal in the field \({\mathbb{Q}}(\theta)\) of values of \(\theta\) over \({\mathbb{Q}}.\)

The corresponding result in the additive case (and this is typical of the analogy between the two) reads as follows. In place of the regulator, one has a generalized resolvent \({\mathfrak R}(\theta,{\mathfrak o}_ N,{\mathbb{Z}}S)\), \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\), and the representation \(\rho\) \(\circ q\) of \(\Omega_ K\) gives rise to an induced representation ind(\(\rho\) \(\circ q)\) of \(\Omega_{{\mathbb{Q}}}\), with Galois Gauss sum \({\tilde \tau}\)(\(\rho\),N)\(=\tau ({\mathbb{Q}},ind(\rho \circ q))\) [cf. the author, Galois module structure of algebraic integers (1983; Zbl 0501.12012), I, § 5 and III, § 2)]. Let L be a \({\mathbb{Z}}\Gamma\)- lattice spanning N over \({\mathbb{Q}}\) and M a \({\mathbb{Z}}\Gamma\)-lattice spanning \({\mathbb{Q}}S\). Then for all \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\) and for the resolvent lattice \({\mathfrak R}(\theta,L,M)\) associated with the pairing and L,M, \[ {\tilde \tau}(\theta,N)\cdot {\mathfrak R}(\theta,L,M)^{-1} \] is an ideal in \({\mathbb{Q}}(\theta).\)

The author defines the notion of factorizability, in which certain homomorphisms from the Burnside ring of a group \(\Gamma\) are expressed as homomorphisms defined on characters. The foregoing is then applied to the module structure of \({\mathfrak o}_ N\) in the global case and of \({\mathfrak U}_{S,N}.\)

The paper concludes with an appendix in which the material on pairings is presented and with some tantalizing questions such as: is there an analogue in the additive theory to the group of cyclotomic units; for example is there a natural free \({\mathbb{Z}}\Gamma\)-lattice whose resolvents are essentially the Galois Gauss sums?

The ‘L-values at zero’ are the leading coefficients of the Taylor expansion at 0 of certain Artin L-functions (see below) and the basic theme of this paper is the interplay between those L-values and the multiplicative Galois module structure of units, especially in the case when Gal(N/K) is abelian and \(K={\mathbb{Q}}.\)

The author obtains novel and explicit results on units in a way which sheds light on why things are as they are. He also obtains integral variants of problems and theorems related to Stark’s conjecture; indeed ‘the general philosophy is that this conjecture has module theoretic consequences and... that the module theory yields stronger integral forms of the conjecture’.

The author develops new and powerful tools based on pairings b: \(X\times Y\to {\mathbb{A}}\), where X and Y are finite-dimensional vector spaces over a field k, the quotient field of some Dedekind domain \({\mathfrak o}\), and \({\mathbb{A}}\) is an algebraically closed field containing k. Those pairings enable one to introduce significant generalizations of the concepts of regulator (in the multiplicative case) and resolvent (in the additive case) and to present the analogy between L-values at 0 in the former case and Galois Gauss sums in the latter in a striking form, having deep connections with the functional equation of the L-functions.

The paper makes demands on the reader, who will be rewarded by rich ideas and fascinating results, some flavour of which is given by the following two theorems.

Let \(\Gamma\) be a group, K a number field and let q: \(\Omega_ K\to \Gamma\) be a continuous surjection of groups, that is an isomorphism \(\Gamma\) \(\cong Gal(N/K)\), where N is the fixed field of Ker q.

In the additive theory, K is a finite extension of the base field k (which is either \({\mathbb{Q}}\) or \({\mathbb{Q}}_ p)\). The set of field embeddings \(N\to {\mathbb{A}}\) (where \({\mathbb{A}}\) is \(k^ c)\) is denoted by S(N). The group \(\Gamma\) acts on N and on \({\mathfrak o}_ N=Int(N)\) on the right and hence on S(N) from the left. Let FS denote the free F-module on S (F is a finite extension of k in which certain representations are realizable). The pairing \(b=b_ N: N\otimes_{k\Gamma}kS\to {\mathbb{A}}\) is defined by \(b(x,ys)=x^ s\cdot y\), \(x\in N\), \(y\in k\), \(s\in S\) and in accordance with the general theory of pairings defines a lattice of the integral closure of \({\mathfrak o}_ N\). Such a lattice is called a generalized resolvent.

In the multiplicative theory, S is a finite set of places of K, including the infinite ones, and S(N) denotes the places of N over S. Let \({\mathfrak U}_{S,N}\) denote the group of S(N) units in N, modulo roots of unity, and let \({\mathfrak S}_{S,N}={\mathbb{Z}}S(N)/(\sum s)\) where \((\sum s)\) denotes the subgroup generated by the sums \(\sum s\), \(s\in S(N)\). The group \(\Gamma\) acts on the right on \({\mathfrak U}_{S,N}\) and on the left on \({\mathbb{Z}}S(N)\) and \({\mathfrak S}_{S,N}\), to yield a pairing \(b=b_{S,N}: {\mathfrak U}_{S,N}{\mathbb{Q}}\times {\mathfrak S}_{S,N}{\mathbb{Q}}\to {\mathbb{C}}\) which extends that given by \(b(u,s)=\log \| u\|_ s\), \(u\in {\mathfrak U}_{S,N}\), \(s\in S(N)\) and the corresponding lattices are the generalized regulators [cf. Tate’s formulation of Stark’s conjecture as formulated by the author in Sémin. Théor. Nombres, Univ. Bordeaux I 1986/87, Exp. No.32 (1987; Zbl 0642.12015)].

Suppose that \(K={\mathbb{Q}}\) and that N is real. The surjection q lifts each representation \(\rho\) of \(\Gamma\) over \({\mathbb{C}}\) to a continuous representation \(\rho\) \(\circ q\) of \(\Omega_ K\), which has Artin L- function \(L_ S(s,\rho \circ q)\), with the Euler factors of places in S removed. The leading non-zero coefficient of the Taylor expansion of \(L_ S\) at \(s=0\) is denoted by \(L_ S(\rho,N)\). If \({\mathfrak U}',{\mathfrak S}'\) are \({\mathbb{Z}}\Gamma\)-lattices spanning \({\mathfrak U}_{S,N}{\mathbb{Q}}\) and \({\mathfrak S}_{S,N}{\mathbb{Q}}\), respectively, and if \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\) with associated generalized regulator \({\mathfrak R}(\theta,{\mathfrak U}',{\mathfrak S}')\) then \[ {\mathfrak R}(\theta,{\mathfrak U}',{\mathfrak S}')^{-1}\cdot L_ S(\theta^{-1},N) \] is an ideal in the field \({\mathbb{Q}}(\theta)\) of values of \(\theta\) over \({\mathbb{Q}}.\)

The corresponding result in the additive case (and this is typical of the analogy between the two) reads as follows. In place of the regulator, one has a generalized resolvent \({\mathfrak R}(\theta,{\mathfrak o}_ N,{\mathbb{Z}}S)\), \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\), and the representation \(\rho\) \(\circ q\) of \(\Omega_ K\) gives rise to an induced representation ind(\(\rho\) \(\circ q)\) of \(\Omega_{{\mathbb{Q}}}\), with Galois Gauss sum \({\tilde \tau}\)(\(\rho\),N)\(=\tau ({\mathbb{Q}},ind(\rho \circ q))\) [cf. the author, Galois module structure of algebraic integers (1983; Zbl 0501.12012), I, § 5 and III, § 2)]. Let L be a \({\mathbb{Z}}\Gamma\)- lattice spanning N over \({\mathbb{Q}}\) and M a \({\mathbb{Z}}\Gamma\)-lattice spanning \({\mathbb{Q}}S\). Then for all \(\theta \in Hom(\Gamma,{\mathbb{C}}^*)\) and for the resolvent lattice \({\mathfrak R}(\theta,L,M)\) associated with the pairing and L,M, \[ {\tilde \tau}(\theta,N)\cdot {\mathfrak R}(\theta,L,M)^{-1} \] is an ideal in \({\mathbb{Q}}(\theta).\)

The author defines the notion of factorizability, in which certain homomorphisms from the Burnside ring of a group \(\Gamma\) are expressed as homomorphisms defined on characters. The foregoing is then applied to the module structure of \({\mathfrak o}_ N\) in the global case and of \({\mathfrak U}_{S,N}.\)

The paper concludes with an appendix in which the material on pairings is presented and with some tantalizing questions such as: is there an analogue in the additive theory to the group of cyclotomic units; for example is there a natural free \({\mathbb{Z}}\Gamma\)-lattice whose resolvents are essentially the Galois Gauss sums?

Reviewer: J.V.Armitage