On the values of abelian \(L\)-functions at \(s=0\). (English) Zbl 0681.12005

Let \(\zeta(s)\) be the zeta function of a global field \(k\), then classically \(\zeta(s)\equiv -hR\omega^{-1}s^{r_ 1+r_ 2-1}\) modulo \(s^{r_ 1+r_ 2}\) near \(s=0\). The author presents a conjecture similar to that.
Let \(S\) be a finite set of places of \(k\) containing archimedean places, and let \(A\) denote the \(S\)-integers, \(U=A^*\), \(h=\# \text{Pic}(A)\), \(n=\#S-1\), \(\omega =\# \text{tors}(U)\). The \(S\)-regulator \(R\) is the absolute value of the determinant of \(\lambda_{{\mathbb R}}: U\to {\mathbb R}\otimes X,\quad \varepsilon \mapsto \sum_{s}\log \| \varepsilon \|_ v\cdot v,\) where \(X=\{\sum a_ vv:\sum a_ v=0\}.\) Then one has \(\zeta (s)\equiv -hR\omega^{- 1}s^ n \pmod {s^{n+1}}.\) Let \(T\) be a finite set of places disjoint from \(S\), and let \(\zeta_ T(s)\) be \(\prod (1-N{\mathfrak p}^{1-s})\zeta (s) \) for \({\mathfrak p}\in T\), \(U_ T\) the units \(\equiv 1 \pmod T,\) \(\text{Pic}(A)_ T\) the group of invertible \(A\)-modules with a trivialization at \(T\). Making \(\omega_ T=\omega \cap U_ T=1\) by restricting \(T\), one thus obtains an integral formula \(\zeta_ T(s)\equiv m\cdot \det_{\mathbb R}(\lambda)s^ n \pmod{s^{n+1}},\) where \(m=\pm h_ T=\pm \# \text{Pic}(A)_ T.\)
To generalize the regulator, the author denotes by \(A\) the adeles of \(k\), \(G\) a finite group, \(f: A^*\to G\) a homomorphism, and defines \(\lambda_ G: U\to G\otimes X,\quad \varepsilon \mapsto \sum_{s}f(1,...,\varepsilon_ v,...,1)v.\) Let \(I=\{\sum_{G}m(g)g :\sum m(g) =0\}\subset {\mathbb Z}[G],\) then \(G\cong I/I^ 2\) (g\(\mapsto g-1)\). So he defines \(\det_ G\lambda =\det ((g_{ij}-1))\in I^ n/I^{n+1}\). Now let \(\theta_ G\) be the unique element of \({\mathbb C}[G]\) such that \(\theta_ G(\chi)=L_ T(\chi,0)\) for all characters \(\chi\in \hat G\), where \(L_ T(\chi,s)\) is the abelian \(L\)-function; the author proves that \(\theta_ G\) is in \({\mathbb Z}[G]\). He then states his conjecture:
\(\theta_ G \equiv m\cdot \det_ G \lambda \pmod{I^{n+1}}\).
If the conjecture holds for \(S,T,\) then it holds for \(S'\supset S\), \(T'\supset T\). Taking \(G\) to be the Galois group of the maximal abelian extension of \(k\) unramified outside \(S\) and \(f\) the reciprocity map of global class field theory, one obtains a conjecture which implies all others. Then in sections 5–6, the author proves the conjecture respectively for the number field \(k\) and for \(G\cong\mathbb Z/\ell\mathbb Z\) with prime \(\ell\) up to a unit \(u\) in \((\mathbb Z/\ell\mathbb Z)^*\). He finally discusses a refinement of Stark’s conjecture for the first derivative [cf. H.M. Stark, Adv. Math. 35, 197–235 (1980; Zbl 0475.12018)] and the \(L\)-functions of quadratic characters.
Reviewer: Zhang Xianke


11R42 Zeta functions and \(L\)-functions of number fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture


Zbl 0475.12018