## 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

### MSC:

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

Zbl 0475.12018