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Zbl 1060.11079
Reid, Michael
Gross' conjecture for extensions ramified over three points of ${\Bbb P}^1$. (English)
[J] J. Math. Sci., Tokyo 10, No. 1, 119-138 (2003). ISSN 1340-5705; ISSN 0040-8980

B. Gross has formulated a conjecture generalizing the class number formula for an abelian extension of a global field with Galois group $G$. A generalized Stickelberger element $\theta\in\Bbb Z[G]$ is constructed from special values of $L$-functions at $s=0$. The conjecture predicts some $I$-adic information about $\theta$, where $I\subseteq\Bbb Z[G]$ is the augmentation ideal. In more detail, let $L$ be a finite extension of a global field $K$ with Galois group $G$, $S$ a finite nonempty set of places of $K$, including all those that ramify in $L$, and $T$ a finite set of places disjoint from $S$. Let $U_{S,T}$ denote the subgroup of units congruent to 1 mod $T$ in the unit group $\Cal O_S^{\ast}$, and assume it is torsionfree. The (opposite) Stickelberger element $\overline{\theta}_{S,T}\in\Bbb Z[G]$ is defined by $\chi(\overline{\theta}_{S,T})= L_{S,T}(\chi,0)$ for all characters $\chi\in\widehat G$, where $L_{S,T}$ is a modified $L$-function. Let $n=\#S-1=\text{rank}\,\Cal O_S^{\ast}$, order the places of $S$ as $v_0, v_1,\dots,v_n$, let $u_1, u_2,\dots,u_n$ be a $\Bbb Z$-basis for $U_{S,T}$ and $r_{v_i}$ the local reciprocity map at the place $v_i$ for the extension $L/K$. Define $\text{det}_G(\lambda_{S,T})= \text{det}(r_{v_i}(u_j)-1)_{1\leq i,j\leq n} \bmod I^{n+1}$, and let $h_{S,T}$ be the order of the ray class group modulo $T$. Gross has conjectured that $\overline{\theta}_{S,T}\equiv\pm h_{S,T}\text{det}_G(\lambda_{S,T}) \bmod I^{n+1}$ (and specified how the sign is chosen). The author proves the conjecture for the extension $K_S/K$, where $K=\Bbb F_q(X)$, $S$ contains three degree one places of $K$, $K_S$ is the maximal abelian extension of $K$ that is unramified outside $S$, and the gcd of the degrees of the places in $T$ is relatively prime to $q-1$ (Theorem 1.7). He remarks that this additional restriction on $T$ is minor, but it would be of interest to eliminate it. It suffices to prove the conjecture for $K_S^{\text{tame}}= \overline{\Bbb F}_q(\root q-1\of{X},\root q-1\of{X-1})$ over $K$, and most work is in proving a form of the conjecture for $L/K$ where $L=\Bbb F_{q^m}(\root q-1\of{X}\root q-1\of{X-1})$ and $T$ contains a single place (Theorem 3.2).
[M. Rafiq Omar (Bellville)]

MSC 2000:: *11R58 Arithmetic theory of algebraic function fields
11G40 L-functions of varieties over global fields

Keywords: Gross' conjecture; L-function; Stickelberger element; Rational function field; ramified place

Cited in: Zbl 1126.11066

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