Gross’ conjecture for extensions ramified over four points of $\mathbb{P}^1$

Huang, Po-Yi

doi:10.5802/jtnb.539

Gross’ conjecture for extensions ramified over four points of

ℙ^{1}

Huang, Po-Yi ¹

¹ Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 183-201.

Résumé
Abstract

Dans le papier ci-après, avec une hypothése modérée, nous prouvons une conjecture de Gross pour l’élément Stickelberger de l’extension abelienne maximale sur le corps des fonctions rationnelles non ramifiée en dehors d’un ensemble des quatre places de degré 1.

In this paper, under a mild hypothesis, we prove a conjecture of Gross for the Stickelberger element of the maximal abelian extension over the rational function field unramified outside a set of four degree-one places.

EuDML MR Zbl

DOI : 10.5802/jtnb.539

Affiliations des auteurs :

Huang, Po-Yi ¹

¹ Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

@article{JTNB_2006__18_1_183_0,
     author = {Huang, Po-Yi},
     title = {Gross{\textquoteright} conjecture for extensions ramified over four points of $\mathbb{P}^1$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {183--201},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.539},
     mrnumber = {2245881},
     zbl = {1126.11066},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.539/}
}

TY  - JOUR
AU  - Huang, Po-Yi
TI  - Gross’ conjecture for extensions ramified over four points of $\mathbb{P}^1$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2006
SP  - 183
EP  - 201
VL  - 18
IS  - 1
PB  - Université Bordeaux 1
UR  - https://www.numdam.org/articles/10.5802/jtnb.539/
DO  - 10.5802/jtnb.539
LA  - en
ID  - JTNB_2006__18_1_183_0
ER  -

%0 Journal Article
%A Huang, Po-Yi
%T Gross’ conjecture for extensions ramified over four points of $\mathbb{P}^1$
%J Journal de théorie des nombres de Bordeaux
%D 2006
%P 183-201
%V 18
%N 1
%I Université Bordeaux 1
%U https://www.numdam.org/articles/10.5802/jtnb.539/
%R 10.5802/jtnb.539
%G en
%F JTNB_2006__18_1_183_0

Huang, Po-Yi. Gross’ conjecture for extensions ramified over four points of $\mathbb{P}^1$. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 183-201. doi : 10.5802/jtnb.539. https://www.numdam.org/articles/10.5802/jtnb.539/

Bibliographie
Cité par

[1] Noboru Aoki, Gross’ Conjecture on the Special Values of Abelian $L$ -Functions at $s = 0$ . Commentarii Mathematici Universitatis Sancti Pauli 40 (1991), 101–124. | MR | Zbl

[2] Noboru Aoki, On Tate’s refinement for a conjecture of Gross and its generalization. J. Théor. Nombres Bordeaux 16 (2004), 457–486. | Numdam | MR | Zbl

[3] David Burns, Congruences between derivatives of abelian $L$ -functions at $s = 0$ . Preprint, 2005.

[4] Henri Darmon, Thaine’s method for circular units and a conjecture of Gross. Canadian J. Math. 47 (1995), 302–317. | MR | Zbl

[5] Benedict H. Gross, On the values of abelian $L$ -functions at $s = 0$ . J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 35 (1988), 177–197. | MR | Zbl

[6] David R. Hayes, The refined $p$ -adic abelian Stark conjecture in function fields. Invent. Math. 94 (1988), 505–527. | MR | Zbl

[7] Po-Yi Huang, Stickelberger elements over Rational Function Fields. In preparation.

[8] Joongul Lee, On Gross’ Refined Class Number Formula for Elementary Abelian Extensions. Journal of Mathematical Sciences, University of Tokyo 4 (1997), 373–383. | MR | Zbl

[9] Joongul Lee, Stickelberger elements for cyclic extensions and the order of vanishing of abelian L-functions at $s = 0$ . Compositio Math. 138, no.2 (2003), 157–163. | MR | Zbl

[10] Joongul Lee On the refined class number formula for global function fields. Math. Res. Lett. 11 (2004), 283–289. | MR | Zbl

[11] Michael Reid, Gross’ Conjecture for extensions ramified over three points on $ℙ^{1}$ . Journal of Mathematical Sciences, University of Tokyo 10 no. 1 (2003), 119–138. | MR | Zbl

[12] Ki-Seng Tan, On the special values of abelian $L$ -functions. J. Math. Sci. Univ. Tokyo 1 (1994), 305–319. | MR | Zbl

[13] M. Yamagishi, On a conjecture of Gross on special values of $L$ -functions. Math. Z. 201 (1989), 391–400. | MR | Zbl

Cité par Sources :