Gross’ conjecture for extensions ramified over four points of ℙ 1

Po-Yi Huang[1]

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

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 183-201
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Huang, Po-Yi. "Gross’ conjecture for extensions ramified over four points of $\mathbb{P}^1$." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 183-201. <http://eudml.org/doc/249610>.

@article{Huang2006,
abstract = {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.},
affiliation = {Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan},
author = {Huang, Po-Yi},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {-functions; Gross' conjecture; Stickelberger element; congruence function fields},
language = {eng},
number = {1},
pages = {183-201},
publisher = {Université Bordeaux 1},
title = {Gross’ conjecture for extensions ramified over four points of $\mathbb\{P\}^1$},
url = {http://eudml.org/doc/249610},
volume = {18},
year = {2006},
}

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
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 183
EP - 201
AB - 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.
LA - eng
KW - -functions; Gross' conjecture; Stickelberger element; congruence function fields
UR - http://eudml.org/doc/249610
ER -

References

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  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. Zbl0742.11055MR1104783
  2. Noboru Aoki, On Tate’s refinement for a conjecture of Gross and its generalization. J. Théor. Nombres Bordeaux 16 (2004), 457–486. Zbl1071.11064MR2144953
  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. Zbl0844.11071MR1335080
  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. Zbl0681.12005MR931448
  6. David R. Hayes, The refined p -adic abelian Stark conjecture in function fields. Invent. Math. 94 (1988), 505–527. Zbl0666.12009MR969242
  7. Po-Yi Huang, Stickelberger elements over Rational Function Fields. In preparation. Zbl1264.11095
  8. Joongul Lee, On Gross’ Refined Class Number Formula for Elementary Abelian Extensions. Journal of Mathematical Sciences, University of Tokyo 4 (1997), 373–383. Zbl0903.11027MR1466351
  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. Zbl1057.11053MR2018824
  10. Joongul LeeOn the refined class number formula for global function fields. Math. Res. Lett. 11 (2004), 283–289. Zbl1112.11053MR2106227
  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. Zbl1060.11079MR1963800
  12. Ki-Seng Tan, On the special values of abelian L -functions. J. Math. Sci. Univ. Tokyo 1 (1994), 305–319. Zbl0820.11069MR1317462
  13. M. Yamagishi, On a conjecture of Gross on special values of L -functions. Math. Z. 201 (1989), 391–400. Zbl0689.12002MR999736

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