On Tate’s refinement for a conjecture of Gross and its generalization

Noboru Aoki[1]

  • [1] Department of Mathematics Rikkyo University Nishi-Ikebukuro, Toshima-ku Tokyo 171-8501, Japan

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 457-486
  • ISSN: 1246-7405

Abstract

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We study Tate’s refinement for a conjecture of Gross on the values of abelian L -function at s = 0 and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.

How to cite

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Aoki, Noboru. "On Tate’s refinement for a conjecture of Gross and its generalization." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 457-486. <http://eudml.org/doc/249258>.

@article{Aoki2004,
abstract = {We study Tate’s refinement for a conjecture of Gross on the values of abelian $L$-function at $s=0$ and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.},
affiliation = {Department of Mathematics Rikkyo University Nishi-Ikebukuro, Toshima-ku Tokyo 171-8501, Japan},
author = {Aoki, Noboru},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Abelian -functions; Gross's conjecture; Stickelberger element},
language = {eng},
number = {3},
pages = {457-486},
publisher = {Université Bordeaux 1},
title = {On Tate’s refinement for a conjecture of Gross and its generalization},
url = {http://eudml.org/doc/249258},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Aoki, Noboru
TI - On Tate’s refinement for a conjecture of Gross and its generalization
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 457
EP - 486
AB - We study Tate’s refinement for a conjecture of Gross on the values of abelian $L$-function at $s=0$ and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.
LA - eng
KW - Abelian -functions; Gross's conjecture; Stickelberger element
UR - http://eudml.org/doc/249258
ER -

References

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  2. N. Aoki, J. Lee, K.S. Tan, A refinement for a conjecture of Gross. In preparation. 
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  18. J.-P. Serre, Local Fields. GTM 67, Springer-Verlag. Zbl0423.12016MR554237
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  20. K.-S. Tan, A note on the Stickelberger elements for cyclic p -extensions over global function fields of characteristic p . To appear in Math. Res. Letters. Zbl1149.11320MR2067472
  21. J. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0 . Progress in Math. 47, Birkhäuser, Boston-Basel-Stuttgart (1984). Zbl0545.12009MR782485
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  23. 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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