On Tate’s refinement for a conjecture of Gross and its generalization
Journal de Théorie des Nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 457-486.

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.

Nous étudions un raffinement dù à Tate de la conjecture de Gross sur les valeurs de fonctions L abéliennes en s=0 et formulons sa généralisation à une extension cyclique abitraire. Nous prouvons que notre conjecture généralisée est vraie dans le cas des corps de nombres. Cela entraine en particulier que le raffinement de Tate est vrai pour tout corps de nombres.

Published online:
DOI: 10.5802/jtnb.456
Noboru Aoki 1

1 Department of Mathematics Rikkyo University Nishi-Ikebukuro, Toshima-ku Tokyo 171-8501, Japan
@article{JTNB_2004__16_3_457_0,
     author = {Noboru Aoki},
     title = {On {Tate{\textquoteright}s} refinement for a conjecture of {Gross} and its generalization},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {457--486},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     doi = {10.5802/jtnb.456},
     zbl = {1071.11064},
     mrnumber = {2144953},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.456/}
}
TY  - JOUR
TI  - On Tate’s refinement for a conjecture of Gross and its generalization
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2004
DA  - 2004///
SP  - 457
EP  - 486
VL  - 16
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.456/
UR  - https://zbmath.org/?q=an%3A1071.11064
UR  - https://www.ams.org/mathscinet-getitem?mr=2144953
UR  - https://doi.org/10.5802/jtnb.456
DO  - 10.5802/jtnb.456
LA  - en
ID  - JTNB_2004__16_3_457_0
ER  - 
%0 Journal Article
%T On Tate’s refinement for a conjecture of Gross and its generalization
%J Journal de Théorie des Nombres de Bordeaux
%D 2004
%P 457-486
%V 16
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.456
%R 10.5802/jtnb.456
%G en
%F JTNB_2004__16_3_457_0
Noboru Aoki. On Tate’s refinement for a conjecture of Gross and its generalization. Journal de Théorie des Nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 457-486. doi : 10.5802/jtnb.456. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.456/

[1] N. Aoki, Gross’ conjecture on the special values of abelian L-functions at s=0. Comm. Math. Univ. Sancti Pauli 40 (1991), 101–124. | MR | Zbl

[2] N. Aoki, J. Lee, K.S. Tan, A refinement for a conjecture of Gross. In preparation.

[3] D. Burns, On relations between derivatives of abelian L-functions at s=0. Preprint (2002).

[4] D. Burns, J. Lee, On refined class number formula of Gross. To appear in J. Number Theory. | MR | Zbl

[5] Pi. Cassou-Nogues, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques. Inv. Math. 51 (1979), 29–59. | MR | Zbl

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

[7] P. Deligne, K. Ribet, Values of Abelian L-functions at negative integers over totally real fields. Inv. Math. 59 (1980), 227–286. | MR | Zbl

[8] L.J. Federer, p-adic L-functions, Regulators, and Iwasawa modules. PhD thesis (1982), Princeton University.

[9] B. Gross, On the values of abelian L-functions at s=0. J. Fac. Univ. Tokyo 35 (1988), 177–197. | MR | Zbl

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

[11] A. Hayward, A class number formula for higher derivatives of abelian L-functions, Compositio Math. 140 (2004), 99–120. | MR | Zbl

[12] S. Lang, Cyclotomic fields II. GTM 69, Springer-Verlag, New-York Heidelberg Berlin (1980). | MR | Zbl

[13] J. Lee, On Gross’s Refined Class Number Formula for Elementary Abelian Extensions. J. Math. Sci. Univ. Tokyo 4 (1997), 373–383. | MR | Zbl

[14] J. Lee, Stickelberger elements for cyclic extensions and the order of zero of abelian L-functions at s=0. Compositio. Math. 138 (2003), 157–163. | MR | Zbl

[15] J. Lee, On the refined class number formula for global function fields. To appear in Math. Res. Letters. | MR | Zbl

[16] M. Reid, Gross’ conjecture for extensions ramified over three points of 1 . J. Math. Sci. Univ. Tokyo 10 (2003), 119–138. | MR | Zbl

[17] K. Rubin, A Stark conjecture “over " for abelian L-functions with multiple zeros. Ann. Inst. Fourier, Grenoble 46, 1 (1996), 33–62. | Numdam | MR | Zbl

[18] J.-P. Serre, Local Fields. GTM 67, Springer-Verlag. | MR | Zbl

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

[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. | MR | Zbl

[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). | MR | Zbl

[22] J. Tate, Refining Gross’s conjecture on the values of abelian L-functions. To appear in Contemporary Math. | MR | Zbl

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

Cited by Sources: