A Stark conjecture “over 𝐙 ” for abelian L -functions with multiple zeros

Karl Rubin

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 1, page 33-62
  • ISSN: 0373-0956

Abstract

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Suppose K / k is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit ϵ of K such that the special values L ' ( χ , 0 ) for all characters χ of Gal / ( K / k ) can be expressed as simple linear combinations of the logarithms of the different absolute values of ϵ .In this paper we formulate an extension of this conjecture, to attempt to understand the values L ( r ) ( χ , 0 ) when the order of vanishing r may be greater than one. This conjecture no longer predicts the existence of individual special global units, but rather of special elements in an exterior power of the Galois module of global units (or S -units). We also discuss connections between this conjecture, class number formulas, and Euler systems.

How to cite

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Rubin, Karl. "A Stark conjecture “over ${\bf Z}$” for abelian $L$-functions with multiple zeros." Annales de l'institut Fourier 46.1 (1996): 33-62. <http://eudml.org/doc/75174>.

@article{Rubin1996,
abstract = {Suppose $K/k$ is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit $\varepsilon $ of $K$ such that the special values $L^\{\prime \}(\chi ,0)$ for all characters $\chi $ of $\{\rm Gal\}/(K/k)$ can be expressed as simple linear combinations of the logarithms of the different absolute values of $\varepsilon $.In this paper we formulate an extension of this conjecture, to attempt to understand the values $L^\{(r)\}(\chi ,0)$ when the order of vanishing $r$ may be greater than one. This conjecture no longer predicts the existence of individual special global units, but rather of special elements in an exterior power of the Galois module of global units (or $S$-units). We also discuss connections between this conjecture, class number formulas, and Euler systems.},
author = {Rubin, Karl},
journal = {Annales de l'institut Fourier},
keywords = {Stark's conjecture; -functions; global units; Euler systems},
language = {eng},
number = {1},
pages = {33-62},
publisher = {Association des Annales de l'Institut Fourier},
title = {A Stark conjecture “over $\{\bf Z\}$” for abelian $L$-functions with multiple zeros},
url = {http://eudml.org/doc/75174},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Rubin, Karl
TI - A Stark conjecture “over ${\bf Z}$” for abelian $L$-functions with multiple zeros
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 1
SP - 33
EP - 62
AB - Suppose $K/k$ is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit $\varepsilon $ of $K$ such that the special values $L^{\prime }(\chi ,0)$ for all characters $\chi $ of ${\rm Gal}/(K/k)$ can be expressed as simple linear combinations of the logarithms of the different absolute values of $\varepsilon $.In this paper we formulate an extension of this conjecture, to attempt to understand the values $L^{(r)}(\chi ,0)$ when the order of vanishing $r$ may be greater than one. This conjecture no longer predicts the existence of individual special global units, but rather of special elements in an exterior power of the Galois module of global units (or $S$-units). We also discuss connections between this conjecture, class number formulas, and Euler systems.
LA - eng
KW - Stark's conjecture; -functions; global units; Euler systems
UR - http://eudml.org/doc/75174
ER -

References

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  2. [2] P. DELIGNE, K. RIBET, Values of abelian L-functions at negative integers over totally real fields, Invent. Math., 59 (1980), 227-286. Zbl0434.12009MR81m:12019
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  5. [5] M. KRASNER, Sur la représentation exponentielle dans les corps relativement galoisiens de nombers p-adiques, Acta Arith., 3 (1939), 133-173. JFM65.0113.01
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  8. [8] K. RUBIN, Stark units and Kolyvagin's Euler systems, J. für die reine und angew. Math., 425 (1992), 141-154. Zbl0752.11045MR93d:11117
  9. [9] J. SANDS, Stark's conjecture and abelian L-functions with higher order zeros at s = 0, Advances in Math., 66 (1987), 62-87. Zbl0631.12006MR89g:11110
  10. [10] H. STARK, L-functions at s = 1 I, II, III, IV, Advances in Math., 7 (1971), 301-343, 17 (1975), 60-92, 22 (1976), 64-84, 35 (1980), 197-235. Zbl0475.12018
  11. [11] J. TATE, Les conjectures de Stark sur les fonctions L d'Artin en s = 0, Prog. in Math., 47, Boston, Birkhäuser (1984). Zbl0545.12009
  12. [12] D. S. RIM, An exact sequence in Galois cohomology, Proc. Amer. Math. Soc., 16 (1965), 837-840. Zbl0166.30604MR31 #3480
  13. [13] R. SWAN, K-theory of finite groups and orders, Lecture notes in Math., 149, New York, Springer (1970). Zbl0205.32105MR46 #7310

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