Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T04:28:40.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  27 August 2014

T. Sano
T. Sano*
Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We formulate a conjecture which generalizes Darmon’s ‘refined class number formula’. We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the ‘except $2$-part’ of Darmon’s conjecture, which was first proved by Mazur and Rubin.

Keywords

refined abelian Stark conjectureRubin–Stark conjectureGross’s conjectureDarmon’s conjectureequivariant Tamagawa number conjecture

MSC classification

Primary: 11R42: Zeta functions and $L$-functions of number fields
Secondary: 11R27: Units and factorization 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 11 , November 2014 , pp. 1809 - 1835
Copyright
© The Author 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burns, D., Congruences between derivatives of abelian L-functions at s = 0, Invent. Math. 169 (2007), 451–499.Google Scholar
Burns, D. and Flach, M., Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570.Google Scholar
Burns, D. and Greither, C., On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), 303–359.CrossRefGoogle Scholar
Darmon, H., Thaine’s method for circular units and a conjecture of Gross, Canad. J. Math. 47 (1995), 302–317.CrossRefGoogle Scholar
Flach, M., On the cyclotomic main conjecture for the prime 2, J. Reine Angew. Math. 661 (2011), 1–36.Google Scholar
Gross, B., On the values of abelian L-functions at s = 0, J. Fac. Sci. Univ. Tokyo 35 (1988), 177–197.Google Scholar
Hayward, A., A class number formula for higher derivatives of abelian L-functions, Compositio Math. 140 (2004), 99–129.Google Scholar
Kolyvagin, V. A., Euler systems, in The Grothendieck Festschrift, vol. II (Birkhäuser, Boston, 1990), 435–483.Google Scholar
Mazur, B. and Rubin, K., Kolyvagin systems, Mem. Amer. Math. Soc. 168(799) (2004).Google Scholar
Mazur, B. and Rubin, K., Refined class number formulas and Kolyvagin systems, Compositio Math. 147 (2011), 56–74.CrossRefGoogle Scholar
Mazur, B. and Rubin, K., Refined class number formulas for $\mathbb{G}_{m}$, Preprint (2013),arXiv:1312.4053v1.Google Scholar
Popescu, C. D., Integral and p-adic refinements of the Abelian Stark conjecture, in Arithmetic of L-functions, IAS/Park City Mathematics Series, vol. 18, eds Popescu, C., Rubin, K. and Silverberg, A. (American Mathematical Society, Providence, RI, 2011), 45–101.CrossRefGoogle Scholar
Rubin, K., The main conjecture, in Appendix to cyclotomic fields I and II, Graduate Texts in Mathematics, vol. 121, ed. Lang, S. (Springer, Berlin, 1990), 397–419.Google Scholar
Rubin, K., A Stark conjecture ‘over ℤ’ for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), 33–62.Google Scholar
Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, Berlin, 1979).Google Scholar
Tate, J., Les conjectures de Stark sur les fonctions L d’Artin en s = 0, Progress in Mathematics, vol. 47 (Birkhäuser, Boston, MA, 1984).Google Scholar