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THE EQUIVALENCE OF RUBIN'S CONJECTURE AND THE ETNC/LRNC FOR CERTAIN BIQUADRATIC EXTENSIONS

Published online by Cambridge University Press:  13 August 2013

PAUL BUCKINGHAM
PAUL BUCKINGHAM*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB, T6G 2G1, Canada e-mail: p.r.buckingham@ualberta.ca
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Abstract

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For an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture (ETNC) at s = 0, which is equivalent to the Lifted Root Number Conjecture (LRNC), implies Rubin's Conjecture by work of Burns [3]. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin's Conjecture in turn implies the ETNC/LRNC. We conclude with some examples.

Keywords

Primary 11R42Secondary 11R70
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 335 - 353
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Bley, W., Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field, Doc. Math. 11 (2006), 73–118 (electronic).Google Scholar
2.Burns, D., Equivariant Tamagawa numbers and Galois module theory. I, Compositio Math. 129 (2) (2001), 203–237.Google Scholar
3.Burns, D., Congruences between derivatives of abelian L-functions at s=0. Invent. Math. 169 (3) (2007), 451–499.CrossRefGoogle Scholar
4.Burns, D., Congruences between derivatives of geometric L-functions, Invent. Math. 184 (2) (2011), 221–256 (with an appendix by Burns et al.).CrossRefGoogle Scholar
5.Burns, D., On derivatives of Artin L-series, Invent. Math. 186 (2) (2011), 291–371.Google Scholar
6.Burns, D. and Flach, M., Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570 (electronic).CrossRefGoogle Scholar
7.Burns, D. and Greither, C., On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2) (2003), 303–359.Google Scholar
8.Chinburg, T., On the Galois structure of algebraic integers and S-units, Invent. Math. 74 (3) (1983), 321–349.CrossRefGoogle Scholar
9.Curtis, C. W. and Reiner, I., Methods of representation theory, vol. II. Pure and applied mathematics (with applications to finite groups and orders) (John Wiley, New York, NY, 1987). A Wiley-Interscience Publication.Google Scholar
10.Dummit, D. S., Sands, J. W. and Tangedal, B.. Stark's conjecture in multi-quadratic extensions, revisited, J. Théor. Nombres Bordeaux 15 (1) (2003), 83–97. (Les XXIIèmes Journées Arithmetiques; Lille, 2001).Google Scholar
11.Emmons, C. J. and Popescu, C. D., Special values of abelian L-functions at s=0, J. Number Theory 129 (6) (2009), 1350–1365.Google Scholar
12.Flach, M., On the cyclotomic main conjecture for the prime 2, J. Reine Angew. Math. 661 (2011), 1–36.Google Scholar
13.Greither, C., Arithmetic annihilators and Stark-type conjectures, in Stark's conjectures: recent work and new directions, Contemporary Mathematics, vol. 358. (American Mathematical Society, Providence, RI, 2004), 55–78.CrossRefGoogle Scholar
14.Gruenberg, K. W., Ritter, J. and Weiss, A., A local approach to Chinburg's root number conjecture, Proc. London Math. Soc., 79 (1) (1999), 47–80.Google Scholar
15.Johnston, H. and Nickel, A., On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results. arXiv:1210.8298.Google Scholar
16.Kim, S. Y., On the equivariant Tamagawa number conjecture for quaternion fields, PhD thesis (King's College London, 2003).Google Scholar
17.Macias Castillo, D., On higher-order Stickelberger-type theorems for multi-quadratic extensions, Int. J. Number Theory 8 (1) (2012), 95–110.Google Scholar
18.Neukirch, J., Schmidt, A. and Wingberg, K., Grundlehren der mathematischen wissen-schaften (Fundamental principles of mathematical sciences), 2nd ed., vol. 323, Cohomology of number fields (Springer-Verlag, Berlin, Germany, 2008).Google Scholar
19.Popescu, C. D., On a refined Stark conjecture for function fields, Compositio Math. 116 (3) (1999), 321–367.Google Scholar
20.Popescu, C. D., Rubin's integral refinement of the abelian Stark conjecture, in Stark's conjectures: recent work and new directions, Contemporary Mathematics, vol. 358. (American Mathematical Society, Providence, RI, 2004), 1–35.Google Scholar
21.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 (American Mathematical Society, Providence, RI, 2011), 45–101.Google Scholar
22.Rubin, K., A Stark conjecture “over Z” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1) (1996), 33–62.CrossRefGoogle Scholar
23.Sands, J. W., Popescu's conjecture in multi-quadratic extensions, in Stark's conjectures: recent work and new directions, Contemporary Mathematics, vol. 358 (American Mathematical Society, Providence, RI, 2004), 127–141.Google Scholar
24.Serre, J.-P., Représentations linéaires des groupes finis (Hermann, Paris, France, 1967).Google Scholar
25.Stark, H. M., L-functions at s=1. IV. First derivatives at s=0, Adv. Math. 35 (3) (1980), 197–235.CrossRefGoogle Scholar
26.Swan, R. G., Algebraic K-theory, Lecture Notes in Mathematics, vol. 76 (Springer-Verlag, Berlin, Germany, 1968).Google Scholar
27.Swan, R. G., K-theory of finite groups and orders, Lecture Notes in Mathematics, vol. 149 (Springer-Verlag, Berlin, Germany, 1970).Google Scholar
28.Tate, J., The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J. 27 (1966), 709–719.Google Scholar
29.Tate, J., Les conjectures de Stark sur les fonctions L d'Artin en s = 0, Progress in Mathematics, vol. 47 (Bernardi, Dominique and Schappacher, Norbert, Editors) (Birkhäuser, Boston, MA, 1984).Google Scholar
30.Vallières, D., The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture (preprint, 2012).Google Scholar
31.Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, UK, 1994).CrossRefGoogle Scholar
32.Weiss, A., Multiplicative Galois module structure, Fields Institute Monographs, vol. 5 (American Mathematical Society, Providence, RI, 1996).Google Scholar