## A generalization of Darmon’s conjecture for Euler systems for general $$p$$-adic representations.(English)Zbl 1296.11143

Summary: H. Darmon’s conjecture [Can. J. Math. 47, No. 2, 302–317 (1995; Zbl 0844.11071)] on a relation between cyclotomic units over real quadratic fields and certain algebraic regulators was recently solved by B. Mazur and K. Rubin [Compos. Math. 147, No. 1, 56–74 (2011; Zbl 1225.11144)] using their theory of Kolyvagin systems. In this paper, we formulate a “non-explicit” version of Darmon’s conjecture for Euler systems defined for general $$p$$-adic representations, and prove it. In the process of the proof, we introduce a notion of “algebraic Kolyvagin systems”, and develop their properties.

### MSC:

 11R29 Class numbers, class groups, discriminants 11R27 Units and factorization 11R37 Class field theory 11F85 $$p$$-adic theory, local fields

### Citations:

Zbl 0844.11071; Zbl 1225.11144
Full Text:

### References:

 [1] Burns, D.; Flach, M., Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math., 6, 501-570, (2001) · Zbl 1052.11077 [2] Burns, D.; Greither, C., On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math., 153, 303-359, (2003) · Zbl 1142.11076 [3] Darmon, H., Thaine’s method for circular units and a conjecture of Gross, Canad. J. Math., 47, 302-317, (1995) · Zbl 0844.11071 [4] Flach, M., On the cyclotomic main conjecture for the prime 2, J. Reine Angew. Math., 661, 1-36, (2011) · Zbl 1242.11083 [5] Gross, B., On the values of abelian L-functions at $$s = 0$$, J. Fac. Sci. Univ. Tokyo, 35, 177-197, (1988) · Zbl 0681.12005 [6] Kolyvagin, V. A., Euler systems, (The Grothendieck Festschrift, vol. II, (1990)), 435-483 · Zbl 0742.14017 [7] Mazur, B.; Rubin, K., Kolyvagin systems, Mem. Amer. Math. Soc., 168, 799, (2004) · Zbl 1055.11041 [8] Mazur, B.; Rubin, K., Refined class number formulas and Kolyvagin systems, Compos. Math., 147, 56-74, (2011) · Zbl 1225.11144 [9] Mazur, B.; Rubin, K., Controlling Selmer groups in the higher core rank case, (2013), preprint [10] Mazur, B.; Rubin, K., Refined class number formulas for $$\mathbb{G}_m$$, (2013), preprint [11] Rubin, K., A Stark conjecture “over $$\mathbb{Z}$$” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble), 46, 33-62, (1996) · Zbl 0834.11044 [12] Rubin, K., Euler systems, Ann. of Math. Stud., vol. 147, (2000), Princeton Univ. Press [13] Sano, T., Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns, Compos. Math., (2014), in press · Zbl 1311.11108 [14] Tate, J., Relations between $$K_2$$ and Galois cohomology, Invent. Math., 36, 257-274, (1976) · Zbl 0359.12011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.