Abstract
Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let r>1 be an odd integer. The p-adic Beilinson conjecture relates the values at s=r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair (h^0(mathrm {Spec}(E))(r), mathbb {Z}[G]). If r>1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.
Highlights
Let E/K be a finite Galois extension of number fields with Galois group G and let r be an integer
The equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Spec(E))(r ), Z[G]) as formulated by Burns and Flach [17] asserts that a certain canonical element T (E/K, r ) in the relative algebraic K -group K0(Z[G], R) vanishes
If r = 0 this might be seen as a vast generalization of the analytic class number formula for number fields, and refines Stark’s conjecture for E/K as discussed by Tate in [75] and the ‘Strong Stark conjecture’ of Chinburg [25, Conjecture 2.2]
Summary
Let E/K be a finite Galois extension of number fields with Galois group G and let r be an integer. The equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Spec(E))(r ), Z[G]) as formulated by Burns and Flach [17] asserts that a certain canonical element T (E/K , r ) in the relative algebraic K -group K0(Z[G], R) vanishes. This element relates the leading terms at s = r of Artin L-functions to natural arithmetic invariants. If r = 0 this might be seen as a vast generalization of the analytic class number formula for number fields, and refines Stark’s conjecture for E/K as discussed by Tate in [75] and the ‘Strong Stark conjecture’ of Chinburg [25, Conjecture 2.2]
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