On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

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 $$r>1$$ r > 1 be an odd integer. The p-adic Beilinson conjecture relates the values at $$s=r$$ 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 $$(h^0(\mathrm {Spec}(E))(r), \mathbb {Z}[G])$$ ( h 0 ( Spec ( E ) ) ( r ) , Z [ G ] ) . If $$r>1$$ r > 1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

A representative of RΓ(N,T) for higher dimensional twists of ℤpr(1)

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

On the relative K-group in the ETNC, Part II

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

LetAbe an abelian variety defined over a number fieldkand letFbe a finite Galois extension ofk. Letpbe a prime number. Then under certain not-too-stringent conditions onAandFwe compute explicitly the algebraic part of thep-component of the equivariant Tamagawa number of the pair(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of thep-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible byp. More generally, our approach leads us to the formulation of certain precise families of conjecturalp-adic congruences between the values ats=1of derivatives of the Hasse–WeilL-functions associated to twists ofA, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field{\mathbb{Q}}, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].

Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture

Let