Let M be a motive which is defined over number field and admits an action of finite dimensional semisimple \mathbb Q -algebra A . We formulate study conjecture for the leading coefficient Taylor expansion at 0 -equivariant L -function This simultaneously generalizes refines Tamagawa Bloch, Kato, Fontaine, Perrin-Riou et al. also central conjectures classical Galois module theory as developed by Fröhlich, Chinburg, M. The precise formulation our depends upon choice order \mathfrak in there...

Let f be a newform of weight k⩾2, level N with coefficients in number field K, and A the adjoint motive M associated to f. We carefully discuss construction realisations A, as well natural integral structures these realisations. then use method Taylor Wiles verify λ-part Tamagawa conjecture Bloch Kato for L(A,0) L(A,1). Here λ is any prime K not dividing Nk!, so that mod representation absolutely irreducible when restricted Galois group over Q((−1)(ℓ−1)/2ℓ) where λ|ℓ. The also establishes...

We establish the equivalence of two definitions invariants measuring Galois module structure K-groups rings integers in number fields (due to Chinburg et al. on one hand and authors other). also make some remarks concerning possibility yet another such definition via Lichtenbaum complexes.

We provide corroborative evidence for the equivariant Tamagawa number conjecture which was formulated in first part of this article.

Abstract We establish various properties of the definition cohomology topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence continuity theorem for compact groups. use these to compute Weil group totally imaginary field, Weil-étale topology number ring recently introduced Lichtenbaum (both with integer coefficients).

We complete the proof of equivariant Tamagawa number conjecture for Tate motives over absolutely abelian fields by proving a refined cyclotomic main at prime 2.

We give a conjectural description of the vanishing order and leading Taylor coefficient Zeta function proper, regular arithmetic scheme \mathcal{X} at any integer n in terms Weil-étale cohomology complexes. This extends work S. Lichtenbaum [Compos. Math. 141, No. 3, 689–702 (2005; Zbl 1073.14024)] T. Geisser [Math. Ann. 330, 4, 665–692 (2004; 1069.14021)] for characteristic p , [Ann. (2) 170, 2, 657–683 (2009; 1278.14029)] \mathcal{X}=\mathrm{Spec}(\mathcal{O}_F) n=0 where F is number field,...

We define and study a Weil-étale topos for any regular, proper scheme \mathcal X over \mathrm{Spec}(\mathbb Z) which has some of the properties suggested by Lichtenbaum such topos. In particular, cohomology with \tilde{\mathbb R} -coefficients expected relation to \zeta(\mathcal X,s) at s=0 if Hasse–Weil L-functions L(h^i(\mathcal X_{\mathbb Q}),s) have meromorphic continuation functional equation. If characteristic p \mathbb Z also our groups recover those previously studied Geisser.

We illustrate the use of Iwasawa theory in proving cases (equivariant) Tamagawa number conjecture.

Suppose that M is a finite module under the Galois group of local or global field. Ever since Tate's papers [17, 18], we have had simple and explicit formula for Euler–Poincaré characteristic cohomology M. In this note are interested in refinement when also carries an action some algebra A, commuting with (see Proposition 5.2 Theorem 5.1 below). This naturally takes shape identity relative K-group attached to A Section 2). We shall deduce such whenever ordinary Euler characteristic, key step...

We prove that the special value conjecture for Zeta function \zeta(\mathcal{X},s) of a proper, regular arithmetic scheme \mathcal{X} we formulated in [ M. Flach and B. Morin , Doc. Math. 23, 1425–1560 (2018; Zbl 1404.14024)] is compatible with functional equation provided rational factor C(\mathcal{X},n) were not able to compute previously has simple explicit form given introduction below.

Abstract We prove that the special-value conjecture for zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to Birch and Swinnerton-Dyer Jacobian generic fibre. There are two key results proof. The first triviality correction factor [6, Conjecture 5.12], which we show arbitrary regular proper schemes. In proof need develop some eh-topology on schemes over finite fields might be independent interest. second result different formula due Geisser,...

The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers L-function associated to a motive over Q.Let f be newform weight k ≥ 2, level N with coefficients in field K. Let M let A adjoint .Let λ finite prime We verify λ-part Bloch-Kato for L(A, 0) 1) when Nk! mod representation is absolutely irreducible restricted Galois group Q (-1) ( -1)/2 where | .

We conjecture the existence of a long exact sequence relating Deninger's conjectural cohomology to Weil-Arakelov cohomology, latter being unconditionally defined. prove this for smooth projective varieties over finite fields whose Weil-etale motivic groups are finitely generated. Then we explain consequences that such an would have.

The aim of this paper is to complement results by Wolfart [ 14 ] about algebraic values the classical hypergeometric series for rational parameters a, b, c and arguments z . essentially determines set ∈ ℚ, ℚ which F(a, c; z) indicates, in a joint with F. Beukers[ 1 ], that some these can be expressed terms special modular forms. This method yields few strikingly explicit identities like but it does not give general statements nature question. In we identify as generator Kummer extension...

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of (global) conjecture on motivic $L$-functions with functional equation. was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ Bloch Kato. We use theory $(\phi, \Gamma_K)$-modules a reciprocity law due to Cherbonnier Colmez provide new proof in case extensions, prove motive $\mathbb{Q}_p(2)$ certain tamely ramified extensions.

We give a conjectural description of the vanishing order and leading Taylor coefficient Zeta function proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms Weil-\'etale cohomology complexes. This extends work Lichtenbaum \cite{Lichtenbaum05} Geisser \cite{Geisser04b} for characteristic $p$, \cite{li04} $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ $n=0$ where $F$ is number field, second author arbitrary \cite{Morin14}. show that our conjecture compatible with Tamagawa...