Number Conjecture Research Articles

Computation of Stark-Tamagawa units

Let K be a totally real number field and let l denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the Equivariant Tamagawa Number Conjecture for the Q[G]-equivariant motive h0(Spec(L)), where L/K is a cyclic extension of degree l and group G. This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet L-functions.

Tamagawa numbers for motives with (noncommutative) coefficients, II

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

On the Equivariant Tamagawa number conjecture for Tate motives

Let L be a finite abelian extension of ℚ and let K be any subfield of L. For each integer r with r≤0 we prove the Equivariant Tamagawa Number Conjecture for the pair (h 0(Spec(L))(r),ℤ[½][Gal(L/K)]).

Toward equivariant Iwasawa theory

Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G ∞ , X ∞ the Galois groups of K ∞ /k and M ∞ /K ∞ , respectively, where K ∞ is the cyclotomic l-extension of K and M ∞ the maximal abelian S-ramified l-extension of K ∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G ∞ ]]-module X ∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when ll[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l.

On the Tamagawa Number Conjecture for CM Elliptic Curves Defined Over ?

In this paper we prove, under the assumption that the Soulé regulator map is injective, that, for all integers k ⩾0, the description by the local Tamagawa number conjecture for CM elliptic curves defined over Q , corresponding to the values of their L -functions at k +2, is true.

On the Tamagawa Number Conjecture for CM Elliptic Curves Defined Over [formula omitted

In this paper we prove, under the assumption that the Soulé regulator map is injective, that, for all integers k⩾0, the description by the local Tamagawa number conjecture for CM elliptic curves defined over Q, corresponding to the values of their L-functions at k+2, is true.

The lifted root number conjecture and Iwasawa theory

Introduction The Tripod Restriction, deflation change of maps, and variance with $S$ Definition of $\mho_S$ $\Omega_\Phi$ as a shadow of $\mho_S$ $\mho_S$ over the maximal order in the case when $G$is abelian Local considerations Towards a representing homomorphism for $\Omega_{\varphi_{\mathcal L}}$ Real cyclotomic extensions tame over $1$ References.

The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems

The lifted root number conjecture for tamely ramified Galois extensions of odd prime degree over the rationals is proved. The main ingredients are as follows: extracting roots of some explicit circular units of the corresponding genus field (the trees are used as a bookkeeping device) and Euler systems.

The Tamagawa number conjecture for CM elliptic curves

In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the $p$-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of $p^r$-torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soul\'e elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology.

Equivariant Tamagawa numbers, fitting ideals and Iwasawa theory

Let L/K be a finite Galois extension of number fields of group G. In [4] the second named author used complexes arising from etale cohomology of the constant sheaf ℤ to define a canonical element TΩ(L/K) of the relative algebraic K-group K 0(ℤ[G],ℝ). It was shown that the Stark and Strong Stark Conjectures for L/K can be reinterpreted in terms of TΩ(L/K), and that the Equivariant Tamagawa Number Conjecture for the ℚ[G]-equivariant motive h 0(Spec L) is equivalent to the vanishing of TΩ(L/K). In this paper we give a natural description of TΩ(L/K) in terms of finite G-modules and also, when G is Abelian, in terms of (first) Fitting ideals. By combining this description with techniques of Iwasawa theory we prove that TΩ(L/ℚ) vanishes for an interesting class of Abelian extensions L/ℚ.

Tamagawa numbers for motives with (non-commutative) coefficients

Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple \mathbb Q -algebra A . We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the A -equivariant L -function of M . This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order \mathfrak A in A for which there exists a 'projective \mathfrak A -structure' on M . The existence of such a structure is guaranteed if \mathfrak A is a maximal order, and also occurs in many natural examples where \mathfrak A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.

The Bloch-Kato conjecture for adjoint motives of modular forms

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

A Local Approach to Chinburg's Root Number Conjecture

A Local Approach to Chinburg's Root Number Conjecture

The Lifted Root Number Conjecture for some cyclic extensions of ℚ

The Lifted Root Number Conjecture for some cyclic extensions of ℚ

The Total Chromatic Number of Graphs of High Minimum Degree

If G is a simple graph with minimum degree (G) satisfying (G) f(|V(G|+1) the total chromatic number conjecture holds; moreover if (G) |V(G| then T(G) (G)+3. Also if G has odd order and is regular with d{G) 7|(G)| then a necessary and sufficient condition for T(G) = (G)+1 is given.