Introduction to motives

This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program “number theory and physics” at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [50] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [42] and Kleiman [36], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [11], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry

Euler characteristic and congruences of elliptic curves

Given two elliptic curves over Q that have good ordinary reduction at an odd prime p, and have equivalent, irreducible mod p Galois representations, we study congruences between the Euler characteristics and special L-values over certain noncommutative extensions of Q

Links between cyclotomic and GL2 Iwasawa theory

We study, in the case of ordinary primes, some connections between the GL2 and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication

Non-commutative Iwasawa theory for modular forms

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.Comment: 40 page

Introduction to motives

This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program “number theory and physics” at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [50] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [42] and Kleiman [36], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [11], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry

STRUCTURE OF FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

This paper studies fine Selmer groups of elliptic curves in abelian p-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Zp-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified

Euler-Poincaré characteristics of abelian varieties

Let F be a finite extension of Q , and let A be an abelian variety defined over F. Let p be a prime, and let Ap∞ be the Galois module of points of A of order a power of p. Let G denote the Galois group of the extension F(Ap∞)/F. Suppose that G contains no element of order p. Then the cohomology groups Hi(G, Ap∞) are finite for all i ≥ 0, and zero for i sufficiently large; let ×(G, Ap∞) be the alternating product of their orders. The principal result of this Note is that we have x(G, Ap∞) = 1