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Search: All articles in the CJM digital archive with keyword abelian varieties

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1. CJM Online first

Murty, V. Kumar; Patankar, Vijay M.
 Tate Cycles on Abelian Varieties with Complex Multiplication We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complex multiplication. We show that there is an effective bound $C = C(A,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $\leq C$. We also show that for a set of primes $v$ of $K$ of density $1$, the space of Tate cycles on the special fibre $A_v$ of the NÃ©ron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself. Keywords:Abelian varieties, complex multiplication, Tate cyclesCategories:11G10, 14K22

2. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
 Modular Abelian Varieties Over Number Fields The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting. Keywords:Modular abelian varieties, $GL_2$-type varieties, modular formsCategories:11G10, 11G18, 11F11

3. CJM 2012 (vol 65 pp. 403)

Van Order, Jeanine
 On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban. Keywords:Iwasawa theory, Hilbert modular forms, abelian varietiesCategories:11G10, 11G18, 11G40