On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.

On the local Tamagawa number conjecture for Tate motives

There is a wonderful conjecture of Bloch and Kato that generalizes both the analytic Class Number Formula and the Birch and Swinnerton-Dyer conjecture. The conjecture itself was generalized by Fukaya and Kato to an equivariant formulation. In this thesis, I provide a new proof for the equivariant local Tamagawa number conjecture in the case of Tate motives for unramified fields, using Iwasawa theory and (φ,Γ)-modules, and provide some work towards extending the proof to tamely ramified fields.

[Exhibition review] Keith Tyson at Tate Britain

This is a review of an exhibition of the work of the twenty-first century artist, Keith Tyson, who specializes in mathematics. He was short-listed for the Turner Prize and his work is included in the exhibition of nominated artists' work at Tate Britain. [Keith Tyson was announced as the winner of the 2002 Turner prize on 8 December 2002.]

Discurs del Dr. Robert Brian Tate com a doctor honoris causa per la Universitat de Girona

Discurs pronunciat pel Dr. Robert Brian Tate (Belfast, 1921), en el decurs de l'acte d'investidura de Doctors Honoris Causa, celebrat a la Universitat de Girona l'octubre de 2004. El seu discurs versa sobre la seva trajectòria professional

Discurs de presentació del professor Robert Brian Tate, a càrrec de la Dra. Mariàngela Vilallonga

En el discurs de concessió del doctorat honoris causa de la Universitat de Girona, la Dra. Mª Vilallonga glossa la lliçó d'història raonada i el llegat de Robert Brian Tate

Discurs del rector de la Universitat de Girona, Dr.Joan Batlle, en l'acte d'investidura com doctors honoris causa de Robert Brian Tate, Joan Bertran Rusca i Jaume Gil Aluja

Discurs d'investidura de doctors honoris causa per la Universitat de Girona, del rector Joan Batlle

Marques de biblioteca: Robert Brian Tate

Explicació del motius que Robert Brian Tate podia tenir per a triar l’humanista quatrecentista Joan Margarit i Pau com a figura del seu ex-libris

Entre el espectáculo y el discurso pedagógico : La Tate Modern.

El artículo forma parte de un monográfico dedicado a la hibridación en las artes plásticas.- Resumen tomado parcialmente de la revista.