• abelian varieties
• varietà abeliane
• Mumford-Tate
• Tate
• Hodge

The Hodge conjecture on the algebraicity of cohomology classes is probably one of the most well-known open problems in modern mathematics, and certainly a topic of central interest in analytic and algebraic geometry.

Despite the remarkable number of contributions, however, the conjecture still seems to be far from a complete solution, even for Abelian varieties, which are relatively well understood.

In this thesis we focus on the Hodge conjecture and related questions in the setting of Abelian varieties, which, while exhibiting most of the richness of the general problem, allows the development of specific tools that have led to important partial results. In this context it is rather natural to introduce the notion of an abstract Hodge structure, a rational vector space equipped with some extra structure at the level of $\mathbb{C}$-points.

In turn, to every Hodge structure we can associate a certain algebraic group, called its Mumford-Tate group, which allows a purely representation-theoretic description of Hodge classes. In principle, once the Mumford-Tate group of a Hodge structure is known, the computation of Hodge classes is reduced to a problem in invariant theory, and this is often enough to determine the whole Hodge ring, sometimes not just for the Abelian variety we started with but for its powers as well.

On the other hand, when the Abelian variety $A$ is defined over a number field $K$, another long-standing conjecture, formulated by Tate, makes predictions about the algebraicity of certain (étale) cohomology classes.

The striking similarities between these two seemingly unrelated statements led Mumford, Tate and Serre to conjecture that a close connection should exist between the Mumford-Tate group of $A$ and the action of the absolute Galois group of $K$ on the $\ell$-adic Tate module $T_\ell(A)$. This statement is now known as the Mumford-Tate conjecture and has an important part to play in this work.

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The final part of this thesis is based on a paper by Moonen and Zarhin, where a criterion is given for the existence, on simple Abelian varieties of dimension 4, of ``exceptional'' classes, namely classes that we expect to be algebraic but do not lie in the algebra generated by divisor classes.

The tools developed along the way to this theorem actually allow the analysis of many other cases, and following Ribet, Serre, Tanke'ev and many others we prove the Hodge and Tate conjectures for Abelian varieties satisfying various combinations of additional requirements on the dimension and on the endomorphism algebra.

After the truth of the so-called minuscule weights conjecture was established by Pink, it has become possible to give unified proofs that work equally well in the complex and $\ell$-adic case; also, in many circumstances the representation-theoretic properties of the Mumford-Tate group allow its precise determination, and the $\ell$-adic counterparts of the same arguments are enough to prove the Mumford-Tate conjecture by computing both sides of the predicted equality. Whenever possible, we try to adopt this kind of approach, in order to emphasize the similarities between the two conjectures.