In this thesis we study Galois representations corresponding to abelian varietieswith certain reduction conditions. We show that these conditions force the imageof 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 isdense 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 badreduction at some prime ϕ, with the dimension of the toric part of the reductionequal 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 subsetof M(Q) corresponding to absolutely simple abelian varieties with a prescribed stablereduction at a large enough prime ϕ of F is dense in M(C) in the complextopology. In particular, the set of simple abelian varieties having bad reductionswith 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 reductionconditions. For example, if A is an abelian variety with End° (A) = Q andthe dimension of the toric part of its reduction is prime to dim( A), then MTholds.
(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 some4-folds of type I.
(5) For some abelian varieties either MT or the Hodge conjecture holds.
