Hecke algebras, Galois representations, and abelian varieties

1.The first part of this thesis treats Hecke algebras for linear algebraic groups over either a number field or a non-archimedean local field of characteristic zero. We study to what extent the representation theory of these Hecke algebras determines the field.
2.The second part of the thesis is concerned with Galois representations attached to three-dimensional abelian varieties. Given a suitable such variety, we give an algorithm which yields a list of prime numbers q for which the corresponding (q-residual) Galois representation is surjective. Conversely, give na prime number q, we construct a principally polarised three-dimensional abelian variety whose (q-residual) Galois representation is surjective.
3.In the third part of the thesis we consider supersingular abelian varieties defined over finite fields, and formulate a characterisation of such varieties, based on the eigenvalues of its Frobenius endomorphism.