Hodge groups of abelian varieties with purely multiplicative reduction

Abelian Varieties Having Purely Additive Reduction

Let E be an elliptic curve over a field K with a discrete valuation v with residue class field k. Suppose E has 'additive reduction' at o, i.e. the connected component A ° of the special fibre A0 of the N6ron minimal model is isomorphic to Q . Then the order of Ao(k)/A°(k) is at most 4 as can be seen by inspection of the usual tables, cf. [9, pp. 124-125] and [5, p. 46]. Thus it follows that if...

Hodge cycles on abelian varieties

This is a TeXed copy of – Hodge cycles on abelian varieties (the notes of most of the seminar “Périodes des Intégrales Abéliennes” given by P. Deligne at I.H.E.S., 1978–79; pp9– 100 of Deligne et al. 1982). somewhat revised and updated. See the endnotes1 for more details.

Character Varieties of Abelian Groups

We prove that for every reductive group G with a maximal torus T and Weyl group W there is a natural normalization map χ from T/W to an irreducible component, X 0 G(Z ), of the G-character variety XG(Z ) of Z . We prove that this component is reduced and that χ is an isomorphism for classical groups. Additionally, we prove that even though there are no irreducible representations of Z into redu...

Good reduction of abelian varieties

Hodge structures on abelian varieties of type III

We show that the usual Hodge conjecture implies the general Hodge conjecture for certain abelian varieties of type III, and use this to deduce the general Hodge conjecture for all powers of certain 4-dimensional abelian varieties of type III. We also show the existence of a Hodge structure M such that M occurs in the cohomology of an abelian variety, but the Tate twist M(1) does not occur in th...