Arithmetic on Abelian and Kummer Varieties

Conditionally on the finiteness of relevant Shafarevich–Tate groups we establish the Hasse principle for certain families of Kummer surfaces.. For i = 1, 2 assume the finiteness of

This section, where an algebraically closed field k of characteristic p, and a prime l different from p are fixed, is devoted to the characterization of l-adic theta

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For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K n A is birationally equivalent to another irreducible symplectic

We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension)

main point in their proof is the following geometrical consequence of these equations: Let P be a point of multiplicity 2 of the theta divisor 03B8 of a (complex)

The linear algebra is rather complicated, because it involves real/complex vector spaces as well as p-adic vector spaces for all p, together with some compatibility among them, but

When E is an imaginary quadratic field and B = A E for a CM elliptic curve A/Q, Theorem 1.1 is a consequence (see Rubin [3]) of the aforementioned result of Bertrand together