3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Selmer Groups of Abelian Varieties in Extensions of Function Fields

Let k be a field of characteristic q, C a smooth connected curve defined over k with function field K := k(C). Let A/K be a non constant abelian variety defined over K of dimension d. We assume that q = 0 or > 2d + 1. Let p 6= q be a prime number and C → C a finite geometrically Galois and étale cover defined over k with function field K ′ := k(C). Let (τ , B) be the K /k-trace of A/K. We give ...

Isogeny Classes of Abelian Varieties over Function Fields

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of d...

Isogeny classes of Hilbert-Blumenthal abelian varieties over finite fields

This paper gives an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L,Q). Finally, let (X/k, ι, α) be a...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

Large Selmer groups over number fields

Let p be a prime number and M a quadratic number field, M , Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least pd.