Class Groups and Selmer Groups

Abstract

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves.

Computing a Selmer group of a Jacobian using functions on the curve

“…Both statements can be shown using the snake lemma and the fact that J(K s ) contains a subgroup of finite index isomorphic to g copies of the ring of integers in K s (see [Ma] and [Sc2,Lemma 3.8,Prop. 3.9]).…”

Root numbers and parity of ranks of elliptic curves

“…(The second equality is an elementary computation using the usual filtration on E(K); e.g. it is spelled out for abelian varieties in [23] Lemma 3.8. )…”

On the height constant for curves of genus two, II

“…For the applications, we will also need the following somewhat technical lemma. 4 ) and y = (y 1 , y 2 , y 3 , y 4 ) be homogeneous coordinates for points on K and suppose that B(x, y) = 0. Let w = (w 1 , w 2 , w 3 , w 4 ) and z = (z 1 , z 2 , z 3 , z 4 ) be homogeneous coordinates such that…”

“…Similarly, we let ∞ (P ) = log max{|δ j (x)| | j ∈ {1, 2, 3, 4}} − 4 log max{|x 1 |, |x 2 |, |x 3 |, |x 4 |}, where again (x 1 , x 2 , x 3 , x 4 ) are homogeneous coordinates for the image of P on K.…”

“…where x = (x 1 , x 2 , x 3 , x 4 ) is some set of Kummer coordinates of P (meaning projective coordinates for the image of P on K). Our main result is the following.…”