Selmer groups and class groups

摘要

Let A be an abelian variety over a global field K of characteristic p >= 0. If A has nontrivial (respectively full) K-rational l-torsion for a prime 1 not equal p, we exploit the fppf cohomological interpretation of the l-Selmer group Sel(l) Lambda to bound # Sel(l) Lambda from below (respectively above) in terms of the cardinality of the l-torsion subgroup of the ideal class group of K. Applied over families of finite extensions of K, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of l-ranks of class groups of quadratic extensions of every K containing a fixed finite field Fp(n) (depending on l). For number fields, it suggests a new approach to the Iwasawa mu = 0 conjecture through inequalities, valid when A(K)[l] not equal 0, between Iwasawa invariants governing the growth of Selmer groups and class groups in a Z(l)-extension.