Duality of Selmer groups of an abelian variety over a number field
Saikat Biswas
TL;DR
The paper establishes a duality between Selmer groups attached to the $p$-torsion $A[p]$ of an abelian variety $A$ over a number field $K$ and the dual $A^{\vee}[p]$, by comparing Euler characteristics of different Selmer structures. It introduces three local Selmer structures on $A[p]$ and proves a global identity: $\chi_{(S)}(K,A[p])/\chi_{\mathrm{Sel}_p}(K,A[p]) = \#\bigoplus_{v\in S}\Phi_{A^{\vee},v}(k_v)[p]$, equivalently relating $\#\mathrm{Sel}_p(A/K)$ and $\#\mathrm{Sel}_p(A^{\vee}/K)$ up to the same product of $p$-torsion component groups. The approach combines local Tate duality, Tamagawa numbers, and Cassels–Poitou– Tate dual sequences to obtain an explicit, computable factorization of the global duality into local contributions, with applications to the elliptic-curve case and semistable Jacobians via Tate’s algorithm, monodromy pairing, or intersection pairings. The results provide a refined framework for understanding Selmer-rank behavior under duality and offer an alternative proof via a dual exact sequence. $A$ and $A^{\vee}$, $p$, and the component groups appear prominently throughout the formulas, highlighting the arithmetic of Tamagawa numbers in Selmer-duality.
Abstract
Let $A$ be an abelian variety defined over a number field $K$ and let $A^{\vee}$ be the dual abelian variety. For an odd prime $p$, we consider two Selmer groups attached to $A[p]$ and relate the orders of these groups along with those of their corresponding duals to the order of the component groups of $A^{\vee}$ at primes $v$.