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Selmer groups and class groups

  • Kęstutis Česnavičius (a1)

Abstract

Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$ . If $A$ has nontrivial (respectively full) $K$ -rational $l$ -torsion for a prime $l\neq p$ , we exploit the fppf cohomological interpretation of the $l$ -Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ 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 $\mathbb{F}_{p^{n}}$ (depending on $l$ ). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$ , between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$ -extension.

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References

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Selmer groups and class groups

  • Kęstutis Česnavičius (a1)

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