Abstract
AbstractThe notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.
Highlights
The Iwasawa theory of Galois representations, especially those arising from elliptic curves and Hecke eigencuspforms, affords deep insights into the arithmetic of such objects
Two elliptic curves E1 and E2 over F are said to be p-congruent if their associated residual representations are isomorphic, i.e., E1[p] and E2[p] are isomorphic as Galois modules. It is of particular interest in Iwasawa theory to study the relationship between Iwasawa invariants of the Selmer groups of p-congruent elliptic curves
Leveraging our results on μ-invariants and imprimitive λ-invariants, we prove congruence relations for the truncated Euler characteristics of multisigned Selmer groups of the p-congruent elliptic curves E1 and E2
Summary
The Iwasawa theory of Galois representations, especially those arising from elliptic curves and Hecke eigencuspforms, affords deep insights into the arithmetic of such objects. When E has good ordinary reduction at the prime p, it was conjectured by Mazur that the Selmer group Sel(E/Qcyc) is cotorsion as a module over the Iwasawa algebra Zp[[Γ]]. This is a celebrated theorem of Kato [12]. It is of particular interest in Iwasawa theory to study the relationship between Iwasawa invariants of the Selmer groups of p-congruent elliptic curves Such investigations were initiated by Greenberg and Vatsal [8], who considered p-congruent, p-ordinary elliptic curves E1 and E2 defined over Q. Leveraging our results on μ-invariants and imprimitive λ-invariants, we prove congruence relations for the truncated Euler characteristics of multisigned Selmer groups of the p-congruent elliptic curves E1 and E2.
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