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The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Let p ≥ 5 be a prime number. We consider the Iwasawa λ-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic Zp-extension of Q. Let g be a p-ordinary cuspidal newform of weight 2 and trivial nebentype. We assume that the µ-invariant of g vanishes, and that the image of the residual representation associated to g is suitably large. We show that for any number greater n greater than or equal to the λ-invariant of g, there are infinitely many newforms f that are p-congruent to g, with λ-invariant equal to n. We also prove quantitative results regarding the levels of such modular forms with prescribed λ-invariant.

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