Euler Characteristics and their Congruences for Multi-signed Selmer Groups

Ray, Anwesh; Sujatha, R.

doi:10.48550/arxiv.2011.05387

Cited by 4 publications

(5 citation statements)

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“…These assumptions apply only to the smaller set of primes Ω 0 and not to the set of all primes Σ 0 . The approach used here is similar to that of the second named author and R. Sujatha, who prove such results for the cyclotomic Z p -extension in [30].…”

Section: Introductionsupporting

confidence: 60%

Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida's Formula

Kundu,

Ray

2021

Preprint

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This paper aims at studying the Iwasawa λ-invariant of the p-primary Selmer group. We study the growth behaviour of p-primary Selmer groups in p-power degree extensions over non-cyclotomic Zp-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic Zp-extension, where all primes are finitely decomposed; in the Zp-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refinements of the results of R. Greenberg-V. Vatsal and K. Kidwell. As an application, we provide an algorithm for constructing elliptic curves with large anticyclotomic λ-invariant. Our results are illustrated by explicit computation.

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Section: Introductionsupporting

confidence: 60%

Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida's Formula

Kundu,

Ray

2021

Preprint

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“…When E has good ordinary reduction at p, this follows from Mazur's control theorem, see [21] or [13, Theorem 1.2]. In the good supersingular reduction case, the result follows from the proof of [10, Lemma 3.9] (see also [28,Proposition 5.1]).…”

Section: Results For a Fixed Elliptic Curve And Varying Primementioning

confidence: 99%

Statistics for Iwasawa invariants of elliptic curves

Kundu,

Ray

2021

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“…In this section, we recall the notion of the truncated Euler characteristic and its relationship with Iwasawa invariants. The reader may consult [2,34,24,25] for a more detailed introduction to the topic. Let E /Q be an elliptic curve and p an odd prime at which E has good ordinary reduction.…”

Section: The Truncated Euler Characteristicmentioning

confidence: 99%

“…If rank E(F ) ≤ 1, the kernel and cokernel of Φ are both finite. Moreover, when rank E(Q) > 1, there is an explicit criterion for the kernel and cokernel of Φ to be finite, see [25,Lemma 3.3…”

Section: The Truncated Euler Characteristicmentioning

confidence: 99%

Arithmetic statistics and diophantine stability for elliptic curves

Ray¹

2021

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We study the growth and stability of the Mordell-Weil group and Tate-Shafarevich group of an elliptic curve defined over the rationals, in various cyclic Galois extensions of prime power order. Mazur and Rubin introduced the notion of diophantine stability for the Mordell-Weil group an elliptic curve E /Q at a given prime p. Inspired by their definition of stability for the Mordell-Weil group, we introduce an analogous notion of stability for the Tate-Shafarevich group, called X-stability. Using methods in arithmetic statistics and Iwasawa theory, we study the diophantine stability of elliptic curves on average. First, we prove results for a fixed elliptic curve E and varying prime p. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is diophantine stable and X-stable at 100% of primes p. Next, we show that standard conjectures on rank distribution give lower bounds for the proportion of rational elliptic curves E that are diophantine stable at a fixed prime p ≥ 11. Related questions are studied for rank jumps and growth of ranks Tate-Shafarevich groups on average in prime power cyclic extensions.

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Euler Characteristics and their Congruences for Multi-signed Selmer Groups

Cited by 4 publications

References 0 publications

Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida's Formula

Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida's Formula

Statistics for Iwasawa invariants of elliptic curves

Arithmetic statistics and diophantine stability for elliptic curves

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