Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions

Beneish, Lea; Kundu, Debanjana; Ray, Anwesh

doi:10.48550/arxiv.2107.09166

Cited by 1 publication

(4 citation statements)

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“…Then, for 100% of the primes p, there are abundantly many p-cyclic extensions in which the p-primary Selmer group grows. More precisely, for any n ∈ Z ≥1 and a finite set of primes Σ, there are infinitely many Z/p n Z-extensions L/Q such that (1) all primes of Σ split in L, (2) Sel p ∞ (E/Q) = 0 and Sel p ∞ (E/Q) = 0.…”

Section: Questionmentioning

confidence: 99%

“…Let E /F be an elliptic curve satisfying the conditions of Theorem 3.2. Then, the following are equivalent (1)…”

Section: The Truncated Euler Characteristicmentioning

confidence: 99%

“…Corollary 3.6] and the Euler characteristic formula that the following conditions are equivalent for a prime p at which E has good ordinary reduction and E[p] is irreducible as a Galois module(1) …”

mentioning

confidence: 99%

“…(1) First consider the case where ℓ is unramified in L. Fix a prime v | ℓ. Since p ≥ 5, the Tamagawa number c(p) v (E/L) = 1 if and only if the Kodaira type of E…”

mentioning

confidence: 99%

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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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Section: Questionmentioning

confidence: 99%

“…Let E /F be an elliptic curve satisfying the conditions of Theorem 3.2. Then, the following are equivalent (1)…”

Section: The Truncated Euler Characteristicmentioning

confidence: 99%

mentioning

confidence: 99%

“…(1) First consider the case where ℓ is unramified in L. Fix a prime v | ℓ. Since p ≥ 5, the Tamagawa number c(p) v (E/L) = 1 if and only if the Kodaira type of E…”

mentioning

confidence: 99%

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