Second derivatives of p-adic L-functions and the Shafarevich--Tate group of rank-two CM elliptic curves

Abstract

For an elliptic curve E/ℚ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of Sha(E/ℚ)[p^∞] at a good ordinary prime p and applied it to five such curves for p < 30,000. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum p-adic L-function at the central point is a p-adic unit if and only if the cyclotomic p-adic regulator is a unit and Sha(E/ℚ)[p^∞] = 0, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke L-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below 30,000: it is a unit at all but three of the 8,050 primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, p = 577 for y² = x³ + 34x, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.

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BibTeX RIS

BibTeX

@misc{banwait2026second,
  title = {{Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves}},
  author = {Barinder S. Banwait},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08431},
  eprint = {2609.08431},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08431},
}

RIS

TY  - UNPB
TI  - Second derivatives of p-adic L-functions and the Shafarevich--Tate group of rank-two CM elliptic curves
AU  - Barinder S. Banwait
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08431
UR  - https://arxiv.org/abs/2609.08431
AB  - For an elliptic curve E/ℚ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of Sha(E/ℚ)[p^∞] at a good ordinary prime p and applied it to five such curves for p < 30,000. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum p-adic L-function at the central point is a p-adic unit if and only if the cyclotomic p-adic regulator is a unit and Sha(E/ℚ)[p^∞] = 0, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke L-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below 30,000: it is a unit at all but three of the 8,050 primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, p = 577 for y^2 = x^3 + 34x, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.
ER  -

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