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Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields

  • Shisong Xu
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Abstract

Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure $2$-Selmer group of a congruent number elliptic curve. When the pure $2$-Selmer dimension is even, we prove that the normalized representation defect modulo $2$ is the Pfaffian of the Cassels pairing matrix; the dimension of its radical yields sharper $2$-adic divisibility and information on the $2$-primary Shafarevich--Tate group. For products of primes congruent to $1$ modulo $8$ that are pairwise quadratic residues, we give explicit Cassels pairing matrices for both En and E2n and express their entries in terms of quartic and Rédei symbols. For each fixed prime p, we construct a governing field of degree $256$ and determine the exact joint distribution of the two Pfaffians by Chebotarev's theorem. In particular, there is a set of primes q of natural density 5/128 for which both E17q and E34q have rank zero and $2$-primary Shafarevich--Tate group isomorphic to (Z/2Z)4.

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Cite

BibTeXRIS
BibTeX
@misc{xu2026representation,
  title = {{Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields}},
  author = {Shisong Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03238},
  eprint = {2609.03238},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03238},
}
RIS
TY  - UNPB
TI  - Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
AU  - Shisong Xu
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.03238
UR  - https://arxiv.org/abs/2609.03238
AB  - Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure $2$-Selmer group of a congruent number elliptic curve. When the pure $2$-Selmer dimension is even, we prove that the normalized representation defect modulo $2$ is the Pfaffian of the Cassels pairing matrix; the dimension of its radical yields sharper $2$-adic divisibility and information on the $2$-primary Shafarevich--Tate group. For products of primes congruent to $1$ modulo $8$ that are pairwise quadratic residues, we give explicit Cassels pairing matrices for both E_n and E_2n and express their entries in terms of quartic and Rédei symbols. For each fixed prime p, we construct a governing field of degree $256$ and determine the exact joint distribution of the two Pfaffians by Chebotarev's theorem. In particular, there is a set of primes q of natural density 5/128 for which both E_17q and E_34q have rank zero and $2$-primary Shafarevich--Tate group isomorphic to (Z/2Z)^4.
ER  -