2019/10/27 by J. E. Cremona, Nuno Freitas, Cremona, John +1 · 1 citation
Computer Science · Mathematics · #11F33 #11F80 #11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1910.12290
openalex publication_date 2019/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe a systematic investigation into the existence of congruences\nbetween the mod p torsion modules of elliptic curves defined over\n\ℚ, including methods to determine the symplectic type of such\ncongruences. We classify the existence and symplectic type of mod p\ncongruences between twisted elliptic curves over number fields, giving global\nsymplectic criteria that apply in situations where the available local methods\nmay fail.\n We report on the results of applying our methods for all primes p\≥7 to\nthe elliptic curves in the LMFDB database, which currently includes all\nelliptic curves of conductor less than 500000. We also show that while such\ncongruences exist for each p\≤17, there are none for p \≥ 19 in the\ndatabase, in line with a strong form of the Frey-Mazur conjecture.\n