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We show that any finite system S in a characteristic zero integral domain can be mapped to Z/pZ, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman isomorphism lemma, which asserts that any finite subset of a torsion-free group can be mapped into Z/pZ, preserving all linear… (More)

We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich–Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank.… (More)

- Jennifer Mun, Young Park, Bjorn Poonen, Matt Baker, Jennifer S Balakrishnan, Manjul Bhargava +30 others
- 2014

Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points [Szp85], XI, §2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed [Col85] that Chabauty's… (More)

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