There is a number \(N(K,g,r)\) such that for any algebraic curve \(C\) defined over a number field \(K\) having genus \(g\) and whose Jacobian variety \(J_C\) has Mordell-Weil rank over \(K\) equal to \(r\), the number of \(K\)-rational points of \(C\) is at most \(N(K,g,r)\).
| Author | Date Published | |
| Barry Mazur | 1997 |
Michael Stoll [2] showed that Mazur's Conjecture B holds for hyperelliptic curves with the additional hypothesis that \(r\leq g-3\).
Q808001 Bombieri-Land Conjecture
Q103002 Erdős–Ulam Problem
Q809001 Uniform Boundedness Conjecture for Rational Points
[1] Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society. 10 (1): 1–35. doi:10.1090/S0894-0347-97-00195-1.
[2] Stoll, Michael (2019). "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank". Journal of the European Mathematical Society. 21 (3): 923–956. doi:10.4171/JEMS/857.
[3] Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David (2016). "Uniform bounds for the number of rational points on curves of small Mordell–Weil rank". Duke Mathematical Journal. 165 (16): 3189–3240. arXiv:1504.00694. doi:10.1215/00127094-3673558.
uniform boundedness; genus; algebraic curve; number field; hyperelliptic curves