Mazur's Conjecture B

Statement:

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

Partial Results:

Michael Stoll [2] showed that Mazur's Conjecture B holds for hyperelliptic curves with the additional hypothesis that $r\le g-3$.

Related Problems:

Q808001 Bombieri-Land Conjecture
Q103002 Erdős–Ulam Problem
Q809001 Uniform Boundedness Conjecture for Rational Points

References:

[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.

Keywords:

uniform boundedness; genus; algebraic curve; number field; hyperelliptic curves