For every number field K and every integer g≥ 1, there exist constants C_K,g,C'_K,g>0 such that every g-dimensional abelian variety A/K satisfies h_F(A)≤ C_K,glog N_A+C'_K,g, where h_F(A) is the Faltings height and N_A is the conductor of A.

Frey’s height conjecture

Frey’s conjecture asserts a uniform logarithmic upper bound for the Faltings height of an abelian variety in terms of its conductor. Frey proposed it in 1989; Mochizuki claimed an elliptic-curve proof in August 2012, but no verification is recorded.

Known results

Frey proved the function-field analogue for elliptic curves (1989).
Geometric analogues in dimensions $g \ge 2$ were established by Faltings, Deligne, and Kim.
For semistable Jacobians, the conjecture implies a related discriminant conjecture for semistable curves.

August 2026 power-saving bounds

A preprint reported on August 24, 2026, proves an unrestricted power-saving regulator bound, with applications to Mordell equations and Frey-height bounds. This is a substantial quantitative advance, but it does not resolve Frey’s conjecture.

Current status (as of August 2026): quantitative bounds have improved, but Frey’s height conjecture remains open; Mochizuki’s 2012 proof claim is unverified.

Frey’s height conjecture

Primary source

Additional references

Progress summary

Solutions 0