Arithmetic Szpiro Conjecture

Establish that for every real number ε>0 there exists an absolute constant C(ε)>0 such that for all elliptic curves E/ℚ, the discriminant and conductor satisfy |Δ_E| ≲ C(ε) · N_E^{6+ε}.

Background

Szpiro’s conjecture is an arithmetic analogue of a geometric inequality originally proved in the function field setting. It posits a uniform quantitative relationship between the minimal discriminant and conductor of elliptic curves over ℚ, and is known to be equivalent to the abc-conjecture.

Within the paper’s program, proving Vojta’s Height Inequality (in the form treated by Mochizuki) and its compactly bounded variant implies the abc-conjecture and therefore the Arithmetic Szpiro Conjecture.

References

The following was conjectured in . Let $0<\varepsilon\in\mathbb{R}$, then there exists an absolute constant $C(\varepsilon)>0$ such that for all elliptic curves $E/\mathbb{Q}$, one has $${\Delta_E}_\ll C(\varepsilon)\cdot N_E{(6+\varepsilon)}.$$

— Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture (2403.10430 - Joshi, 15 Mar 2024) in Section 2.2 (Arithmetic Szpiro Conjecture)