From Genus-Two Heights to a Weak abc Theorem
This presentation explores a striking application of arithmetic intersection theory to number theory's abc conjecture. The authors construct a genus-six Kodaira fibration over a genus-two curve and use Arakelov-theoretic techniques to derive an effective discriminant-dependent height bound. While the resulting exponent of 24 plus epsilon falls far short of the conjectural 1 plus epsilon, the work demonstrates how geometric machinery can yield unconditional, uniform Diophantine inequalities with explicit constants, and further derives consequences for elliptic curve discriminants and zero-free regions of quadratic L-functions.Script
Three coprime integers satisfy a plus b equals c. The abc conjecture predicts that c cannot be too large compared to the product of primes dividing abc, unless epsilon room is allowed. After decades, no proof exists, but this paper delivers something new: an unconditional exponent of 24 plus epsilon, derived purely from the geometry of curves.
The authors study rational points on genus-two curves using Arakelov intersection theory. For every such curve and every rational point, they prove that the logarithmic height is bounded by 12 plus epsilon times the normalized discriminant, plus a constant. This coefficient of 12 is explicit and effective, though Vojta's conjecture predicts a coefficient of 1.
The geometric engine is a genus-six Kodaira fibration built over a finite cover of the original genus-two curve. By analyzing intersection numbers of the relative dualizing sheaf and two branch divisors, the authors compute that the square of the canonical bundle on the fiber equals 12 times a covering degree. This coefficient 12 eventually controls the discriminant dependence.
The proof uses successive minima for adelic vector spaces of global sections, combined with an intersection inequality relating section heights to bundle self-intersections. When the fiber has only two independent sections, their heights can be controlled by the discriminant of the field. This mechanism converts local sup norms into a global discriminant bound, with the constant dictated by the bundle's numerical invariants.
Applying the genus-two estimate to the modular curve X zero of 23, which has genus 2 and a degree-24 j-map, yields a discriminant-conductor inequality for semistable elliptic curves. The large coefficient reflects both the modular degree and the preceding factor of 12. For the abc conjecture itself, the authors embed a plus b equals c into the curve y squared plus y equals x to the fifth, introduce a degree-five extension, and carefully track discriminants to obtain the final exponent of 24 plus epsilon.
The exponent 24 plus epsilon is far from the conjectural 1 plus epsilon, yet it is the first unconditional, uniform abc-type bound with an explicit dependence on the radical. The paper also derives a classical zero-free region for quadratic Dirichlet L-functions by controlling the height of CM j-invariants. To learn more about this blend of Arakelov geometry and Diophantine consequences, visit EmergentMind.com and create your own video summary.
