NSF PAR Search | NSF Public Access Repository

Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$

≫ \frac{X^{\frac{1}{3}}}{{(log X)}^{2}}

imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$

| d_{k} | \leq X

and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$

≫ X^{\frac{1}{4}}

in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$

Q

such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author.

A Schmidt–Nochka theorem for closed subschemes in subgeneral position

https://doi.org/10.1515/crelle-2024-0085

Heier, Gordon; Levin, Aaron (November 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

In previous work, the authors established a generalized version of Schmidt’s subspace theorem for closed subschemes in general position in terms of Seshadri constants.We extend our theorem to weighted sums involving closed subschemes in subgeneral position, providing a joint generalization of Schmidt’s theorem with seminal inequalities of Nochka.A key aspect of the proof is the use of a lower bound for Seshadri constants of intersections from algebraic geometry, as well as a generalized Chebyshev inequality.As an application, we extend inequalities of Nochka and Ru–Wong from hyperplanes in 𝑚-subgeneral position to hypersurfaces in 𝑚-subgeneral position in projective space, proving a sharp result in dimensions 2 and 3, and coming within a factor of 3/2 of a sharp inequality in all dimensions.We state analogous results in Nevanlinna theory generalizing the second main theorem and Nochka’s theorem (Cartan’s conjecture).

Full Text Available

Search for: All records