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Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ 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.
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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).more » « less
