activity
20232026
collaborators

13 papers

math.NT2026

Equidistribution and the torsor method

Christian Bernert, Ulrich Derenthal, Florian Wilsch

We prove equidistribution and Manin's conjecture for rational points outside the lines on smooth split quintic del Pezzo surfaces over number fields with respect to any anticanonic…

math.AG2026

Motivic counting of curves on split quintic del Pezzo surfaces

Christian Bernert, Loïs Faisant, Jakob Glas

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields…

math.CO2026

The inverse problem for the Steiner--Wiener index via additive number theory

Christian Bernert, Joshua Shaw

We show that, for any given , every sufficiently large number appears as the Steiner--Wiener index of a graph.

math.NT2026

Side lengths of cubes with vertices in

Christian Bernert, Jens Reinhold

We determine the set of side lengths of -dimensional cubes with vertices in using Witt's cancellation theorem from the algebraic theory of quadratic forms.

math.NT2026

Integral points over number fields: a Clemens complex jigsaw puzzle

Christian Bernert, Ulrich Derenthal, Judith Ortmann +1

We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with resp…

math.NT2025

The exceptional set in the abc conjecture

Christian Bernert

We study the size of the exceptional set in the conjecture, improving on and simplifying work of Browning, Lichtman and Teräväinen.

math.NT2025

Integral points of bounded height on quintic del Pezzo surfaces over number fields

Christian Bernert, Ulrich Derenthal

We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to…

math.NT2025

Points of bounded height on quintic del Pezzo surfaces over the rational numbers

Christian Bernert, Ulrich Derenthal

We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.

math.NT2024

Bounds on the exceptional set in the conjecture

Christian Bernert, Tim Browning, Jared Duker Lichtman +1

We study solutions to the equation , where form a triple of coprime natural numbers. The conjecture asserts that, for any , such triples satisfy $\mathrm{…

math.NT2024

Points of bounded height on quintic del Pezzo surfaces over number fields

Christian Bernert, Ulrich Derenthal

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to…

math.NT2023

Small solutions to homogeneous and inhomogeneous cubic equations

Christian Bernert

We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the -invariant of $C…

math.NT2023

The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma

Christian Bernert

We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least , we prove the absolute convergence (and hence positiv…

math.NT2023

Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms

Christian Bernert, Leonhard Hochfilzer

We show that every cubic form with coefficients in an imaginary quadratic number field in at least variables represents zero non-trivially. This builds on the c…