arXiv subjects
Publications and source records attributed to Christian Bernert.
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 anticanonical height. The proof is based on a general theorem that is broadly usable to deduce equidistribution when using the torsor method with several equivalent height functions to treat variants of Manin's problem.
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 in positive characteristic and for the motivic version over a general base field.
We show that, for any given $k \ge 2$, every sufficiently large number appears as the Steiner--Wiener $k$ index of a graph.
We determine the set of side lengths of $d$-dimensional cubes with vertices in $\mathbb Z^n$ using Witt's cancellation theorem from the algebraic theory of quadratic forms.
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 respect to the largest admissible boundary divisor. The resulting Clemens complex is more complicated than usual, and leads to particularly interesting effective cone constants, associated with exponentially many polytopes whose volumes appear in the expected formula. Like a jigsaw puzzle, these polytopes fit together to one large polytope. The volume of this polytope appears in the asymptotic formula that we obtain using the universal torsor method via o-minimal structures.
We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Ter\"av\"ainen.
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 one of the lines as the boundary divisor.
We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.
We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many exceptions. In this article we obtain a power-saving bound on the size of the exceptional set of triples. The proof is based on a combination of upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis.
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 universal torsors, we first show that we may restrict the torsor variables to their typical sizes, and then we can solve the counting problem in the framework of o-minimal structures.
We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least $10$, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric Condition, improving on a result of Heath-Brown. For the case of $9$ variables, we give a conditional treatment. We also provide a new short and elementary proof of Davenport's Shrinking Lemma which has been a crucial tool in previous literature on this and related problems.
We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the $h$-invariant of $C$ is at least $14$, improving on work of Davenport-Lewis and generalizing the method from Heath-Brown's seminal work in the homogeneous case. We also provide an upper bound on the smallest solution, polynomially in the height of the coefficients. The method also yields new results in the homogeneous case where we generalize and improve on previous work of Browning, Dietmann and Elliott.
We show that every cubic form with coefficients in an imaginary quadratic number field $K/\mathbb{Q}$ in at least $14$ variables represents zero non-trivially. This builds on the corresponding seminal result by Heath-Brown for rational cubic forms. As an application we deduce that a pair of rational cubic forms has a non-trivial rational solution provided that $s \geq 627$. Furthermore, we show that every rational cubic hypersurface in at least $33$ variables contains a rational line, and that every rational cubic form in at least $33$ variables has "almost-prime" solutions.