with H. Pasten

Abstract: Let X be a smooth projective variety over a number field k. The Green-Griffiths-Lang conjecture relates the question of finiteness of rational points in X to the triviality of rational maps from abelian varieties to X and to complex hyperbolicity. Here we investigate the phenomenon of sparsity of rational points in X - roughly speaking, when there are very few rational points if counted ordered by height. We are interested in the case when sparsity holds over every finite extension of k, in which case we say that the variety is xeric. We initiate a systematic study of the relation of this property with the non-existence of rational curves in X as well as with certain notion of p-adic hyperbolicity.

with H. Pasten and T. Pheidas

Abstract: Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine.

with H. Pasten, T. Pheidas, and X. Vidaux

Abstract: The analogue of Hilbert's tenth problem over ℚ asks for an algorithm to decide the existence of rational points in algebraic varieties over this field. This remains as one of the main open problems in the area of undecidability in number theory. Besides the existence of rational points, there is also considerable interest in the problem of effectivity: one asks whether the sought rational points satisfy determined height bounds, often expressed in terms of the height of the coefficients of the equations defining the algebraic varieties under consideration. We show that, in fact, Hilbert's tenth problem over ℚ with (finitely many) height comparison conditions is undecidable.

with J. Caro

Abstract: For an elliptic curve E defined over the field ℂ of complex numbers, we classify all translates of elliptic curves in E³ such that the x-coordinates satisfy a linear equation. This classification enables us to establish a relation between the rank of finite rank subgroups of E and triples in E whose x-coordinates are linearly related. The method of proof integrates complex analytic techniques on elliptic curves with results of Gao, Ge and Kühne on Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties.

with H. Pasten

Abstract: It is an old problem in the area of Diophantine definability to determine whether ℚ is Diophantine in ℚ(z). We provide a positive answer conditional on two standard conjectures on elliptic surfaces.

with H. Pasten

Abstract: In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves E₁ and E₂ along with even degree-2 maps π_ij:E_j→ℙ¹ having different branch loci, the intersection of the image of the torsion points of E₁ and E₂ under their respective π_ij is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. As it has been observed in the literature, the recent proof of the Uniform Manin-Mumford conjecture implies a full solution of the Bogomolov-Fu-Tschinkel conjecture. In this work we prove a generalization of the Bogomolov-Fu-Tschinkel conjecture where instead of even degree-2 maps one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin-Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

with H. Pasten

Abstract: We give a general criterion for Zariski degeneration of integral points in the complement of a divisor D with n components in a variety of dimension n defined over ℚ or over a quadratic imaginary field. The key condition is that the intersection of the components of D is not well-approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a GCD bound for algebraic points in varieties which can be of independent interest.

with H. Pasten

Abstract: In the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia-Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.

with D. Chompitaki, H. Pasten, T. Pheidas, X. Vidaux

Abstract: One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.

with H. Pasten

Abstract: For any family of elliptic curves over the rational numbers with fixed j-invariant, we prove that the existence of a long sequence of rational points whose x-coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for y-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings.

with H. Pasten

Abstract: For a positive proportion of primes p and q, we prove that ℤ is Diophantine in the ring of integers of ℚ(∛p,√-q). This provides a new and explicit infinite family of number fields K such that Hilbert's tenth problem for O_K is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

with G. Urzua

Abstract: We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular case, obtaining results for the surface of cuboids, the generalized surfaces of cuboids, and other families of Diophantine surfaces of general type. In particular, we produce explicit families of smooth complete intersection surfaces of multidegrees (m₁,...,m_n) in ℙⁿ⁺² which are hyperbolic, for any n ≥ 8 and any degrees m_i ≥ 2. We also show similar results for complete intersection surfaces in ℙⁿ⁺² for n = 4,5,6,7. These families give evidence for [Dem18, Conjecture 0.18] in the case of surfaces.

Abstract: By explicitly finding the complete set of curves of genus 0 or 1 in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound M > 0 such that there are only finitely many sequences of length M formed by k-th rational powers with second differences equal to 2. Moreover, we prove the unconditional analogue of this result for function fields, with M depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.

with H. Pasten

Abstract: We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is uniformly positive existentially interpretable in the class of {0,1,t,+,‧,=}-structures consisting of positive characteristic rings of entire functions on the variable t. From this we deduce uniform undecidability results for the positive existential theory of such structures. As a key intermediate step, we prove a rationality result for the solutions of certain Pell equation (which a priori could be transcendental entire functions).

Abstract: Given a field F and polynomials a and b in F[t], we prove that in general the sets {aλ+b : λ∈F} and {λ2+aλ+b : λ∈F} contain only finitely many powers and find bounds that are uniform from various points of view. We derive from this analysis various first-order definability and undecidability results. For example, we prove that there is no algorithm to decide whether or not an arbitrary system of linear equations over the integers, together with conditions of the form 'x is a power' and of the form 'x is non-constant' on some of the variables, has a solution in F[t].