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Natalia Garcia-Fritz

Publications and source records attributed to Natalia Garcia-Fritz.

At least 19 recordsLinked to original sources
arXiv2026

A note on Bremner's conjecture and uniformity

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang theorem of Gao--Ge--Kühne. Thus, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a much more direct proof of this last statement, using the height-uniform Mordell theorem of Dimitrov--Gao--Habegger. The method is flexible and we give a new application of these ideas to $x$-coordinates in finitely generated multiplicative groups and geometric progressions; connections to a possible semiabelian uniform Mordell--Lang are also discussed.

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arXiv2026

Patterns on elliptic curves beyond Bremner's conjecture

In the late 1990's, Bremner conjectured that long arithmetic progressions among the $x$-coordinates of rational points of an elliptic curve $E$ over $\mathbb{Q}$ should force the rank of $E$ to be large. This conjecture (and a broad generalization of it) was proved by the authors two decades later, by combining Nevanlinna theory and the Uniform Mordell--Lang theorem of Gao--Ge--Kühne. The proof inspired subsequent work by the authors where a generalization of the Bogomolov--Fu--Tschinkel conjecture was proved by similar means. In this note we isolate a flexible pattern principle implicit in the latter work, obtaining rank-dependent (but otherwise uniform) bounds for more general patterns in the image of finite rank subgroups of elliptic curves under maps to the projective line. These patterns include, for instance, arithmetic progressions, geometric progressions, additive shifts, multiplicative shifts, and Möbius orbits.

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arXiv2026

Towards Lang--Vojta via Degeneration

Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of $S$-integral points of varieties over number fields $k$, including many cases with geometrically irreducible boundary divisors. Our approach builds on the study of arithmetic and geometric properties of moduli spaces of curves with extra structure. As an application, we provide families of explicit examples of geometrically irreducible divisors on the projective plane (such as the dual of any smooth curve of degree at least $3$), with respect to which the sets of $S$-integral points are finite. Answering a question of Achenjang and Morrow, we show that, other than the case of curves, every normal projective variety admits a geometrically irreducible divisor $D$ for which finiteness of $(D,S)$-integral points holds over every finite extension of $k$.

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arXiv2025

Xeric varieties

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 \emph{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.

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arXiv2025

Effective Mordell for curves with enough automorphisms

We prove a completely explicit and effective upper bound for the Néron--Tate height of rational points of curves of genus at least $2$ over number fields, provided that they have enough automorphisms with respect to the Mordell--Weil rank of their jacobian. Our arguments build on Arakelov theory for arithmetic surfaces. Our bounds are practical, and we illustrate this by explicitly computing the rational points of a certain genus $2$ curve whose jacobian has Mordell--Weil rank $2$.

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arXiv2024

Algebroid maps and hyperbolicity of symmetric powers

Given a complex projective algebraic variety $X$ we define $ h(X)$ as the largest $n$ such that the $n$-th symmetric power of $X$ is (Brody) hyperbolic. Using Nevanlinna theory for algebroid maps, we give non-trivial lower bounds for $ h(X)$. From an arithmetic point of view, the problem is closely related to the finiteness of algebraic points of bounded degree in varieties over number fields. We provide explicit applications of our results in the case of curves embedded in surfaces and in the case of subvarieties of abelian varieties.

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arXiv2023

A criterion for non-density of integral points

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 $\mathbb{Q}$ 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.

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arXiv2023

Effectivity for existence of rational points is undecidable

The analogue of Hilbert's tenth problem over $\mathbb{Q}$ 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 $\mathbb{Q}$ with (finitely many) height comparison conditions is undecidable.

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arXiv2023

Intersecting the torsion of elliptic curves

In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $π_j\colon E_j\to \mathbb{P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $π_j$ 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.

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arXiv2023

Linear $x$-coordinate relations of triples on elliptic curves

For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ 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.

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arXiv2023

Hilbert's tenth problem for lacunary entire functions of finite order

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.

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arXiv2022

Non-Diophantine sets in rings of functions

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.

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arXiv2022

On the conjectures of Vojta and Campana over function fields with explicit exceptional sets

We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result towards Campana's conjecture over complex function fields of curves. Our methods rely on a local study of $ω$-integral varieties.

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arXiv2019

Elliptic curves with long arithmetic progressions have large rank

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.

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arXiv2019

Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points

For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. 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.

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arXiv2018

Families of explicit quasi-hyperbolic and hyperbolic surfaces

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 $(m_1,\ldots,m_n)$ in $\mathbb{P}^{n+2}$ which are hyperbolic, for any $n \geq 8$ and any degrees $m_i \geq 2$. We also show similar results for complete intersection surfaces in $\mathbb{P}^{n+2}$ for $n=4,5,6,7$. These families give evidence for [Dem18, Conjecture 0.18] in the case of surfaces.

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arXiv2016

Quadratic sequences of powers and Mohanty's Conjecture

We prove under the Bombieri-Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by J. Browkin and J. Brzezinski, and independently by E. Gonzalez-Jimenez and X. Xarles. We also show that under the Bombieri-Lang conjecture for surfaces, for every $k\geq 3$ there is an absolute bound on the length of sequences formed by $k$-th powers with constant second differences. This gives a conditional result on one of Mohanty's conjectures on arithmetic progressions in Mordell's elliptic curves $y^2=x^3+b$. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form $y^2=x^k+b$. These results are proved by unconditionally finding all curves of genus $0$ or $1$ on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

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