1 citations · 1 across the 18 of their papers we have counts for
21 papers
A note on Bremner's conjecture and uniformity
Natalia Garcia-Fritz, Hector Pasten
In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose -coordinates are in arithmetic progression, have lar…
Patterns on elliptic curves beyond Bremner's conjecture
Natalia Garcia-Fritz, Hector Pasten
In the late 1990's, Bremner conjectured that long arithmetic progressions among the -coordinates of rational points of an elliptic curve over should force the r…
Towards Lang--Vojta via Degeneration
Ryan C. Chen, Natalia Garcia-Fritz, Siddharth Mathur +1
Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of -integral points of varieties over number fields , including many cases with geom…
Xeric varieties
Natalia Garcia-Fritz, Hector Pasten
Let be a smooth projective variety over a number field . The Green--Griffiths--Lang conjecture relates the question of finiteness of rational points in to the triviality…
Effective Mordell for curves with enough automorphisms
Natalia Garcia-Fritz, Hector Pasten
We prove a completely explicit and effective upper bound for the Néron--Tate height of rational points of curves of genus at least over number fields, provided that they have e…
Algebroid maps and hyperbolicity of symmetric powers
Natalia Garcia-Fritz, Hector Pasten
Given a complex projective algebraic variety we define as the largest such that the -th symmetric power of is (Brody) hyperbolic. Using Nevanlinna theory for…
Hilbert's tenth problem for rings of holomorphic functions of bounded order
Hector Pasten
One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable. In the direction of…
A criterion for non-density of integral points
Natalia Garcia-Fritz, Hector Pasten
We give a general criterion for Zariski degeneration of integral points in the complement of a divisor with components in a variety of dimension defined over $\mathbb{Q…
Effectivity for existence of rational points is undecidable
Natalia Garcia-Fritz, Hector Pasten, Xavier Vidaux
The analogue of Hilbert's tenth problem over asks for an algorithm to decide the existence of rational points in algebraic varieties over this field. This remains as o…
Hilbert's tenth problem for lacunary entire functions of finite order
Natalia Garcia-Fritz, Hector Pasten
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…
Intersecting the torsion of elliptic curves
Natalia Garcia-Fritz, Hector Pasten
In 2007, Bogomolov and Tschinkel proved that given two complex elliptic curves and along with even degree- maps having different bran…
Propagation of Zariski Dense Orbits
Hector Pasten, Joseph H. Silverman
Let be a smooth projective variety defined over a number field, and let be a morphism defined over . We formulate a number of statements of varying strengths a…
On the fibres of an elliptic surface where the rank does not jump
Jerson Caro, Hector Pasten
For a non-constant elliptic surface over defined over , it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the…
Non-Diophantine sets in rings of functions
Natalia Garcia-Fritz, Hector Pasten, Thanases Pheidas
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 contr…
A Diophantine definition of the constants in
Natalia Garcia-Fritz, Hector Pasten
It is an old problem in the area of Diophantine definability to determine whether is Diophantine in . We provide a positive answer conditional on two st…
A Chabauty-Coleman bound for surfaces
Jerson Caro, Hector Pasten
Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve of genus defined over a number field ,…
Elliptic curves with long arithmetic progressions have large rank
Natalia Garcia-Fritz, Hector Pasten
For any family of elliptic curves over the rational numbers with fixed -invariant, we prove that the existence of a long sequence of rational points whose -coordinates form a…
Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points
Natalia Garcia-Fritz, Hector Pasten
For a positive proportion of primes and , we prove that is Diophantine in the ring of integers of . This provides a new and e…
Bivariate polynomial injections and elliptic curves
Hector Pasten
For every number field , we construct an affine algebraic surface over with a Zariski dense set of -rational points, and a regular function on inducing an inj…
Bounded ranks and Diophantine error terms
Hector Pasten
We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very…
Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic
Natalia Garcia-Fritz, Hector Pasten
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 characteri…