Browsing by Author "Pasten, Hector"

Now showing 1 - 13 of 13
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    A Chabauty-Coleman bound for surfaces
    (2023) Caro, Jerson; Pasten, Hector
    Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve C of genus g = 2 defined over a number field F, with Jacobian of rank at most g - 1. Namely, in the case F = Q, if p > 2g is a prime of good reduction, then the number of rational points of C is at most the number of F-p-points plus a contribution coming from the canonical class of C. We prove a result analogous to Coleman's bound in the case of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a p-adic analytic subgroup with a subvariety of A by means of overdetermined systems of differential equations in positive characteristic.
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    A CONJECTURE OF WATKINS FOR QUADRATIC TWISTS
    (2021) Esparza-Lozano, Jose A.; Pasten, Hector
    Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r, the modular degree of E is divisible by 2(r). If E has non-trivial rational 2-torsion, we prove the conjecture for all the quadratic twists of E by squarefree integers with sufficiently many prime factors.
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    A criterion for nondensity of integral points
    (2024) Garcia-Fritz, Natalia; Pasten, Hector
    We give a general criterion for Zariski degeneration of integral points in the complement of a divisor D$D$ with n$n$ components in a variety of dimension n$n$ defined over Q$\mathbb {Q}$ or over a quadratic imaginary field. The key condition is that the intersection of the components of D$D$ is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest.
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    A Derivation of the Infinitude of Primes
    (2024) Pasten, Hector
    The well-known analogy between polynomials and integers breaks down when it comes to considering the polynomial derivative. This is rather unfortunate since derivatives are a powerful tool for doing arithmetic with polynomials. Nevertheless, there are some proposals in the literature for arithmetic analogues of derivatives. In this article we use one of these arithmetic derivatives to give a proof of the infinitude of primes which is analogous to an argument that will be presented for polynomials using polynomial derivatives. We hope that this "differential" proof of the infinitude of primes will help to motivate the reader to look for good notions of arithmetic derivatives.
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    Arithmetic derivatives through geometry of numbers
    (2021) Pasten, Hector
    We define certain arithmetic derivatives on Z that respect the Leibniz rule, are additive for a chosen equation a + b = c, and satisfy a suitable nondegeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the abc Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the abc Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighties, relating Geometry of Numbers in arithmetic to derivatives in function fields and Nevanlinna theory. In addition, our construction formalizes the widespread intuition that the abc Conjecture should be related to arithmetic derivatives of some sort.
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    Hilbert's tenth problem for lacunary entire functions of finite order
    (2024) Garcia-Fritz, Natalia; Pasten, Hector
    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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    Intersecting the torsion of elliptic curves
    (2023) García-Fritz, Natalia; Pasten, Hector
    Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves E1 and E2 along with even degree-2 maps πj: Ej → P1 having different branch loci, the intersection of the image of the torsion points of E1 and E2 under their respective πj is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, 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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    Non-thin rank jumps for double elliptic K3 surfaces
    (2024) Pasten, Hector; Salgado, Cecilia
    For an elliptic surface pi:X -> P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :X\rightarrow \mathbb {P}<^>1$$\end{document} defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}. When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}<^>1$$\end{document} for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.
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    NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES
    (2022) Pasten, Hector
    A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.
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    On the Chevalley-Warning Theorem When the Degree Equals the Number of Variables
    (2022) Pasten, Hector
    Let f be a degree d polynomial in n variables defined over a finite field k of characteristic p and let N be the number of zeros of f in k(n). The Chevalley-Warning theorem asserts that if d, then N is divisible by p. In this note we show a version of the result for d = n.
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    Shimura curves and the abc conjecture
    (2024) Pasten, Hector
    In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields. Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L2-norm of integral quaternionic modular forms.The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations.& COPY; 2023 Published by Elsevier Inc.
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    The largest prime factor of n2+1and improvements on subexponential ABC
    (2024) Pasten, Hector
    We combine transcendental methods and the modular approaches to the ABC conjecture to show that the largest prime factor of n(2)+1isatleastofsize(log(2)n)(2)/log(3)nwhere log k is the k-th iterate of the logarithm. This gives a substantial improvement on the best available estimates, which are essentially of size log2ngoing back to work of Chowla in 1934. Using the same ideas, we also obtain significant progress on sub expoential bounds for the ABC conjecture, which in a case gives the first improvement on a result by Stewart and Yu dating back over two decades. Central toour approach is the connection between Shimura curves and the ABC conjecture developed by the author.
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    WATKINS'S CONJECTURE FOR ELLIPTIC CURVES WITH NON-SPLIT MULTIPLICATIVE REDUCTION
    (2022) Caro, Jerson; Pasten, Hector
    Let E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487-502 (2003)] conjectured that the rank of E is bounded by the 2-adic valuation of the modular degree of E. We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.