Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn a question of Mordell
105
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We make several improvements to methods for finding integer solutions to $x^3+y^3+z^3=k$ for small values of $k$. We implemented these improvements on Charity Engines global compute grid of 500,000 volunteer PCs and found new representations for several values of $k$, including $k=3$ and $k=42$. This completes the search begun by Miller and Woollett in 1954 and resolves a challenge posed by Mordell in 1953.
rate research
Read More
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(bP^1)^g$ has only finite intersection with any curve contained in $(bP^1)^g$. We also show that our result holds for indecomposable polynomials $phi$ with coefficients in $bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(phi,phi)$ on $bA^2$.
In this paper, we prove the Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties. As a byproduct, we prove the Uniform Bogomolov Conjecture for subvarieties in abelian varieties.
We give an effective proof of Faltings theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields. We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of $mathrm{GL}_2$-type over an odd-degree totally real field. We deduce for example an effective height bound for $K$-points on the curves $C_a : x^6 + 4y^3 = a^2$ ($ain K^times$) when $K$ is odd-degree totally real. (Over $overline{mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an {e}tale cover dominating $C_1$.)
This expository survey is based on my online talk at the ICCM 2020. It aims to sketch key steps of the recent proof of the uniform Mordell-Lang conjecture for curves embedded into Jacobians (a question of Mazur). The full version of this conjecture is proved by combining Dimitrov-Gao-Habegger (https://annals.math.princeton.edu/articles/17715) and K{u}hne (arXiv:2101.10272). We include in this survey a detailed proof on how to combine these two results, which was implicitly done in another short paper of Dimitrov-Gao-Habegger (arXiv:2009.08505) but not explicitly written in existing literature. At the end of the survey we state some future aspects.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Neron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
