Schedule for: 17w5045 - Diophantine Approximation and Algebraic Curves

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r < codim s, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. After a small survey on these problems, I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of curves in families of abelian varieties. This gives also applications to the study of the solvability of some polynomial Diophantine equations.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Let $k$ be a number field and $O_{S}^{\times}$ the group of $S$-units, for a given set $S$ of places of $k$. Let $h(.)$ denote the Weil height. Assume $\Gamma$ is a subset of $O_{S}^{\times}$. I will discuss the following problem: if $\beta \neq 0$ is an $S$-integer, what is a reasonable estimate for $$ \min \{ h(\beta \gamma) : \gamma \in \Gamma\} ? $$ I will also talk about some applications of such estimates in solving some Diophantine equations. This talk is based on joint work in progress with Jeffery Vaaler.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In 2002, Corvaja and Zannier obtained a new proof of Siegel's theorem (on integral points on curves) based on Schmidt's celebrated Subspace Theorem. Soon after that (and based on earlier work), Evertse and Ferretti applied Schmidt's theorem to give diophantine results for homogeneous polynomials of higher degree on a projective variety in $\Bbb P^n$. This has led to further work of A. Levin, P. Autissier, M. Ru, G. Heier, and others. In particular, Ru has defined a number, $\Nev(D)$, that concisely describes the best diophantine approximation obtained by this method, where $D$ is an effective Cartier divisor on a projective variety $X$. In this talk, I will give an overview of variants of $\Nev(D)$ as developed by Ru and myself, and indicate how an example of Faltings can be derived using these constants.

Many finiteness and enumerative problems in number theory rely on the finiteness/enumeration of the set of solutions to the equation x+y=1 over the group of S-units in a number field, where S is a finite set of primes. In 1995, Nigel Smart solved certain S-unit equations to enumerate all genus 2 curves defined over the rationals with good reduction away from p=2. Smart's work build on that of of Baker, de Weger, Evertse, Yu, and many others. In 2016, following Smart's methods, Malmskog and Rasmussen found all Picard curves over Q with good reduction away from p=3, and Angelos Koutsianas described methods for enumerating, and in some cases explicitly describes, all elliptic curves defined over a number field with good reduction outside S. Both projects required Sage implementation of special cases of Smart's general method. In January 2017, Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, and Mckenzie West combined these implementations and created new functions to solve the equation x+y=1 over the S-units of a general number field K for any finite set S of primes in K. The code is available on SageTrac and is under review for inclusion in future releases of Sage. This talk will give an overview of motivating problems and applications, the methods involved, and the current state of the implementation.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.