# Schedule for: 17w5045 - Diophantine Approximation and Algebraic Curves

Beginning on Sunday, July 2 and ending Friday July 7, 2017

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, July 2
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
17:30 - 19:30 Dinner
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.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (Corbett Hall Lounge (CH 2110))
Monday, July 3
07:00 - 08:45 Breakfast
Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
08:45 - 09:00 Introduction and Welcome by BIRS Station Manager (TCPL 201)
09:00 - 09:30 Cameron Stewart (TCPL 201)
09:30 - 10:00 Michael Coons: A problematic excursion at the interface of number theory and analysis (TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Laura Capuano: Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations
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.
(TCPL 201)
11:30 - 13:00 Lunch (Vistas Dining Room)
13:00 - 14:00 Guided Tour of The Banff Centre
Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.
(Corbett Hall Lounge (CH 2110))
14:00 - 14:20 Group Photo
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!
(TCPL Foyer)
14:30 - 15:00 Shabnam Akhtari: An inequality for units in the ring of $S$-integers
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.
(TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:30 Benjamin Matschke: Solving S-unit, Mordell, Thue, Thue–Mahler and generalized Ramanujan–Nagell equations via Shimura–Taniyama conjecture (TCPL 201)
16:30 - 17:00 Ariyan Javanpeykar (TCPL 201)
17:30 - 19:30 Dinner
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.
(Vistas Dining Room)
Tuesday, July 4
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:00 Paul Vojta: Birational Nevanlinna Constants and an Example of Faltings
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.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:00 Beth Malmskog: Solving S-unit equations in Sage and Applications to Algebraic Curves
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.
(TCPL 201)
11:00 - 11:30 Sebastian Troncoso: $K$-rational preperiodic hypersurfaces on $\mathbb{P}^n$ (TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:00 Adela Gherga: Implementing algorithms to compute elliptic curves over $\mathbb{Q}$ (TCPL 201)
14:00 - 15:00 Fabien Pazuki: Elliptic curves and isogenies (TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:30 Amos Turchet: Uniformity of integral points in curves and surfaces (TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Wednesday, July 5
07:00 - 09:00 Breakfast (Vistas Dining Room)
10:00 - 10:30 Coffee Break (TCPL Foyer)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 17:30 Free Afternoon (Banff National Park)
17:30 - 19:30 Dinner (Vistas Dining Room)
Thursday, July 6
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 10:00 Noriko HIRATA-Kohno: New Pade approximation related to a series with periodic coefficients (TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:00 Lajos Hajdu: Finding well approximating lattices for a finite set of points (TCPL 201)
11:00 - 11:30 Jan-Hendrik Evertse: S-parts of values of binary forms and decomposable forms (TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:00 Yann Bugeaud: On the digital representation of smooth numbers (TCPL 201)
14:30 - 15:00 Richard Guy: Quadration (TCPL 201)
15:00 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:30 Samuel Le Fourn (TCPL 201)
16:30 - 17:30 talk (TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Friday, July 7
07:00 - 09:00 Breakfast (Vistas Dining Room)
10:00 - 10:30 Coffee Break (TCPL Foyer)
11:30 - 12:00 Checkout by Noon
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.
(Front Desk - Professional Development Centre)
12:00 - 13:30 Lunch from 11:30 to 13:30 (Vistas Dining Room)