Sage Days 21: Function Fields

Logistics

Organizers: William Stein, Chris Hall, Craig Citro, Salman Baig

Location: The Shuey House, 5218 16th Ave NE, Seattle, WA 98105

Dates: May 24 - 28, 2010 (check in on May 23rd after 4:00pm, check out on May 29th by 10:30am)

Mailing list: Group page.

Wireless Account

UW NetID: event0210

Password: 77Uw-52Kf-54Ga

Housing

A handful of visitors will be staying at the Shuey House, while the remaining visitors have rooms on hold for them at Hotel Deca (4507 Brooklyn Avenue NE, Seattle, WA 98105). The tentative housing arrangement is as follows:

The Shuey House

• Second floor (layout)

  • Master suite bedroom: K. Wilson and S. Miller
  • Bedroom 2: J. Balakrishnan
  • Bedroom 3: C. Hall
  • Bedroom 4: F. Hess

• Third Floor (layout)

  • Bedroom 1: G. Moehlmann, R. Butenuth
  • Bedroom 2: T. Occhipinti

Hotel Deca

• G. Boeckle
• D. Ulmer
• F. Voloch

Getting to the Shuey House/Hotel Deca from Seattle-Tacoma

Here are some options to get to the Shuey House or Hotel Deca from the airport:

• A-ABC Transportation offers a flat rate ($30 typically, before tip) for door-to-door car service. I (Salman) have used them multiple times, and they are my preferred transportation to/from the airport if I'm traveling by myself. They are a private car service (though they don't have the nicest fleet of cars) and are less expensive than a cab. Call for a reservation.

• Taxis from the airport cost about $40 (before tip) and can be found by following the signs in the airport.

• Shuttle Express picks up passengers at the inner curb on the third floor of the Airport Garage. Call to make a reservation.

• There is a new light-rail service from the airport to downtown Seattle, but you will then need to take a taxi to the house/hotel (about 5 miles north of downtown). This is not a good option unless you want to spend time in downtown Seattle.

Getting Around

Both the Shuey House and Hotel Deca are close to the UW campus as well as plenty of restaurants, coffee shops, etc. You will find that you can walk to nearly anything you like from the house or hotel, so a car is probably unnecessary. If you do plan on renting a car, please let me (Salman) know, and I will provide you information on parking, getting to/from the hotel, etc.

Seattle also has a well-connected bus system, which can be used if you want to get out of the University District and explore the rest of the city. Talk to Craig or Salman if you have any questions about riding the buses.

Map

Here's a Google map with the major locations marked.

This is a campus map that shows the location of Padelford (the Math department) and Savery and Thomson Halls (where talks on Monday will be).

Participants

• Salman Baig (Washington)
• Jen Balakrishnan (MIT) []
• Rob Beezer (Univ of Puget Sound)
• Gebhard Boeckle (Universitat Duisburg-Essen) []
• Tom Boothby (Washington)
• Robert Bradshaw (Washington)
• Ralf Butenuth (Universitat Duisburg-Essen) []
• Craig Citro (Washington)
• Alyson Deines (Washington)
• Mike Hansen
• Chris Hall (Wyoming)
• Florian Hess (Technische Universitat Berlin) []
• Jacob Lewis (Washington)
• Robert Miller (Washington)
• Steven J Miller (Williams) []
• Gerriet Moehlmann (Technische Universitat Berlin)
• Thomas Occhipinti (Arizona) []
• Dan Shumow (Washington)
• William Stein (Washington)
• Doug Ulmer (Georgia Tech) []
• Felipe Voloch (Texas) []
• Wenhan Wang (Washington)
• Kevin Wilson (Princeton)

Projects

• L-functions and the Birch and Swinnerton-Dyer Conjecture over function fields

  • ELLFF (Baig, Hall)
  • BSD (Baig, Balakrishnan, Stein, Occhipinti, Wilson, Ulmer)

    • \#Sha

    • Mordell-Weil
    • heights

• Sage's view of elliptic curves and function fields (Baig, Hall, Stein, Citro, Bradshaw)

• The Mordell-Weil group (descent, computing the algebraic rank, finding generators, torsion)

  • Generalizing (global function fields, ...) D. Robert's work on 2-descent over \mathbb{F}_q(t)

    • porting from Magma to Sage (Moehlmann, Balakrishnan, Deines, R. Miller)

  • p-descent in characteristic p (Ulmer, Moehlmann, Voloch, Hall)

• Drinfeld modular forms

  • Porting Butenuth's code to Sage (Butenuth, Stein, some Sage devs)
  • Comparing Hall's code to Butenuth's (Boeckle, Hall, Butenuth)

  • Gekeler's \theta functions

• Statistics over function fields (Baig, Bradshaw, Hall, Hansen, S. Miller)

  • excess rank, repulsion near central point, random matrix models
  • families of quadratic twists, one parameter families
  • making data available

• "Cremona's Tables" for function fields (Bradshaw, Stein, S. Miller, Occhipinti, Hall, Voloch)

  • enumerating isogeny classes

• Small Characteristic (2 - 7)

  • Boothby

• Global function field class (Stein, Hall, Voloch) -- connected to Curves over Finite Fields project

• Curves over Finite Fields in Sage

  • William Stein, Mike Hansen, Florian Hess, Michael Stoll (remote)

Informal Talks

• Guided ELLFF tour

Schedule

We will have one full day of organized talks, followed by working sessions and status reports the rest of the week. Informal talks are also welcome and will be left to individuals and working groups to organize.

Monday, May 24

Please let me know as soon as possible if you will be able to attend the dinner on Monday night for the group.

Talk Abstracts

Function Fields and Number Fields (Ulmer): I will try to explain some of the analogies between arithmetic in these two domains and also why we can often do more in the function field setting.

An Introduction to Sage for Number Theorists (Citro): I will give a short introduction to Sage, highlighting aspects that are particularly relevant for number theory. You are welcome to come armed with questions, especially of the "can Sage do <insert your favorite thing here>" variety.

Drinfeld Modular Forms and Harmonic Cocycles (Boeckle): In this talk I shall introduce the main concepts needed to understand Drinfeld cusp forms and the combinatorics that allows their computation: The Bruhat-Tits tree, Drinfeld's symmetric space, the cusp forms and harmonic cocycles. I shall state many of the basic theorems including some of Teitelbaum's work relating Drinfeld cusp forms and harmonic cocycles. I will also present a (very incomplete and subjective) list of open questions.

Computing Drinfeld Modular Forms (Butenuth): Drinfeld modular forms can be related to harmonic cochains, which are functions on the edges of the Bruhat-Tits tree fulfilling certain properties. In my talk I will try to explain how to relate Drinfeld modular forms to these objects and how to explicitly compute Hecke operators on them. The slides to the talk can be downloaded here.

Calculating L-functions over F_q(t) (Hall): We discuss solutions to the problem of computing the L-function of a non-constant elliptic curve E/F_q(t). Concretely it is a polynomial with coefficients in Z and can be computed in O(q^m) operations for some integer m=m(E) depending on E. The naive approach via point counting works but at the expense of a 'large' m(E). However, given enough information about one E/F_q(t), the relative cost of computing a 'related' elliptic curve's L-function is smaller. One can consider a quadratic twist or, more generally, a 'pullback' and the 'primitive' part of its L-function. We'll elaborate on these themes in the talk and introduce a library we are developing for calculating L-functions.

Tuesday, May 25

Free day to work or organize talks.

Wednesday, May 26

Thursday, May 27

Free day to work or organize talks.

Friday, May 28

Reading List

A reading list can be found here where participants can add items as well.