Simons Collaborationon Arithmetic Geometry, Number Theory, and Computation

June 3-7 (ICERM)

Arithmetic of low-dimensional abelian varieties

In this workshop, we will explore a number of themes in the arithmetic of abelian varieties of low dimension (typically dimension 2–4), with a focus on computational aspects. Topics will include the study of torsion points, Galois representations, endomorphism rings, Sato-Tate distributions, Mumford-Tate groups, complex and p-adic analytic aspects, L-functions, rational points, and so on.

Our goal is for the workshop to bring together researchers working on abelian varieties in a number of facets to establish collaborations, develop algorithms, and stimulate further research.

August 20-24 (MIT)

Conference on Arithmetic Geometry, Number Theory, and Computation

The collaboration’s first conference will take place at MIT.

July 16-20 (University of Wisconsin, Madison)

ANTS XIII

Many collaboration members will attend this bi-annual conference on algorithmic number theory.

May 10-14 (ICERM)

Birational Geometry and Arithmetic

Many collaboration members will attend this ICERM workshop on birational geometry. Abstract

Number field fragments and Fermat’s last theorem
Simons Collaboration Seminar (open to all) by Bjorn Poonen (MIT)
3:30-4:30pm in MIT room 6-120

We describe an attempt (unsuccessful so far) to give a new proof of Fermat’s last theorem, involving geometry of numbers and the isotypic components of a Galois number field viewed as a representation of its Galois group.

March 5

Rigorous computation of the endomorphism ring of a Jacobian Collaboration PI John Voight (Dartmouth) speaks in the Brown University Algebra seminar on algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field. Abstract

Computing Belyi Maps Collaboration PI John Voight gives the colloquium at Temple University, Philadelphia. Abstract

Strong Arithmetic Equivalence Andrew Sutherland (MIT), a collaboration PI, discusses research results that have applications to the construction of curves with isomorphic Jacobians and number fields with the same zeta function in the Brown University Algebra seminar. Abstract

Modularity of K3 surfaces: explicit methods and computational questions

This public collaboration lecture will discuss the predictions of the Langlands program for K3 surfaces, presenting some examples, open problems, and computational challenges. At 4:00pm at MIT. Abstract

Collaboration PI John Voight (Dartmouth) gives the Math Colloquium at Penn State University on Heuristics for Boundedness of Ranks of Elliptic Curves. He also speaks on Mock Heegner points and Sylvester’s Conjecture in the Number Theory Seminar.

January 12 (Marriott Marquis San Diego Marina Hotel)

ICERM Mixer at the Joint Mathematics MeetingsTo learn more about our collaboration, join past and future ICERM participants 6pm-8pm in the Catalina room. Refreshments will be served.
Please RSVP at info@icerm.brown.edu.