## Math 125: Explicit Methods for Hilbert Modular Surfaces

### Winter 2018

Course Info:

• Lectures: Tuesday, Thursday, block 10A (10:10 a.m. - 12:00 noon)
• x-period: Wednesday, 3:30 - 4:20 p.m.
• Dates: 3 January 2018 - 6 March 2018
• Room: 343 Kemeny Hall

• Instructor: John Voight
• Office: 341 Kemeny Hall
• E-mail: jvoight@gmail.com
• Office hours: Thursdays 1:00 - 2:00 p.m., or by appointment
• Course Web Page: http://www.math.dartmouth.edu/~m125w18/

• Prerequisites: None
• Required Texts: None
• Recommended Texts:
• Grading: Grade will be based on either homework or on a final project (in each case, 100%).

Syllabus:

[PDF] Syllabus

In this specialized topics course in number theory, we will compute equations for Hilbert modular surfaces.

Classical modular curves, the quotients of the upper half-plane by congruence subgroups of group of integer 2-by-2 matrices with determinant 1, parametrize elliptic curves with level (torsion) structure. Equations for modular curves may be obtained by computing modular forms and multiplying their q-expansions.

One dimension up, and in an analogous way, Hilbert modular surfaces parametrize abelian surfaces with endomorphisms by an order in a real quadratic field and with level structure. In this case, it is more complicated (but by now, standard) to compute the Hecke eigenvalues for the associated Hilbert modular forms, but it is more complicated both to understand the structure of the ring of modular forms (e.g., the degrees of generators and relations), and even to multiply the corresponding q-expansions, now series indexed by totally positive elements of the inverse different of the order.

In this course, we will explore the above problem by first diving into the relevant mathematics and surveying the literature where the first few examples are worked out. Then we will design, implement, and run an algorithm to automate the generation of equations and models for Hilbert modular surfaces, including the maps between them and a description of the (stacky) universal abelian surface over them.

The course will be offered in two tracks. In one track, students who have taken previous courses in algebraic number theory, the geometry of discrete groups, modular forms, and elliptic curves, will be involved in a "final project" which is a (hopefully awesome and publishable) research paper containing what we discover. In a second track, students without this background but who still would like to learn whatever they can are invited to come to lecture and follow along; their grade will be determined by doing exercises with the background material, primarily coming from parts IV and V from my quaternion algebras book.

 1 4 Jan (R) Introduction: rings of classical modular forms Infosheet 2 9 Jan (T) Canonical maps, totally real fields - 10 Jan (W) Office hour - 11 Jan (R) No class, JMM 3 16 Jan (T) Hilbert modular forms 4 18 Jan (R) Different, q-expansions HW 1 due 5 23 Jan (T) Multiplying q-expansions, project description 6 25 Jan (R) Hecke operators, project work 7 30 Jan (T) Modularity, project work 8 1 Feb (R) Project work (JV at Penn State) 9 6 Feb (T) Hecke recurrence, moduli interpretation of modular curves 10 7 Feb (W, 4:00-4:50 p.m.) "Homework track" meeting 11 8 Feb (R) Moduli interpretation of Hilbert modular varieties, project reports 12 13 Feb (T) Project work 13 15 Feb (R) Project work HW 2 due 14 20 Feb (T) Project work 15 22 Feb (R) Project work 16 27 Feb (T) Mutiplication coefficients discussion, project work 17 28 Feb (W) Higher class number 18 1 Mar (R) Igusa invariants, project work 19 6 Mar (T) Project presentations, discussion,planning Final project due

Homework:

For those on the homework track, details about assignments will be discussed in class and will likely be made on an individual basis (or posted above).

Late homework will be accepted but all homework must be turned in by the last day of class.

Cooperation on homework is permitted (and encouraged), but please write up the solution on your own and indicate on your assignment the names of any other collaborators you worked with.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.