Math 206A - Topics in Algebraic Geometry (Spring 2016)

Course description: This course will broadly be about zeta functions of algebraic varieties over finite fields and L-functions of algebraic varieties over number fields. The focal point will be the L-Functions and Modular Forms Database (LMFDB), a recently developed web site that catalogs many types of objects of interest in arithmetic algebraic geometry and the links between them. The LMFDB is scheduled for a public release on May 10; feedback from the course before and after the release will help shape the future direction of the LMFDB project.

The exact goals of the course will not be fixed too thoroughly in advance, in order to be able to respond to the interests of the students. However, some possible topics (based on material currently included in LMFDB or planned for future development) include:

• Number fields, especially with small discriminant and/or ramification set
• Curves over finite fields with many rational points (see ManYPoints.org)
• Newton polygons
• Elliptic curves and classical modular forms
• Other modular forms (Hilbert, Siegel, Bianchi)
• K3 surfaces
• Euclidean lattices
• Genus 2 curves and abelian surfaces
• Applications of the Langlands correspondence (Lafforgue's theorem)
• Group theory and equidistribution results (Sato-Tate conjecture and variations)

Instructor: Kiran Kedlaya, kedlaya [at] math [etcetera], APM 7202.

Lectures: MWF 12-1, APM B412. There will be no lectures the weeks of May 9-13 or May 16-20.

Textbook: No required text. In addition to the LMFDB, relevant readings will be posted here.

Prerequisites: Math 203 (may take 203C simultaneously) or equivalent.

Grading: This being an advanced topics course, there will be no formal assignments, but some participation is expected.

Announcements:

• No lectures the weeks of May 9-20.

Readings:

• Weil conjectures: article by Brian Osserman.
• Modular forms: F. Diamond and J. Shurman, A First Course in Modular Forms.
• Computing elliptic curves from modular forms: J. Cremona, Algorithms for Modular Elliptic Curves (available online).
• Not technically a reading, but: SageMathCloud.
• Cartier operator: Stöhr and Voloch, A formula for the Cartier operator on plane algebraic curves (available online from UCSD). For more information about computational aspects, see Harvey and Sutherland, Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time (arXiv version, published version (download from UCSD)).
• Sato-Tate groups: Serre, Abelian l-adic representations and elliptic curves, appendix to chapter 1 (pdf); Fite, Kedlaya, Rotger, Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2 (available online from UCSD; see also errata). See also Sutherland's 2016 Arizona Winter School lectures and notes.
• Abelian varieties over finite fields: Waterhouse and Milne, Abelian varieties over finite fields (pdf).
• L-functions associated to algebraic varieties: Bucur and Kedlaya, An application of the effective Sato-Tate conjecture (pdf).
• Bad reduction Euler factors for hyperelliptic curves: Bouw and Wewers lecture notes.

Calendar of lecture topics:

• Monday, March 28: Introduction to the LMFDB (computer demo).
• Wednesday, March 30: Zeta functions of varieties over finite fields, the Weil conjectures.
• Friday, April 1: Modular forms (Diamond-Shurman, Chapter 1).
• Monday, April 4: Modular curves (Diamond-Shurman, Chapter 2)
• Wednesday, April 6: Hecke operators (Diamond-Shurman, Chapter 5).
• Friday, April 8: Hecke operators and Fourier coefficients (Diamond-Shurman, Chapter 5).
• Monday, April 11: Eichler-Shimura relation (Diamond-Shurman, Chapter 8).
• Wednesday, April 13: elliptic curves and modular forms in LMFDB (computer demo).
• Friday, April 15: modular symbols (Cremona, Chapter 2).
• Monday, April 18: continuation
• Wednesday, April 20: modular symbols in Sage (computer demo).
• Friday, April 22: number fields, elliptic curves over number fields (computer demo).
• Monday, April 25: genus 2 curves (computer demo).
• Wednesday, April 27: Cartier operators (Stöhr-Voloch).
• Friday, April 29: Sato-Tate conjecture in genus 1 and 2 (computer demo from Andrew Sutherland's web page).
• Monday, May 2: Equidistribution and L-functions (Serre).
• Wednesday, May 5: Sato-Tate groups of abelian varieties (FKRS).
• Friday, May 7: continuation.
• May 9-20: NO LECTURES
• Monday, May 23: Abelian varieties over finite fields (Waterhouse-Milne).
• Wednesday, May 25: Behind the scenes: beta functionality of LMFDB (computer demo).
• Friday, May 27: L-functions associated to algebraic varieties (BK, section 2).
• Monday, May 30: NO LECTURE (university holiday).
• Wednesday, June 1: Bad reduction Euler factors (Bouw-Wewers).
• Friday, June 3: continuation.