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 Lfunctions of algebraic varieties
over number fields. The focal point will be the
LFunctions 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 (SatoTate conjecture and variations)
These topics will not be presented in a highly cumulative fashion; there will
be frequent "resets" when we change topics.
Instructor: Kiran Kedlaya,
kedlaya [at] math [etcetera], APM 7202.
Lectures: MWF 121, APM B412. There will be no lectures the weeks of May 913 or May 1620.
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 920.
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 HasseWitt matrices of hyperelliptic curves in average polynomial time (arXiv version,
published version (download from UCSD)).

SatoTate groups: Serre, Abelian ladic representations and elliptic curves, appendix to chapter 1 (pdf); Fite, Kedlaya, Rotger, Sutherland, SatoTate 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).

Lfunctions associated to algebraic varieties: Bucur and Kedlaya, An application of the effective
SatoTate 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 (DiamondShurman, Chapter 1).

Monday, April 4: Modular curves (DiamondShurman, Chapter 2)

Wednesday, April 6: Hecke operators (DiamondShurman, Chapter 5).

Friday, April 8: Hecke operators and Fourier coefficients (DiamondShurman, Chapter 5).

Monday, April 11: EichlerShimura relation (DiamondShurman, 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 (StöhrVoloch).

Friday, April 29: SatoTate conjecture in genus 1 and 2 (computer demo from
Andrew Sutherland's web page).

Monday, May 2: Equidistribution and Lfunctions (Serre).

Wednesday, May 5: SatoTate groups of abelian varieties (FKRS).

Friday, May 7: continuation.

May 920: NO LECTURES

Monday, May 23: Abelian varieties over finite fields (WaterhouseMilne).

Wednesday, May 25: Behind the scenes: beta functionality of LMFDB (computer demo).

Friday, May 27: Lfunctions associated to algebraic varieties (BK, section 2).

Monday, May 30: NO LECTURE (university holiday).

Wednesday, June 1: Bad reduction Euler factors (BouwWewers).

Friday, June 3: continuation.