# VaNTAGe

### a virtual math seminar on open conjectures in

### number theory and arithmetic geometry

In spring 2020, a new math seminar was started: VaNTAGe, a virtual seminar on open conjectures in number theory and arithmetic geometry (NT&AG). The seminar provides open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG. The purpose of the seminar is to provide a viable way for researchers to be involved with cutting-edge research in NT&AG without the expense and environmental impact of travel. Another aim of the seminar is to advance understanding on the most exciting open problems in this field. As the goal of the seminar is to build communication among researchers developing the fields of NT&AG, speakers and participants will be expected to uphold the highest standards for clear exposition and respectful interactions.

The seminar is organized by Rachel Pries and Andrew Sutherland.

The 7th topic is **Manin conjectures and rational points. **

The following survey paper provides background for this topic: Algebraic varieties with many rational points

4/6 Jordan Ellenberg Counting points on (some) stacks: progress and problems (slides, video)

4/20 Marta Pieropan The split torsor method for Manin's conjecture (paper, slides, Zoom link)

5/4 Will Sawin The freeness alternative to thin sets in Manin's conjecture

5/11 Yuri Tschinkel Height zeta functions

5/18 Damaris Schindler Campana points on toric varieties

The intended audience includes graduate students and faculty with some background in NT&AG; for practical reasons the seminar will be presented in English at (1 pm Eastern US)=(10 am Pacific US), every other Tuesday. Live access links are posted the night before each talk and also sent to the seminar mailing list which you can subscribe to via

https://groups.google.com/forum/#!forum/vantageseminar

**Video recording**: Most VaNTAGe meetings are recorded and posted to our YouTube channel a week after the talk (this gives us time to add high quality captions, which are included with all our video recordings). Participants should be aware that if the join the live meeting both the audio and video will be recorded. Participants who do not wish to be recorded should keep their video camera and microphone off and use the chat to ask questions.

VaNTAGe is currently using Zoom. At other times, we used BlueJeans.

Spring 2020 is the initial development period of the VaNTAGe math seminar. If you have suggestions, please send them to vantageseminar@gmail.com or rachelpries@gmail.com or drew@math.mit.edu.

Rachel Pries would like to thank the National Science Foundation DMS-19-01819 for its support of the VaNTAGe math seminar, and Andrew Sutherland is grateful to the Simons Foundation (grant 550033) for its support. We thank MIT for hosting the virtual zoom platform and the ACNS video conference services at Colorado State University for hosting the virtual BlueJeans platform and Chris Chagnon for providing technical support.

The first topic of the seminar was **Class groups of number fields**. The following survey paper provides background for this topic:

*On a conjecture for â„“-torsion in class groups of number fields: from the perspective of moments*

Jan 21: Lillian Pierce. On some questions in number theory, from the perspective of moments (video not available)

Feb 4: Melanie Matchett-Wood. Conjectures for number field counting (video)

Feb 18: Caroline Turnage-Butterbaugh. Moments of zeta and the vertical distribution of its zeros (video)

Mar 3: David Zureick-Brown. Moduli spaces and arithmetic statistics (video)

The second topic was the **Sato-Tate conjecture for abelian varieties**. The following survey paper provides background for this topic:

Mar 24: Kiran Kedlaya. The Sato-Tate conjecture and its generalizations (slides, video)

Apr 7: Francesc Fite. Sato-Tate groups of abelian varieties of dimension up to 3 (paper, slides, video)

Apr 28: Andrew Sutherland. Arithmetic L-functions and their Sato-Tate distributions (slides, video)

May 5: David Zywina. Computing Sato-Tate and monodromy groups (slides, video)

May 19: Alina Bucur. Effective Sato-Tate and applications (paper, slides)

We would like to thank Jun Bo Lau for editing the closed captions for the videos in this series.

The third topic was **Arithmetic dynamics**. The following survey paper provides background for this topic:

*Current trends and open problems in arithmetic dynamics *

May 26: Holly Krieger. Equidistribution and unlikely intersections in arithmetic dynamics (slides, video)

June 9: Patrick Ingram. The critical height of an endomorphism of projective space (slides, video)

June 23: Joseph Silverman. Moduli problems and moduli spaces in algebraic dynamics (slides, video).

July 7: Nicole Looper. The ABC conjecture and arithmetic dynamics

The fourth topic was **Rational points on elliptic curve**s . A good survey paper to prepare for this series of talks is:

*Heuristics for the arithmetic of elliptic curves*

September 1: Bjorn Poonen. Heuristics for the arithmetic of elliptic curves (slides, video)

September 15: Noam Elkies. Rank speculation (slides, video)

September 29: Alvaro Lozano-Robledo. The distribution of ranks of elliptic curves and the minimalist conjecture: reconciling conjectures and data (slides, video)

October 13: Wei Ho. Integral points on elliptic curves (slides, video)

October 27: Arul Shankar. Ordering elliptic curves by conductor (slides, video)

The fifth topic was **An update on ICERM workshop on ****Arithmetic Geometry, Number Theory, and Computation** .

Some of the postdocs involved with the June ICERM workshop will give updates about their group's research project.

We are sorry we could only accommodate six such talks because there were many postdocs at this conference.

November 10: Nicholas Triantafillou Computing isolated points on modular curves (slides, video)

November 10: Raymond van Bommel Cluster pictures of hyperelliptic curves (slides, video)

November 24: Jeremy Booher Can you hear the shape of a curve? (slides, video)

November 24: David Corwin Kim's conjecture and effective Faltings (slides, video)

December 8: Padma Srinivasan Computing exceptional primes associated to Galois representations of abelian surfaces (slides, video)

December 8: Manami Roy Challenges and usefulness of creating a database of groups in LMFDB (slides, video)

The sixth topic was **K3 surfaces. **

The following survey paper provides background for this topic: Arithmetic of K3 surfaces

1/26 Edgar Costa From counting points to rational curves on K3 surfaces (slides, video)

2/9 Alessandra Sarti Old and new on the symmetry groups of K3 surfaces (slides, video)

2/23 Bianca Viray The Brauer group and the Brauer-Manin obstruction on K3 surfaces (slides, video)

3/9 Francesca Balestrieri The arithmetic of zero-cycles on products of K3 surfaces and Kummer varieties (slides, video)

3/23 Tony Varilly-Alvarado Descent on K3 surfaces: Brauer group computations and challenges (slides, video)