2025: The 21st topic is Fundamental groups of curves and their arithmetic applications
April 29: Romyar Sharifi, Arithmetic through the geometry of locally symmetric spaces (video)
May 13: Chris Rasmussen, Heavenly elliptic curves over quadratic fields (slides, video)
May 20: special time 9 am ET, Shun Ishii, Galois action on pro-p fundamental groups of punctured CM elliptic curves (slides, video)
May 20: special time 10 am ET, Benjamin Collas, Oda's problem for the arithmetic homotopy of moduli spaces of curves and stack symmetries (video)
2025: The 20th topic is Rational points on higher genus curves and the Chabauty method.
Feb 11: Jennifer Balakrishnan, Rational points on modular curves and quadratic Chabauty (slides, video)
Feb 25: Alex Betts, Bounding rational points using non-abelian Chabauty (paper, slides, video )
March 4: Sachi Hashimoto, Rational points via p-adic L-functions (paper, slides, video)
March 11: Netan Dogra, Rational points on curves via algebraic cycles on surfaces (computational aspects) (paper, slides, video)
March 25: Minhyong Kim, Arithmetic topology, path integrals, and the analogy between function fields and number fields (slides, video)
April 8: Aashraya Jha, Quadratic Chabauty over number fields (paper, slides, video)
2024: The 19th topic is The Ceresa Class and Algebraic Cycles.
Aug 27: Ari Shnidman, Vanishing criteria for Ceresa classes and examples (paper, paper2, slides, video)
Sept 10: Padma Srinivasan, Algorithms for certifying nontriviality of Ceresa cycles (slides, video)
Sept 24: Wanlin Li, The Ceresa and Gross--Kudla--Schoen cycles associated to modular curves (paper, slides, video)
Oct 1: Richard Hain, What can one say about the loci where the normal function of the Ceresa cycle is constant? (slides, video)
Oct 15 Valia Gazaki, Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for 0-cycles (paper, slides, video)
Oct 22 Shou-Wu Zhang, Heights of homologically trivial cycles in families (paper, zoom)
2024: The 18th topic is Lean in algebraic number theory and arithmetic geometry.
April 9: Alex Best Formalization and Arithmetic Geometry; past, present, and future (slides, video)
April 16: Maria Inés de Frutos-Fernández Local fields in Lean (paper, slides, video)
April 23: Michael Stoll How to translate a proof into Lean (video)
April 30: Kevin Buzzard On the ingredients for Fermat (slides, video)
May 7: Alex Kontorovich: Polymath-type Projects in the age of Formalized Mathematics (slides, video)
2024: The 17th topic is Heights and Diophantine Geometry
Jan 23: Hector Pasten A criterion for algebraic degeneracy of integral points (slides, video)
Feb 6: Philipp Habegger Conjectures on Unlikely Intersections: What is known and what is open? (slides, video)
Feb 20: Ziyang Gao Sparsity of rational points on curves: What is known and what is expected? (slides, video)
Feb 27: Sara Checcoli On fields with finitely many points of bounded height: around property (N) (slides, video )
2023: The 16th topic is Perspectives on Galois groups
Oct 3: David Harbater An overview of the inverse Galois problem (slides, video)
Oct 17: Lior Bary-Soroker On the discriminant of random polynomials (slides, video)
Oct 31: Sam Schiavone In search of 17T7: explicit realizations of Galois groups (slides, video)
Nov 7: Jesse Wolfson Perspectives on Hilbert's 13th Problem (slides, video)
Nov 14: Jen Paulhus Automorphism groups of compact Riemann surfaces (slides, video)
Dec 5: Daniel Litt Galois theory of local systems (slides, video)
Dec 12: Andrew Obus Abhyankar's conjectures and fundamental groups (slides, video)
Dec 19: Noémie Combe Avatars of Grothendieck—Teichmuller groups and the absolute Galois group (slides, video)
2023: The 15th topic is Shimura varieties in positive characteristic and related topics
4/4: Bryden Cais, Iwasawa theory for class group schemes in characteristic p (slides, video)
4/18: Elena Mantovan, Density of primes of ordinary reduction for abelian varieties with simple signature (slides, video)
4/25: Maria Fox, Supersingular loci of some unitary Shimura varieties (slides, video)
5/2: Sug Woo Shin, The Langlands-Rapoport conjecture and related topics (slides, video)
5/23: Abhishek Oswal, Lifts of supersingular and almost ordinary abelian varieties (video)
2022: The 14th topic is Developments in isogeny-based cryptosystems
9/20 Kristin Lauter, Supersingular isogeny graphs in cryptography (slides, video)
9/20 Steven Galbraith, Isogeny graphs, computational problems, and applications to cryptography (paper, slides, video)
10/4 Benjamin Smith, Isogenies in genus 2 for cryptographic applications (paper, slides, video)
10/18 Wouter Castryck, An efficient key recovery attack on supersingular isogeny Diffie-Hellman (paper, slides, video)
11/8 Chloe Martindale, Torsion-point attacks on the supersingular isogeny Diffie-Hellman key exchange protocol (paper, video)
11/15: Luca De Feo, Proving knowledge of isogenies, quaternions and signatures (paper, slides, video)
12/6: Damien Robert, Applications of isogenies between abelian varieties to elliptic curves cryptosystems (slides, paper)
Resources on this topic can be found here: by Luca de Feo, Mathematics of Isogeny-based cryptography
or video
2022: The 13th topic is Arithmetic Statistics II
5/31 Manjul Bhargava, Integers that are the sum of two rational cubes (slides, video not available)
6/14 Ila Varma, Counting quartic number fields and predicting asymptotics (paper, slides, video)
6/28 Stephanie Chan, Integral points in families of elliptic curves (paper, slides, video)
7/12 Robert Lemke Oliver, Upper bounds on number fields (paper, slides, video)
Resources for arithmetic statistics can be found here: https://alozano.clas.uconn.edu/arithmetic-statistics/
2022: The 12th topic is In memory of Bas Edixhoven: some topics on modularity and Serre's conjecture.
Possible references: Edixhoven, 1992, The weight in Serre's conjectures on modular forms, Inventiones Math 109, #3.
Edixhoven lecture notes: an introduction to Serre's conjecture and overview of Khare's work
Edixhoven: Rational elliptic curves are modular, Asterisque, 2002.
Here is a schedule
4/5 Chandrashekhar Khare, Serre's conjecture and computational aspects of the Langlands program (slides, video)
4/12 Hanneke Wiersema, Minimal weights of mod-p Galois representations (slides, paper, video)
4/26 Fred Diamond, Geometric Serre weight conjectures and Θ-operators (slides, paper, video)
5/10 Ken Ribet, Ogg's conjecture for J0(N) (slides, video)
5/24 Ana Caraiani, Modularity over CM fields (slides, video)
2022: The 11th topic is Curves and abelian varieties over finite fields
The following on-line notes by Gouvea on a book by Serre provide background for this topic:
Here is a schedule
1/18 Valentijn Karemaker Mass formulae for supersingular abelian varieties (paper, slides, video)
2/1 Stefano Marseglia Computing isomorphism classes of abelian varieties over finite fields (slides, video)
2/15 Christelle Vincent Exploring angle rank using the LMFDB (slides, video)
2/22 Jeff Achter Equidistribution counts abelian varieties (paper, slides, video)
3/1 Everett Howe Deducing information about a curve over a finite field from its Weil polynomial (paper, slides, video)
2021: The 10th topic is Complex multiplication and reduction of curves and abelian varieties.
10/26 Noam Elkies Supersingular reduction of elliptic curves (paper, slides, video)
11/9 Wanlin Li A generalization to Elkies' Theorem on infinitely many supersingular primes (slides, video)
11/23 Ananth Shankar Picard ranks of K3 surfaces and the Hecke orbit conjecture (paper1, paper2, video)
12/7 Jacob Tsimerman Heights and the André Oort conjecture (slides, video)
12/14 Ben Moonen Jacobians with complex multiplication (slides, video)
2021: The 9th topic is Belyi maps and Hurwitz spaces.
The following survey papers provide background for this topic:
Aug 17 1 pm ET: John Voight, Belyi maps in number theory: a survey (slides, video)
Aug 31 1 pm ET: Edray Goins, Critical points of toroidal Belyi maps (slides, video)
Sept 14 1 pm ET: Ozlem Ejder, Dynamical Belyi maps (slides, video)
Sept 21 1 pm ET: Sam Schiavone, Belyi maps: Computation and data (slides, video)
Sept 28 1 pm ET: Irene Bouw, Belyi maps in positive characteristic (slides, video)
Oct 12 1 pm ET: David Roberts, Hurwitz-Belyi maps (slides,video)
2021: The 8th topic was Modular curves and Galois representations.
The following survey papers provide background for this topic:
Rational points on modular curves (posted with permission)
Torsion subgroups of elliptic curves over number fields
6/8 noon ET: Ekin Ozman Quadratic points on modular curves and Fermat-type equations (paper, slides, video)
6/8 1 pm ET: Lori Watson Odd degree isolated points on X1(N) with rational j-invariant (paper, slides, video)
6/22 noon ET: Pete Clark The torsion subgroup of a CM elliptic curve over a number field (paper, slides, video)
6/22 1 pm ET: Jeremy Rouse l-adic images of Galois for elliptic curves over Q (paper, slides, video)
6/29 noon ET: Barinder Banwait Explicit isogenies of prime degree over number fields (paper, slides, video)
6/29 1 pm ET: Filip Najman Q-curves over odd degree fields and sporadic points (paper, paper, slides, video)
2021: The 7th topic was 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, video)
5/4 Will Sawin The freeness alternative to thin sets in Manin's conjecture (slides, video)
5/11 Yuri Tschinkel Height zeta functions (slides, video)
5/18 Damaris Schindler Density of rational points near/on compact manifolds (paper, slides)
2021: 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)
2020: 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)
2020: The fourth topic was Rational points on elliptic curves .
The following survey paper provides background for this topic:
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)
2020: 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 (video not available)
2020: 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.
2020: 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 ℓ-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 (slides, video)
Mar 3: David Zureick-Brown. Moduli spaces and arithmetic statistics (video)
We would like to thank the ACNS video conference services at Colorado State University for hosting the virtual BlueJeans platform and Chris Chagnon for providing technical support for the first topic.