Graduate Student Seminar

Organiser: Henri Darmon.
Time:

Thursdays when there is no QVNTS, we will convene in the Concordia CICMA room from 2:00-3:00 PM after the Séminaire Iovita which meets in the mornings.
Thursdays when there is a seminar, we will convene right after the afternoon talk, in the Concordia CICMA room (from around 4:00-5:00).

This seminar is aimed primarily at the graduate students, postdocs and faculty in the arithmetic geometry group. It will be comprised mainly of two types of talks:

1. Talks by graduate students presenting the questions they are thinking about in order to share their progress, as well as their doubts and confusions, with others in the group.

2. Short series of talks on articles that we would like to understand better. Three such articles that I would like to focus on are:
2a. A lovely article by Akshay Venkatesh on class field theory from a fairly concrete and classical viewpoint.
2b. Mumford's theory of theta functions and the role that is played by the finite Heisenberg group.

2c. An article by Marcus Appleby, Steven Flammia, and Gene Kopp describing a striking relation between Stark units and Symmetric Informationally Complete Positive Operator Valued Measures (SIC-POVMs).

Talks will typically run for 50 minutes. Suggestions by students, post-docs and faculty of talks that are both self-contained and of broad appeal are invited and strongly encouraged.

Here is the schedule so far:

September 12. Organisational meeting.

Roughly 4-6 30 minute talks by whatever willing victims we can muster, on the general theme of "what I did over the summer". Volunteers are encouraged to step up.

10:30-11:30. Hazem Hassan. Higher Green's functions for Stark-Heegner cycles.

11:30-12:00. Marti Roset Julia. Cocycles for SL(n).

2:00-2:30. Shousen Lu. Periods of cusp forms over imaginary quaratic fields.

2:30-3:00. Antoine Giard. Cocycles of higher levels.
September 19. Meeting at Concordia after QVNTS

4:00-5:00. Arihant Jain. Diagonal restrictions of rigid cocycles: a progress report.
September 26. We will meet at McGill from 10:30--12:00, after a seance at Concordia from 9:00-10:00 for the diehard afficionados of Katz-style $p$-adic $L$-functions. Recall that there will also be a picnic that day!!

9:00-10:00. Henri Darmon Katz style $p$-adic $L$-functions.

10:30-12:00. Arihant Jain. Diagonal restrictions of rigid cocycles: anextended progress report. (Cancelled due to covid.)
October 3. QVNTS Day.

I might be unable to meet in person that day, and we might use that afternoon for Arihant to meet with Eyal at McGill at 4:00 PM where Arihant will give a more generalist overview of his project. All are welcome to attend.
October 10. Non-QVNTS day.

Elias Caeiro. Venkatesh's paper on class field theory, I.
(Recall, as per the instructions above, that we will convene at Concordia at 2 PM after a morning that will, for the Iovita disciples, feature lectures by Paola and Hazem.)
October 17. QVNTS day.

Shousen Lu. Venkatesh's paper on class field theory, I.
(Recall, as per the instructions above, that we will convene at Concordia at 2 PM after a morning that will, for the Iovita disciples, feature lectures by Paola and Hazem.)
October 24. non-QVNTS day.

Rudy Ariaz. Venktatesh's paper on class field theory, II.
October 31. QVNTS day.

Highes Bellemare Venkatesh's paper on class field theory, III.
November 7. non-QVNTS day.

Arihant Jain Venkatesh's paper on class field theory, IV .
November 14. QVNTS day.

Antoine Giard Venkatesh's paper on class field theory, V .

Nikita Lvov has kindly written up an alternate description of some of the results in Chapter 7 of Venkatesh's paper.
November 21. non-QVNTS day.

Hao Fu Arithmetic level raising.
November 28. QVNTS day.

Shousen Lu CFT d'après Venkatesh, VI.
December 5. non-QVNTS+ non-Iovita day.
We will meet in Bunside 10th floor lounge, from 3-4:30.
Hazem Hassan CFT d'après Venkatesh, VII.
January 23. non-QVNTS day.
Concordia, from 2-3:30.
Hazem Hassan Introduction to the paper by Kopp et al. . Basic facts about SIC-POVMs.
January 30. QVNTS day.
Concordia, from 4-5:30.
Shousen Lu Introduction to the paper by Kopp et al. .
February 6. No seminar on that day .
February 13. No seminar on that day .
February 20. non-QVNTS, non-Iovita day.
McGill, from 2-3:45.
Antoine Giard The paper by Kopp et al., cont'd. . The Shintani-Faddeev cocycle.
February 27. QVNTS day.
Concordia, from 4-5:30.
Hazem Hassan The paper by Kopp et al., cont'd. . The Shintani-Faddeev cocycle.
March 6. McGill break, no seminar.
March 13. QVNTS day.
Concordia, from 4-5:30.
Arihant Jain Intersection of cycles on hyperbolic spaces and coefficients of modular forms (d'après Kudla, Funke-Millson). .
March 20. Tate memorial conference.
Arihant Jain Intersection of cycles, cont'd
March 27. QVNTS day.
Concordia, 4:00-5:30.
Marti Roset Julia More about rigid cocycles for $SL(n)$. .
April 3. non-QVNTS day.
McGill, 10:00-12:00.
Romain Branchereau TBA .
Note: On that day there will be a lecture about the work of Viazovska -- in the probablilty seminar, strangely enough -- at 11:30 in BH 920. So we should try to go from 10 to 11:30 with Romain's talk so those of us who are interested can transer to that talk at 11:30, and then we can aim to have lunch after that.
April 10. QVNTS day.
Concordia, 4:00-5:00.
Muhammad Manji. TBA .
April 17. Non-QVNTS day.
McGill, 10:30-12:00.
Shousen Lu. Research update. .

Antoine Giard. Rigid cocycles of higher level: a computational post-mortem. .
April 24. QVNTS day.
Concordia, 4:00-5:00.
Steven Groen (Lehigh University).
Title: The Schottky problem in characteristic 2.
Abstract: This talk explores the relation between curves and abelian varieties in characteristic 2. Abelian varieties are ubiquitous objects in number theory and algebraic geometry, possessing the structure both of a projective variety and of a group. Important examples of abelian varieties are Jacobians of curves, but most abelian varieties are not Jacobians. Hence a natural ambition, called the Schottky problem, is to characterize the Jacobians among abelian varieties. It can be beneficial to approach this problem from the angle of p-torsion group schemes in characteristic p. Equivalently, it is fruitful to study which p-torsion group schemes can occur as the p-torsion of the Jacobian of a (specific type of) curve. In this talk, we treat the 2-torsion group schemes of Jacobians of curves in characteristic 2 that admit a double cover to another curve. Through an analysis of the first De Rham cohomology, we prove that the 2-torsion group scheme of a double cover of an ordinary curve is determined by the ramification invariants of the cover, generalizing a result of Elkin and Pries. Moreover, when the base curve is not ordinary, we prove restrictions on the possible 2-torsion group schemes of the double cover. As an application, we obtain asymptotics for the 2-torsion group scheme of a one point cover whose ramification invariant goes off to infinity. .
April 31. non-QVNTS day.
There will be no seminar that day since I will be at Columbia.
May 7. non-QVNTS day.
McGill, 10:30-12:00.
Hazem Hassan: Intermediate jacobians of modular curves.