# Workshop on Elliptic Curves, Torsors, and L-functions

## March 24-27, 2017

### University of Virginia, Charlottesville, VA

#### The workshop will be followed by the lectures of B. H. Gross in the distinguished lecture series Virginia Mathematics Lectures March 27-29, 2017

 Speakers: Mirela Çiperiani University of Texas Vladimir Chernousov University of Alberta Pete Clark University of Georgia Wei Ho University of Michigan Kiran Kedlaya University of California, San Diego Alexander Merkurjev University of California, Los Angeles Raman Parimala Emory University Gopal Prasad University of Michigan Sujatha Ramdorai University of British Columbia Arul Shankar Harvard Univeristy Lucien Szpiro City University of New York John Voight Dartmouth College

# Participants

 Nivedita Bhaskhar UCLA Kestutis Cesnavicius UC Berkeley Valdimir Chernousov University of Alberta Michael Chou University of Connecticut Mirela Çiperiani University of Texas Pete Clark University of Georgia Joseph Gunther City University of New York Bastian Haase Emory University Wei Ho University of Michigan Kiran Kedlaya University of California, San Diego Bao V. Le Hung University of Chicago Brandon Levin University of Chicago Eli Matzri Bar-Ilan University Alexander Merkurjev University of California, Los Angeles Sumit Chandra Mishra Emory University Jackson Morrow Emory University Raman Parimala Emory University Gopal Prasad University of Michigan Sujatha Ramdorai University of British Columbia Igor Rapinchuk Michigan State University Arul Shankar Harvard Univeristy Lucien Szpiro City University of New York Sergey Tikhonov Belarusian State University Ila Varma Harvard University John Voight Dartmouth College

# Titles and Abstracts

### Talks

Vladimir Chernousov (University of Alberta) Classification of torsors over Laurent polynomial rings

Abstract. We will discuss in the talk the classification of torsors of reductive group schemes over Laurent polynomial rings and its applications in infinite dimensional Lie algebras. Joint work with P. Gille and A. Pianzola.

Mirela Çiperiani (UT Austin) Divisibility questions for genus one curves

Abstract. Genus one curves with a fixed Jacobian can be viewed as elements of the Weil-Chatelet group. We will discuss divisibility questions within this group. This leads us to analyzing the divisibility properties of the Tate-Shafarevich group. There are two related questions:
1. Are the elements of the Tate-Shafarevich group divisible within the Weil-Chatelet group? (Cassels' question)
2. How does the Tate-Shafarevich group intersect the maximal divisible subgroup of the Weil-Chatelet group? (Bashmakov's question)
We will discuss our answers to these questions. This is joint work with Jakob Stix.

Pete Clark (University of Georgia) Hasse Principle Violations for Quadratic Twists

Abstract. If $X/K$ is a nice curve over a number field endowed with a $K$-rational involution $\iota$, then to every quadratic extension $L/K$ one can consider the twist of $X$ by $L/K$ and $\iota$. Some years ago I showed that for all squarefree $N > 163$, there are infinitely many primes $p$ such that the Atkin-Lehner twist of $X_0(N)$ by $\mathbb{Q}(\sqrt{p})/\mathbb{Q}$ violates the Hasse Principle (HP). This led to a Twist Anti-Hasse Principle (TAHP): criteria on $X/K$ and $\iota$ in order for infinitely many quadratic twists to violate HP. Recently, in joint work with J.H. Stankewicz we extended TAHP so as to apply to Atkin-Lehner twists of Shimura curves $X^D$. Here I will also discuss some results involving Atkin-Lehner twists of Shimura curves $X^D_0(N)$. The hypotheses of TAHP are never satisfied when $\iota$ is a hyperelliptic involution. But we will show that for any hyperelliptic curve $X/\mathbb{Q}$ of genus $g \geq 3$, infinitely many quadratic twists violate the Hasse Principle iff $X$ has no rational Weierstrass point.... conditionally on the ABC conjecture.

Wei Ho (University of Michigan) Noncommutative Galois closures and moduli problems

Abstract. In this talk, we will discuss the notion of a Galois closure for a possibly noncommutative algebra. We will explain how this problem is related to certain moduli problems involving genus one curves and torsors for Jacobians of higher genus curves. This is joint work with Matt Satriano.

Kiran Kedlaya (UCSD) Sato-Tate groups

Abstract. Given an arithmetic L-function, the distribution of its normalized Euler factors is conjecturally always controlled by random matrices in a certain compact Lie group. This assertion includes a few known statements (the Chebotarev density theorem, the Sato-Tate conjecture for elliptic curves) and a great many statements which remain open. Indeed, in many cases, even a precise conjectural statement can be elusive; for instance, the possible groups occurring for curves of genus 2 are known, but not for curves of genus 3 or higher.

Alexander Merkurjev (UCLA) Rationality problem for classifying spaces of algebraic groups

Abstract. Many classical objects in algebra such as simple algebras, quadratic forms, algebras with involutions, Cayley algebras etc, have large automorphism groups and correspond to torsors of algebraic groups. Each type of algebraic objects (torsors) is given by points of the classifying space of an algebraic group. The rationality property for the classifying space yields description of objects by means of algebraically independent parameters. In the talk we will discuss the rationality property of classifying spaces of various algebraic groups.}

Raman Parimala (Emory University) Indices of a principal homogeneous spaces under a connected linear algebraic group

Abstract. In this talk we discuss the question of whether a principal homogeneous space under a connected linear algebraic group admits a closed point of degree dividing its index.}

Gopal Prasad (University of Michigan) A new approach to unramified descent in Bruhat-Tits theory

Abstract. We will present a new approach to unramified (or etale) descent in Bruhat-Tits theory of reductive groups over a discretely valued field k with Henselian local ring, which appears to be conceptually simpler, and more geometric, than the original approach of Bruhat and Tits. We are able to derive the main results of the theory over $k$ from the theory over the maximal unramified extension $K$ of $k$. Even in the most interesting case for number theory and representation theory, where $k$ is a locally compact nonarchimedean field, the geometric approach described in this paper appears to be considerably simpler than the original approach.}

Sujatha Ramdorai (University of British Columbia) On the fine Selmer group of elliptic curves

Abstract. We shall talk about the conjectures that were proposed jointly with J. Coates on the fine Selmer groups of elliptic curves, and the information that can be gleaned from the residual representations in attempts to tackle these conjectures.}

Arul Shankar (University of Toronto) Subconvex bounds on the 2-torsion in the class groups of number fields (joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao)

Abstract. Given a number field K of fixed degree $n$ over $\mathbb{Q}$, a classical theorem of Brauer--Siegel asserts that the size of the class group of $K$ is bounded by $O_\epsilon(|Disc(K)|^{(1/2+\epsilon)})$. However, it is conjectured that the $p$-torsion subgroup of the class group of $K$ is bounded by $O_\epsilon(|Disc(K)|^\epsilon)$, for any fixed prime $p$. Only the case $n=p=2$ of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" Brauer--Siegel bound. In this talk, we will prove a subconvex bound on the size of the 2-torsion in the class groups of number fields, for all degrees $n$. We will also discuss an application of this result towards proving improved bounds on the ranks of elliptic curves, integral points on elliptic curves, and ranks of the Jacobians of hyperelliptic curves.}

Lucien Szpiro (CUNY Graduate Center) Proof of the Shafarevich conjecture'' for self maps of $\mathbb{P}^1$ (joint with Tom Tucker and Lloyd West)

Abstract. Let $S$ be a finite set of places of $K$ a number field or a function field over a field $k.$ We prove finiteness up to conjugacy for the set of nonisotrivial, separable, tame, self maps of $\mathbb{P}^1_K$ of a given degree and ramified at least in 3 points, with 'critically excellent reduction' outside of $S$. The simple notion of good reduction (conservation of the degree in reduction) is not adequate, as one can see by looking at monic polynomial maps. Critically excellent reduction says that the ramification and branch locus are étale. Apply to the Lattès map associated to the multiplication by 2 in an elliptic curve this reproves Shafarevich's original result on elliptic curves. We will recall the different 'Shafarevich conjectures' already proved (Finite maps, Curves over function fields, Abelian varieties over number fields, K3 surfaces).

John Voight (Dartmouth College) Semi-arithmetic points

Abstract. We present a method for constructing algebraic points on elliptic curves defined over number fields, combining the theory of Belyi maps and quaternionic Shimura varieties; our method generalizes the construction of Heegner points arising from classical modular curves. In particular, we report on some computational investigations of these points.

### Short Communications

Nivedita Bhaskhar (UCLA) Reduced Whitehead groups of prime exponent algebras over $p$-adic curves

Abstract. The question of whether every reduced norm one element of a central simple algebra A is a product of commutators was formulated in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group $\mathrm{SK}_1(A) = \mathrm{SL}_1(A)/[A^*,A^*]$. In 1991, Suslin conjectured that if the index of the central simple algebra $A/K$ is not square free, then $\mathrm{SK}_1(A)$ is generically non-trivial, i.e, there exists a field extension $F/K$ such that $\mathrm{SK}_1(A_F)$ is non-trivial. This conjecture is supported by evidence provided by the affirmative answer of Merkurjev for algebras with indices divisible by 4.
However it is a theorem of Merkurjev/Rost that for central simple algebras of degree 4, the Whitehead group is trivial over fields of cohomological dimension 3. This is a consequence of an injection of $\mathrm{SK}_1(A)$ into a sub-quotient of the degree 4 Galois cohomology group which led Suslin to ask whether the same was true for index $\ell^2$ algebras for any prime $\ell$ over cohomological dimension 3 fields. In this talk, we address this question for $\ell$-torsion, degree $\ell^2$ algebras over function fields of $p$-adic curves where $\ell$ is any prime not equal to $p$. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields to show triviality of the reduced Whitehead group.

Kestutis Cesnavicius (UC Berkeley) The Manin constant in the semistable case

Abstract. For an optimal modular parametrization $J_0(n) \to E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^0(E, \Omega^1)$. Multiple authors generalized his conjecture to higher dimensional newform quotients. We will discuss the semistable cases of the Manin conjecture and of its generalizations using a technique that establishes general relations between the integral $p$-adic etale and de Rham cohomologies of abelian varieties over $p$-adic fields.

Michael Chou (University of Connecticut) Growth of torsion on elliptic curves from $\mathbb{Q}$ to the maximal abelian extension

Abstract. Torsion of an elliptic curve over a number field is finite due to the Mordell-Weil theorem. However, even in certain infinite extensions of $\mathbb{Q}$ we have that torsion is finite. We will discuss known results for torsion over infinite extensions and present an almost classification over the maximal abelian extension of $\mathbb{Q}$.

Joseph Gunther (CUNY) Unexpected quadratic points on random hyperelliptic curves

Abstract. On a hyperelliptic curve over $\mathbb{Q}$, there are infinitely many points defined over quadratic fields: just pull back rational points of the projective line through the degree two map. But for a positive proportion of genus $g$ odd hyperelliptic curves over $\mathbb{Q}$, when ordered by height, we give a bound on the number of quadratic points not arising in this way. The proof uses tropical geometry and work of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Jackson Morrow.

Bao V. Le Hung (University of Chicago) Recent work on the weight part of Serre's conjecture (Part I)

Abstract. (Joint with Brandon Levin.) We will discuss recent results towards the weight part of Serre's conjecture for $\mathrm{GL}_n$ as formulated by Herzig. The conjecture predicts the set of weights where an odd $n$-dimensional mod $p$ Galois representation will appear in cohomology in terms of the restriction of the representation to the decomposition group at $p$. This is joint work with Daniel Le and Stefano Morra.

Brandon Levin (University of Chicago) Recent work on the weight part of Serre's conjecture (Part II)

Abstract. (Joint with Bao V. Le Hung.) We will discuss recent results towards the weight part of Serre's conjecture for $\mathrm{GL}_n$ as formulated by Herzig. The conjecture predicts the set of weights where an odd $n$-dimensional mod $p$ Galois representation will appear in cohomology in terms of the restriction of the representation to the decomposition group at $p$. This is joint work with Daniel Le and Stefano Morra.

Eli Matzri (Bar-Ilan University) A birational interpretation of the Severi-Brauer variety of a division algebra

Abstract. Let $F$ be a field and $D/F$ a finite dimensional $F$-central division algebra of index $n$. The Severi-Brauer variety $SB(D)$ of $D$ is the variety of minimal left ideals, determining D up to isomorphism. In particular, for any field extension, $K/F$ the algebra $D \otimes_F K$ is split (i.e isomorphic to a matrix algebra) if and only if $SB(D)(K)$ is non-empty. In the talk we will give a birational description of SB(D) as the elements of reduced norm zero in a "generic" linear subspace of D of dimension n+1. If time permits, we will show how to prove Amitsur's conjecture for symbol algebras stating that for two symbol algebras $D_i$ $(i = 1, 2)$ of the same degree, the function fields, $F(SB(D_1))$ and $F(SB(D_2))$, are isomorphic if and only if the subgroups generated by their classes in $\mathrm{Br}(F)$ coincide: $\langle [D_1] \rangle = \langle [D_2] \rangle$.

Jackson Morrow (Emory University) Sporadic cubic torsion

Abstract. In Mazur's celebrated 1978 Inventiones paper, he classified the torsion subgroups which can occur in the Mordell-Weil group of an elliptic curve over $\mathbb{Q}$. His result was extended to elliptic curves over quadratic number fields by Kamienny, Kenku, and Momose, with the full classification being completed in 1992. What both of these cases have in common is that each subgroup in the classification occurs for infinitely many elliptic curves; however, this no longer holds for cubic number fields. In 2012, Najman showed that there exists a unique (up to $\overline{\mathbb{Q}}$-isomorphism) elliptic curve whose torsion subgroup over a particular cubic field is $\mathbb{Z}/21 \mathbb{Z}$. This curve yielded the first sporadic example of a torsion subgroup. In this talk, we will recall previous literature concerning torsion subgroups of elliptic curves over number fields, introduce new results about sporadic points on the modular curves $X_1(N)$ and $X_1(M,N)$, and discuss some of the tools used in the analysis of cubic points on these modular curves.

Igor Rapinchuk (Michigan State University) Finiteness results for the genus of division algebras over finitely generated fields

Abstract. In this talk, I will discuss a finiteness result for the genus of finite-dimensional central division algebras over finitely generated fields. Some explicit upper bounds on the size of the genus will be given for division algebras defined over function fields of curves over number fields. This is joint work with V. Chernousov and A. Rapinchuk.

Sergey Tikhonov (Belarusian State University) Division algebras with infinite genus

Abstract. The genus ${\bf gen}(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D']\in Br(F)$, where $D'$ is a central division $F$-algebra having the same maximal subfields as $D$. In the talk we will discuss the construction of algebras with infinite genus.

Ila Varma (Harvard University) The average size of 2-torsion elements in ray class groups of cubic fields

Abstract. In 2005, Bhargava computed the average size of the 2-torsion subgroup in the class groups of cubic fields ordered by discriminant by proving asymptotics on nowhere overramified quartic fields. Class field theory gives a relationship between the number of 2-torsion ideal classes of cubic fields and the number of nowhere overramified quartic fields. I will describe a generalization of this correspondence to 2-torsion elements of the ray class groups of cubic fields, which can enumerated using certain pairs of quartic and quadratic fields satisfying explicit ramification conditions. I will illustrate how one can apply Bhargava's asymptotics on the number of quartic fields of bounded discriminant to obtain the mean number of 2-torsion elements in ray class groups with fixed conductor of cubic fields ordered by discriminant.

# Schedule

All talks on Friday - Sunday (3/24 - 3/26) will be in KER 317, and tea/coffee breaks in the lounge across the hall.

The breakfasts on Saturday and Sunday (3/25 & 3/26) will also be in the lounge.

Short communications on Monday (3/27) will be in Dell 2 100.

### Friday, March 24 (please note changes!)

 1:30 - 2:20 Kiran Kedlaya (UCSD) Sato-Tate groups 2:30 - 3:15 Tea 3:15 - 4:05 Lucien Szpiro (CUNY Graduate Center) Proof of the 'Shafarevich conjecture' for self maps of $\mathbb{P}^1$

### Saturday, March 25

 9:00 - 9:30 Light breakfast 9:30 - 10:20 Raman Parimala (Emory University) Indices of principal homogeneous spaces under a connected linear algebraic group 10:30 - 11:20 Alexander Merkurjev (UCLA) Rationality problem for classifying spaces of algebraic groups 11:30 - 12:00 Tea 12:00 - 12:50 Vladimir Chernousov (University of Alberta) Classification of torsors over Laurent polynomial rings 1:00 - 2:45 Lunch 2:45 - 3:35 Sujatha Ramdorai (University of British Columbia) On the fine Selmer group of elliptic curves 3:45 - 4:15 Tea 4:15 - 5:05 Mirela Çiperiani (UT Austin) Divisibility questions for genus one curves 5:15 - 6:05 John Voight (Dartmouth College) Semi-arithmetic points 6:30 - 8:30 Dinner at Afghan Kabob (400 Emmet St.)

### Sunday, March 26

 9:15 - 10:00 Light breakfast 10:00 - 10:50 Arul Shankar (University of Toronto) Subconvex bounds on the 2-torsion in the class groups of number fields 11:00 - 11:50 Wei Ho (University of Michigan) Noncommutative Galois closures and moduli problems 12:00 - 12:50 Gopal Prasad (University of Michigan) A new approach to unramified descent in Bruhat-Tits theory 1:00 - 3:00 Lunch

#### Short communications

 3:00 - 3:20 Viet Bao Le Hung (University of Chicago) Recent work on the weight part of Serre's conjecture (Part I) 3:30 - 3:50 Brandon Levin (University of Chicago) Recent work on the weight part of Serre's conjecture (Part II) 4:00 - 4:30 Tea 4:30 - 4:50 Michael Chou (University of Connecticut) Growth of torsion on elliptic curves from $\mathbb{Q}$ to the maximal abelian extension 5:00 - 5:20 Igor Rapinchuk (Michigan State University) Finiteness results for the genus of division algebras over finitely generated fields 5:30 - 5:50 Sergey Tikhonov (Belarusian State University) Division algebras with infinite genus

### Monday, March 27 (in Dell 2 100)

#### Short communications

 9:00 - 9:20 Nivedita Bhaskhar (UCLA) Reduced Whitehead groups of prime exponent algebras over p-adic curves 9:30 - 9:50 Eli Matzri (Bar-Ilan University) A birational interpretation of the Severi-Brauer variety of a division algebra 10:00 - 10:20 Ila Varma (Harvard University) The average size of 2-torsion elements in ray class groups of cubic fields 10:30 - 11:00 Tea 11:00 - 11:20 Jackson Morrow (Emory University) Sporadic cubic torsion 11:30 - 11:50 Joseph Gunther (CUNY) Unexpected quadratic points on random hyperelliptic curves 12:00 - 12:20 Kestutis Cesnavicius (UC Berkeley) The Manin constant in the semistable case

# Practical Tips

### 1. Hotel information.

The hotel is The Graudate Charlottesville (1309 West Main Street, (434) 295-4333, website). It is located at the intersection of West Main Street and 13th Street NW. If you are coming from the airport, you should take a taxi or Uber and ask for a receipt (for travel reimbursement). If you are coming from the train station, it is an easy walk from there (less than half a mile). Go up the staircase opposite the station house. The street you see is West Main Street. Cross to the far side and walk left along West Main Street until you reach the hotel.

### 2. Wifi.

You can get a passcode to access wifi on the UVA grounds (i.e. campus) from Raven James in the math office in Kerchof Hall (see below for directions to Kerchof Hall). Please pick up your reimbursement form at the same time.

### 3. UVA grounds information.

The easiest way to find your way around UVA is to use the interactive web map available here: map (also click links below). The buildings of relevance as far as the workshop are:
(1) Kerchof Hall - #22 on the McCormick Road Area map - this is where the Math Department is located, and where all of the lectures on Friday, Saturday, and Sunday will take place (official address 141 Cabell Drive);
(2) Physics - #41 on the McCormick Road Area map, where Prof. Gross's Monday and Wednesday lectures will take place;
(3) Monroe Hall - #26 on the Central Grounds map, where Prof. Gross's Tuesday lecture will take place;
(4) Dell 2 - #52 on the McCormick Road Area map, where Monday's short communications will take place.

Googlemaps knows the way to Kerchof Hall (by name or as 141 Cabell Drive). Specifically, to reach Kerchof from the hotel, exit onto West Main Street, turn left, cross West Main Street at the first crosswalk, and then turn right onto Jefferson Park Avenue. Walk down Jefferson Park Avenue until you reach Brandon Avenue. At this point, cross Jefferson Park avenue and walk up the ramp to the left. Once up the ramp, continue to walk straight until you reach a staircase. When you get to the top of this staircase, you will be in a small parking lot. Walk along the parking lot as it curves to your left. Eventually, you will see Kerchof Hall on your left. The whole walk is about three-quarters of a mile.

The departmental office is right across from the main entrance (this is where you need to find Raven James and get your wifi password as well as your reimbursement forms). Our offices as well as the lounge are one level above: KER 208 (Andrew), KER 229 (Lloyd), and KER 307 (Andrei). The departmental phone number is 434-924-4919. Another building that may be useful is Newcomb Hall - there is a food court in this building which can be used for lunch (it will be open on weekdays). The neighborhood around the hotel also has lots of restaurants for lunch and dinner.

### 4. Giving a Talk.

We will provide an overhead projector and a computer projector. If you wish you can connect your own laptop, but you can also copy your talk on a flash drive and use ours. You may, of course, use the chalkboard as well.

### 5. Dinner.

The conference dinner will be at the Afghan restaurant Afghan Kabob (400 Emmet Street N, website ), on Saturday, March 25. Despite the name, the restaurant has lots of vegetarian options. The last lecture will end at 6:05, and our dinner reservation is for 6:30, so we will leave directly from Kerchof. It is a bit under a mile from Kerchof. Some people will walk, and we will be able to offer rides for the others.

Very important: we need to tell the restaurant how many people will come, so please e-mail Andrew (andrewobus@gmail.com) at your earliest convenience to confirm that you intend to come to dinner. The restaurant does not serve alcohol, but Andrew will lead an outing to a bar after dinner for those who would like to go out (unfortunately, we cannot reimburse alcohol).