#### Recent Developments in Number Theory

May 28-30, 2019

### Plenary Speakers

- Adebisi Agboola (University of California Santa Barbara)
- Jennifer S. Balakrishnan (Boston University)
~~Alexandru Buium (University of New Mexico)~~- Bryden Cais (University of Arizona)
- Taylor Dupuy (University of Vermont)
- Kirsten Eisenträger (Pennsylvania State University)
- Matilde Lalín (Université de Montréal)
- Jennifer Park (Ohio State University)
**Bradley Lecture**: Felipe Voloch (University of Canterbury)

**Note:** Alexandru Buium will be unable to attend.

### Other Speakers

- Roberto Alvarenga (Universidade de São Paulo and UC Irvine)
- Jeremy Booher (University of Arizona)
- Ricardo Conceição (Gettysburg College)
- Edgar Costa (MIT)
~~Daniel Hast (Rice University)~~- Wanlin Li (University of Wisconsin-Madison)
- Allysa Lumley (York University)
- Travis Morrison (Univeristy of Waterloo)
- Jackson Morrow (Emory University)
- Padmavathi Srinivasan (Georgia Tech)
- Isabel Vogt (MIT and Stanford)

**Note:** Daniel Hast will be unable to attend.

### History of the Barrett Lectures

The John Barrett Memorial Lectures began in 1970 as a tribute to Dr. John H. Barrett, an expert on oscillation and disconjugacy theory of linear ordinary differential equations and influential Mathematics Department Head at the University of Tennessee. Originally dedicated to topics in Barrett’s field, ordinary differential equations, the lectures have been held annually since 1970 (except in 1990 and 2004). Since the 1980’s, however, the lecture themes have traversed the mathematical landscape: from mathematics education through computational and applied mathematics, discrete mathematics and stochastics to general relativity, nonlinear partial differential equations and topological quantum field theory.

During their history, the Barrett Lectures have been graced by a succession of distinguished mathematicians, with lectures given by Garrett Birkoff (1972); Shreeram Abhyankar (1986); Sir Michael Atiyah, Isadore Singer and Karen Uhlenbeck (1989); Sir John Ball and Lawrence C. Evans (1991); Sergiu Klainerman, Fang-Hua Lin and Michael Struwe (1995); Alice Chang, Tobias Colding and Karsten Grove (2000); Yasutaka Ihara and Yujiro Kawamata (2003); John Baez (2006); Richard Schoen, Robert Wald and Igor Rodnianski (2011); and Fernando Marques and André Neves (2013), to name a few.

A list of past Barrett Lectures and speakers can be found at Barrett Lectures History.

### Registration

Registration for all (including speakers) who would like to request financial support (see below) have to be **completed by April 15**. You can register here: ~~Registration Page~~.

Anyone who wants to attend should register, even if after the deadline, but we cannot guarantee financial support in that case. If registering late, please also send us an e-mail, so that we can *try* make the proper adjustments to the budget and organization.

All participants will also have to register on site between 8am and 8:45am on the first day of the conference. See the Schedule page for more information.

### Financial Support

This conference is sponsored by the University of Tennessee‘s Mathematics Department, as well as its College of Arts and Sciences, its Office of Research and Engagement.

This conference is also supported in part by:

- The National Science Foundation.
- The Institute for Mathematics and its Applications (IMA) through its Participating Institution (PI) Program. PI members may use IMA/PI funds to support personnel’s travel to this conference.
- The Barrett Memorial Fund
*.*

Limited funds are available to help support participant lodging and travel costs, particularly to those who are graduate students, early-career researchers, and/or members of groups that are under-represented in mathematics.

Financial support can be requested in the registration page. The conference will pay for lodging (at the Hilton Knoxville — see Accommodations below) of participants who request it in their registration (and registered by the deadline of April 15). Some extra support for travel for those not giving talks *might* also be available, but cannot be guaranteed and the actual amounts will only be known *after* the conference.

Note that all *invited speakers* will have their travel expenses (plane tickets or mileage) reimbursed by the conference.

### Travel

All participants, including invited speakers, are responsible for making their own travel arrangements.

The conference will be at University of Tennessee, located in Knoxville, TN. The closest major airport is McGhee Tyson Airport (TYS). The closest major hub for connections is the Hartsfield-Jackson Atlanta International Airport (ATL).

**Important:** international airfare *must* be purchased through a U.S. flag carrier airline in order for the reimbursement to be paid (this is a stipulation of the NSF grant).

*Under the Fly America Act, federal grant sponsors require that any participant wishing to receive travel reimbursement fly with a U.S. flag carrier airline. For international travelers, funding may be limited if airfare is not purchased through a U.S. flag air carrier (i.e., Delta, American, United) or through a foreign flag air carrier under an “Open Skies” agreement between the United States and the foreign government.*

### Accommodations

All participants who requested financial support from the conference, including speakers, will stay at the Hilton Knoxville, which is within walking distance from the conference site. (Here are directions. Google says 0.5 mile and a 12 minute walk.)

The reservations will be made directly by the math department and include parking. (If the reservation was *not* made by us, you might be charged for parking.)

Note that graduate students and post-docs might have to share rooms. You can request a roommate in the registration form. If you don’t have a request, we can assign you one.

Although we expect all participants to stay at the Hilton, those who do not need support for lodging and desire to make their own hotel reservations have also the following hotels within walking distance of the conference. **Note that most of these charge extra for parking!**

- Four Points by Sheraton. Closest to the conference site.
- The Tennessean.
- Holiday Inn Knoxville downtown.
- Hampton Inn & Suites.

### Parking

Participants staying at any of the above hotels are encouraged to walk to the conference location. On-campus hourly parking is available at Volunteer Hall Parking Garage at 1545 White Avenue. The cost is $1.00 for every half-hour. For more information, check UT’s Visitor Parking Page.

Note also that hourly parking is available at several parking garages in downtown Knoxville. For more information, check Downtown Knoxville Parking.

### Location

The Barrett Lectures will be held at the Ken And Blaire Mossman Laboratory Building (map), room 212, at the UT campus. Note that this is *not* the math department building!

Walking directions to the conference (from the Hilton Hotel) can be found in the Points of Interest Map, as well as walking directions to on-site registration and the banquet. These directions do not give you the shortest path (which would be to take Cumberland Ave), but they avoid going up and down steep hills.

Here is the Campus Map.

### Food

Coffee and light snacks will be served before the talks and during the breaks. They will be available in room 211, right beside the conference room (212).

Besides coffee and snacks, lunch will also be available (for free) to all participants at the atrium of the Mossman Building (where the talks will take place).

Please let us know in the *Additional Comments* of the registration form if you have any dietary restriction (e.g., vegetarian, vegan, allergies, gluten-free, etc.) and we will do our best to accommodate you for the lunches and banquet.

For other options for dinner, check the list of Restaurants Near Campus/Hotels.

#### Banquet

There will also be a (free) Conference Banquet on **Wednesday 05/29** at Calhoun’s On the River (Map).

Calhoun’s is within walking distance from the Hilton, so transportation will not be provided. Here are walking directions. (Google estimates 0.4 miles and 9 minutes.) Keep in mind that the way is quite hilly, going up on the way back to the hotel. If there is interest in sharing cabs/Ubers, we can try to help organizing it.

Please let us know if you would like to attend the banquet in your registration.

### WiFi

Participants can connect to the internet via eduroam if that is available from your home institution.

If you cannot connect with eduroam, you will be able to connect to UT’s WiFi as a guest. You will register as an *“Sponsored Guest”*. The username and password necessary to register will be given at the conference.

### Links

- Flyer
- Schedule
- Map of Points of Interest
- Titles and Abstracts
- List of Registered Participants
- Restaurants Near Campus/Hotels
- University of Tennessee
- Math Department

#### Contact Information

Feel free to contact Luís Finotti or Marie Jameson (co-organizers) for more information or assistance.

### On Site Registration

All participants must registered on **Tuesday (May 28) between 8am and 8:45am** at the main office (Room 227) of the Math Department, located on the second floor of Ayres Hall (map).

Ayres Hall

Ayres Hall is located on top and at the center of “The Hill”. It’s the tall building with the clock tower. If you take the main entrance of Ayres and go up the stairs right in front of it, you will arrive on the second floor right in front of the main office.

Math Department’s Office

**Note that the first talk is at 9am in a different building!** Talks will be at the Mossman Building (map), not on Ayres Hall. Mossman is right across Ayres Hall: you just need to cross Cumberland Ave. (See the Points of Interest Map to see where both buildings are located.) You should plan on about 5 minutes to walk from the registration to the conference room.

Mossman Building

Although we ask you to please register at this scheduled time, if for any reason you miss it or cannot make it, you can go to the main office at a different time (like lunch or coffee break) during regular hours.

### Schedule of Talks

(Plenary talks are in **bold**.)

Tuesday (05/28) | Wednesday (05/29) | Thursday (05/30) | |
---|---|---|---|

8:30am-9:00am | Registration and Coffee | Coffee | Coffee |

9:00am-10:00am | Matilde Lalín | Adebisi Agboola | Jennifer Park |

10:10am-10:40am | Allysa Lumley | Isabel Vogt | Ricardo Conceição |

10:40am-11:00am | Coffee | Coffee | Coffee |

11:00am-12:00pm | Jennifer Balakrishnan | Bryden Cais | Kirsten Eisenträger |

12:00pm-1:30pm | Lunch | Lunch | Lunch |

1:30am-2:00am | Edgar Costa | Wanlin Li | Roberto Alvarenga |

2:10pm-3:10pm | Taylor Dupuy | Felipe Voloch (Bradley Lecture) | Jackson Morrow (ends at 2:40) |

3:10pm-3:30pm | Coffee | Coffee | Coffee |

3:30pm-4:00pm | Jeremy Booher | Padmavathi Srinivasan | |

4:10pm-4:40pm | Travis Morrison | ||

6:30pm-9:00pm | Banquet |

### Schedule by Day

#### Tuesday (05/28)

8:00 – 8:45 | Registration | |

9:00 – 10:00 | Matilde Lalín | The Mean Value of Cubic L-Functions over Function Fields |

10:10 – 10:40 | Allysa Lumley | Complex Moments and the Distribution of Values of L(1,χD) over Function Fields with Applications to Class Numbers |

10:40 – 11:00 | Coffee | |

11:00 – 12:00 | Jennifer Balakrishnan | Rational Points on the Cursed Curve |

12:00 – 1:30 | Lunch | |

1:30 – 2:00 | Edgar Costa | Variation of Néron-Severi Ranks of Reductions of Algebraic Surfaces |

2:10 – 3:10 | Taylor Dupuy | A User’s Guide to Mochizuki’s Inequality |

3:10 – 3:30 | Coffee | |

3:30 – 4:00 | Jeremy Booher | a-Numbers of Curves in Artin-Schreier Covers |

4:10 – 4:40 | Travis Morrison | Isogeny Graphs in Cryptography |

#### Wednesday (05/29)

8:30 – 9:00 | Coffee | |

9:00 – 10:00 | Adebisi Agboola | Relative K-Groups and Rings of Integers |

10:10 – 10:40 | Isabel Vogt | Low Degree Points on Curves |

10:40 – 11:00 | Coffee | |

11:00 – 12:00 | Bryden Cais | Iwasawa Theory for Function Fields |

12:00 – 1:30 | Lunch | |

1:30 – 2:00 | Wanlin Li | Newton Polygon Stratification of the Torelli Locus in PEL-type Shimura Varieties |

2:10 – 3:10 | Felipe Voloch | Hearing Algebraic Curves and Factoring Polynomials |

3:10 – 3:30 | Coffee | |

3:30 – 4:00 | Padmavathi Srinivasan | Conductors and Minimal Discriminants of Hyperelliptic Curves in Odd Residue Characteristic |

6:30 – 9:00 | Banquet |

#### Thursday (05/30)

8:30 – 9:00 | Coffee | |

9:00 – 10:00 | Jennifer Park | The Proportion of a Certain Family of Everywhere Locally Soluble Genus 1 Curves |

10:10 – 10:40 | Ricardo Conceição | Solutions of the Hurwitz-Markoff Equation Over Polynomial Rings |

10:40 – 11:00 | Coffee | |

11:00 – 12:00 | Kirsten Eisenträger | Using Supersingular Elliptic Curves for Cryptography |

12:00 – 1:30 | Lunch | |

1:30 – 2:00 | Roberto Alvarenga | On Graphs of Hecke Operators |

2:10 – 2:40 | Jackson Morrow | Non-Archimedean Hyperbolicity and Applications |

2:40 – 3:10 | Coffee |

## Titles and Abstracts

**Relative \(K\)-Groups and Rings of Integers**

Adebisi Agboola (UCSB)

**Abstract:** Suppose that \(F\) is a number field and \(G\) is a finite group. I shall discuss a conjecture in relative algebraic \(K\)-theory (in essence, a conjectural Hasse principle applied to certain relative algebraic \(K\)-groups) that implies an affirmative answer to both the inverse Galois problem for \(F\) and \(G\) and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of \(F\). It also implies the weak Malle conjecture on counting tame \(G\)-extensions of \(F\) according to discriminant. The \(K\)-theoretic conjecture can be proved in many cases (subject to mild technical conditions), e.g. when \(G\) is of odd order, giving a partial analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield substantial new results concerning both the Galois module structure of rings of integers and the weak Malle conjecture.

Much of what we shall discuss is joint work with Leon McCulloh.

**On Graphs of Hecke Operators**

Roberto Alvarenga (USP and UC Irvine)

**Abstract:** Some of recent development in number theory is related to Hecke operators and automorphic forms (e.g., Langlands correspondence). In Bombay 1979, Don Zagier observes that if the kernel of certain operators on automorphic forms turns out to be an unitarizable representation, over the field of rational numbers \(\mathbb{Q}\), a formula of Hecke implies the Riemann hypothesis. Zagier calls the elements of this kernel toroidal automorphic forms. Moreover, Zagier asks what happens if \(\mathbb{Q}\) is replaced by a global function field and remarks that the space of unramified toroidal automorphic forms can be expected to be finite dimensional. Motivated these questions, Oliver Lorscheid introduces, in 2012, the *graphs of Hecke operators* for global function fields. This theory allowed him to prove, among other things, that the space of unramified toroidal automorphic forms for a global function field is indeed, finite dimensional. The graphs of Hecke operators introduced by Lorscheid encode the action of Hecke operators on automorphic forms.

On the other hand, Ringel (1990), Kapranov (1997), Schiffmann (2012) et al. have been developing the theory of *Hall algebra* of coherent sheaves over a smooth geometric irreducible projective curve over a finite field (in general for a finitary category).

For this talk we discuss the connection between graphs of Hecke operators and Hall algebras. In the elliptic case, Atiyah’s work on vector bundles (1957) allow us to describe (explicitly) these graphs.

**Rational Points on the Cursed Curve**

Jennifer Balakrishnan (Boston University)

**Abstract:** The split Cartan modular curve of level 13, also known as the “cursed curve,” is a genus 3 curve defined over the rationals. By Faltings’ proof of Mordell’s conjecture, we know that it has finitely many rational points. However, Faltings’ proof does not give an algorithm for finding these points. We discuss how to determine rational points on this curve using “quadratic Chabauty,” part of Kim’s nonabelian Chabauty program. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

**\(a\)-Numbers of Curves in Artin-Schreier Covers**

Jeremy Booher (University of Arizona)

**Abstract:** Let \(\pi : Y \rightarrow X\) be a branched \(\mathbf{Z}/p \mathbf{Z}\)-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic \(p>0\). We investigate the relationship between the \(a\)-numbers of \(Y\) and \(X\) and the ramification of the map \(\pi\). This is analogous to the relationship between the genus (respectively \(p\)-rank) of \(Y\) and \(X\) given the Riemann-Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the \(a\)-number of \(Y\) is not determined by the \(a\)-number of \(X\) and the ramification of the cover, so we instead give bounds on the \(a\)-number of \(Y\). We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator. This is joint work with Bryden Cais.

**Arithmetic Levi-Civita Connection**

Alexandru Buium (University of New Mexico)

**Abstract:** We present existence and uniqueness results for certain remarkable Frobenius lifts on the \(p\)-adic completions of the general linear group over the integers; these Frobenius lifts will be attached to a given symmetric matrix with integer coefficients. We will then consider the problem of defining and computing commutators of the Frobenius lifts corresponding to various primes \(p\). From a conceptual viewpoint, the above collection of Frobenius lifts attached to a symmetric integral matrix may be viewed as an arithmetic analogue, for the spectrum of the integers, of the Levi-Civita connection attached to a metric on a manifold; the collection of commutators of these Frobenius lifts can then be viewed as an arithmetic analogue of Riemannian curvature. We will show that this arithmetic Riemannian curvature satisfies congruences that are analogous to the symmetries of the classical Riemannian tensor. We will also explain how, in order for these analogies to operate, one needs to revisit some of the main concepts of classical Riemannian geometry.

**Iwasawa Theory for Function Fields**

Bryden Cais (University of Arizona)

**Abstract:** Let \(\{X_n\}\) be a \(\mathbf{Z}_p\)-tower of smooth projective curves over a perfect field \(k\) of characteristic \(p\) that totally ramifies over a finite, nonempty set of points of \(X_0\) and is unramified elsewhere. In analogy with the case of number fields, Mazur and Wiles studied the growth of the \(p\)-parts of the class groups \(\mathrm{Jac}(X_n)[p^\infty](\overline{k})\) as \(n\) varies, and proved that these naturally fit together to yield a module that is finite and free over the Iwasawa algebra. We introduce a novel perspective by proposing to study growth of the full \(p\)-divisible group \(G_n:=\mathrm{Jac}(X_n)[p^\infty]\), which may be thought of as the \(p\)-primary part of the *motivic class group* \(\mathrm{Jac}(X_n)\). One has a canonical decomposition \(G_n = G_n^{et} x G_n^{m} x G_n^{ll}\) of \(G\) into its etale, multiplicative, and local-local components, as well as an equality \(G_n(\overline{k}) = G_n^{et}(\overline{k})\). Thus, the work of Mazur and Wiles captures the etale part of \(G_n\), so also (since Jacobians are principally polarized) the multiplicative part: both of these \(p\)-divisible subgroups satisfy the expected structural and control theorems in the limit. In contrast, the local-local components \(G_n^{ll}\) are far more mysterious (they can not be captured by \(\overline{k}\)-points), and indeed the tower they form has no analogue in the number field setting. This talk will survey this circle of ideas, and will present new results and conjectures on the behavior of the local-local part of the tower \(\{G_n\}\).

**Solutions of the Hurwitz-Markoff Equation Over Polynomial Rings**

Ricardo Conceição (Gettysburg College)

**Abstract:** Let \(A\) and \(n\) be positive integers. The structure of the set of integral solutions of the equation \[(1) \qquad x_1^2+\cdots+x_n^2=Ax_1\cdots x_n \] was first studied by Hurwitz, as a generalization of Markoff’s equation (the case \(n=A=3\)). Hurwitz showed that all integral solutions can be generated by the action of certain automorphisms of the hypersurface defined by (1) on finitely many solutions. Ever since, several authors have extended Hurwitz’s work to the study of solutions of (1) over finite fields and number fields. Our goal is to discuss some progress made in understanding the solutions of (1) over the polynomial ring \(k[t]\), where \(k\) is a field.

**Variation of Néron-Severi Ranks of Reductions of Algebraic Surfaces**

Edgar Costa (MIT)

**Abstract:** We study the behavior of geometric Picard rank of a \(K3\) surface over \(\mathbb{Q}\) under reduction modulo primes. We compute these ranks for reductions of representative examples, investigate the resulting statistics and discuss the implications.

**A User’s Guide to Mochizuki’s Inequality**

Taylor Dupuy (University of Vermont)

**Abstract:** The aim of this talk is to give just the statement of Mochizuki’s inequality. We hope to make the statement of Corollary 3.12 accessible and the ideas behind its application to Szpiro-like inequalities accessible to analytic number theorist. This is joint work with Anton Hilado.

**Using Supersingular Elliptic Curves for Cryptography**

Kirsten Eisentraeger (Pennsylvania State University)

**Abstract:** Cryptosystems based on supersingular isogenies have been proposed recently for use in post-quantum cryptography. Three problems have emerged related to their hardness: computing an isogeny between two supersingular elliptic curves, computing the endomorphism ring of a supersingular elliptic curve, and computing a maximal order associated to it. We give reductions between these problems, describe the cryptosystems and discuss their security.

**The Unipotent Albanese Map and Rational Points on Varieties**

Daniel Hast (Rice University)

**Abstract:** Given a curve of genus at least 2 over a number field, Faltings’ theorem tells us that its set of rational points is finite. Provably computing the set of rational points is a major open problem, as is the question of whether the number of rational points can be uniformly bounded. We will survey some recent progress and ongoing work using the Chabauty–Kim method, which uses the fundamental group to construct \(p\)-adic analytic functions that vanish on the set of rational points. In particular, we present a new proof of Faltings’ theorem for superelliptic curves over the rational numbers (due to joint work with Jordan Ellenberg), and a conditional generalization of the Chabauty–Kim method to number fields and higher dimensions.

**The Mean Value of Cubic \(L\)-Functions over Function Fields**

Matilde Lalín (Université de Montréal)

**Abstract:** We present results about the first moment of \(L\)-functions associated to cubic characters over \(\mathbb{F}_q(T)\) when \(q\) is congruent to 1 modulo 3. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. We will explain how to obtain an asymptotic formula with a main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields. We will also discuss the case \(q\) congruent to 2 modulo 3.

**Newton Polygon Stratification of the Torelli Locus in PEL-type Shimura Varieties**

Wanlin Li (University of Wisconsin-Madison)

**Abstract:** A fundamental problem in arithmetic geometry is to determine which abelian varieties arise as Jacobians of (smooth) curves. In positive characteristic \(p\), we study this problem from the moduli perspective by asking which Newton strata intersect the Torelli locus in the moduli of abelian varieties. In this talk, I will introduce a general picture where we try to answer his question by replacing \(\mathcal{A}_g\) with a Shimura variety of PEL-type, and \(\mathcal{M}_g\) with a Hurwitz space of cyclic covers of \(\mathbb{P}^1\). Using an inductive method, when \(p = 2 \pmod{3}\), for all \(g\), we prove the existence of a smooth curve of genus \(g\) whose Newton polygon has about \(2g/3\) slopes of \(1/2\). This work is joint with Mantovan, Pries and Tang.

**Newton Polygon Stratification of the Torelli Locus in PEL-type Shimura Varieties**

Wanlin Li (University of Wisconsin-Madison)

**Abstract:** A fundamental problem in arithmetic geometry is to determine which abelian varieties arise as Jacobians of (smooth) curves. In positive characteristic \(p\), we study this problem from the moduli perspective by asking which Newton strata intersect the Torelli locus in the moduli of abelian varieties. In this talk, I will introduce a general picture where we try to answer his question by replacing \(\mathcal{A}_g\) with a Shimura variety of PEL-type, and \(\mathcal{M}_g\) with a Hurwitz space of cyclic covers of \(\mathbb{P}^1\). Using an inductive method, when \(p = 2 \pmod{3}\), for all \(g\), we prove the existence of a smooth curve of genus \(g\) whose Newton polygon has about \(2g/3\) slopes of \(1/2\). This work is joint with Mantovan, Pries and Tang.

**Complex Moments and the Distribution of Values of \( L(1,\chi_D) \) over Function Fields with Applications to Class Numbers**

Allysa Lumley (York University)

**Abstract:** In 1992, Hoffstein and Rosen proved a function field analogue to Gauss’ conjecture (proven by Siegel) regarding the class number, \( h_D \), of a discriminant \( D \) by averaging over all polynomials with a fixed degree. In this case \( h_D=|\text{Pic}(\mathcal{O}_D)| \), where \( \text{Pic}(\mathcal{O}_D) \) is the Picard group of \( \mathcal{O}_D \). Andrade later considered the average value of \( h_D \), where \( D \) is monic, squarefree and its degree \( 2g+1 \) varies. He achieved these results by calculating the first moment of \( L(1,\chi_D) \) in combination with Artin’s formula relating \( L(1,\chi_D) \) and \( h_D \). Later, Jung averaged \( L(1,\chi_D) \) over monic, squarefree polynomials with degree \( 2g+2 \) varying. Making use of the second case of Artin’s formula he gives results about \( h_DR_D \), where \( R_D \) is the regulator of \( \mathcal{O}_D \).

For this talk we discuss the complex moments of \( L(1,\chi_D) \) , with \( D \) monic, squarefree and degree \( n \) varying. Using this information we can describe the distribution of values of \( L(1,\chi_D) \) and after specializing to \( n=2g+1 \) we give results about \( h_D \) and specializing to \( n=2g+2 \) we give results about \( h_DR_D \).

If time permits, we will discuss similar results for \( L(\sigma,\chi_D) \) with \( 1/2<\sigma<1 \).

**Isogeny Graphs in Cryptography**

Travis Morrison (University of Waterloo)

**Abstract:** A large enough quantum computer will be able to break RSA and elliptic curve cryptography, so several “post-quantum” cryptosystems are under consideration for standardization in a process run by NIST. One submission, SIKE, uses isogenies of supersingular elliptic curves in a public key cryptosystem. Private keys are paths in isogeny graphs. In this talk, I will discuss the structure of these graphs, how they are used in cryptography, and how they might be used to compute endomorphism rings of supersingular elliptic curves.

**Non-Archimedean Hyperbolicity and Applications**

Jackson Morrow (Emory University)

**Abstract:** The conjectures of Green-Griffiths-Lang predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, groupless, and more. In his thesis (1993), W. Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the “correct” version, as it does not mirror complex hyperbolicity.

In recent work, A. Javanpeykar and A. Vezzani introduced a new non-Archimedean notion of hyperbolicity, which ameliorates this issue, and also stated a non-Archimedean variant of the Green-Griffiths-Lang conjecture.

In this talk, I will discuss complex and non-Archimedean notions of hyperbolicity as well as some recent progress on the non-Archimedean Green-Griffiths-Lang conjecture. This is joint work with Ariyan Javanpeykar (Mainz) and Alberto Vezzani (Paris 13).

**The Proportion of a Certain Family of Everywhere Locally Soluble Genus 1 Curves**

Jennifer Park (Ohio State University)

**Abstract:** Poonen and Voloch proved that the Hasse principle holds for either 100% or 0% of most families of hypersurfaces (specified by degrees and the number of variables). In this joint work with Tom Fisher and Wei Ho, we study one of the special families of hypersurfaces not accounted for by Poonen and Voloch, and we show that the explicit proportion of everywhere locally soluble (1,1)-curves in \( P^1×P^1 \) is about 87.4%.

**Conductors and Minimal Discriminants of Hyperelliptic Curves in Odd Residue Characteristic**

Padmavathi Srinivasan (Georgia Tech)

**Abstract:** Conductors and minimal discriminants are two measures of degeneracy in a family of hyperelliptic curves. We will outline recent progress in extending Liu’s inequality in genus 2 relating these two invariants to hyperelliptic curves of arbitrary genus when the residue characteristic is odd.

**Low Degree Points on Curves**

Isabel Vogt (MIT and Stanford University)

**Abstract:** In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer *e* such that the points of residue degree bounded by *e* are infinite. By work of Faltings, Harris–Silverman and Abramovich–Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre–Fahlaoui these criteria do not generalize to *e* at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality. This is joint work with Geoffrey Smith.

**Hearing Algebraic Curves and Factoring Polynomials**

Felipe Voloch (University of Canterbury)

**Abstract:** The possibility of telling apart algebraic curves over a finite field by their zeta function is a problem analogous to the classical question of hearing the shape of a drum. Just like drums, this is not always possible but often is. We discuss this problem and approaches to telling algebraic curves apart by looking at zeta functions of their étale covers. This problem has a surprising connection with the question of factoring polynomials over finite fields in deterministic polynomial time. We will also discuss this connection and a conjectural resolution.

## List of Registered Participants

**Adebisi Agboola**(University of California Santa Barbara), Professor,**Plenary Speaker****Vasilios Alexiades**(University of Tennessee Knoxville), Professor**Roberto Alvarenga**(Universidade de São Paulo and University of California Irvine), Postdoc,**Speaker****Ahmad Asiri**(Long Island University), Graduate Student**Jennifer Balakrishnan**(Boston University), Assistant Professor,**Plenary Speaker****Lea Beneish**(Emory University), Graduate Student**Ghanshyam Bhatt**(Tennessee State University), Associate Professor**William Bitting**(University of Tennessee Knoxville), Graduate Student**Jeremy Booher**(University of Arizona), Postdoc,**Speaker****Theodora Bourni**(University of Tennessee Knoxville), Assistant Professor**Will Bowling**(University of Tennessee Knoxville), Graduate Student**Alexandru Buium**(University of New Mexico), Professor,**Plenary Speaker****Bryden Cais**(University of Arizona), Associate Professor,**Plenary Speaker****Dustin Cartwright**(University of Tennessee), Assistant Professor**Holly Paige Chaos**(Wake Forest University), Graduate Student**Alexander Clifton**(Emory University), Graduate Student**Ricardo Conceição**(Gettysburg College), Assistant Professor,**Speaker****Edgar Costa**(MIT), Research Scientist,**Speaker****Jack Dalton**(University of South Carolina), Graduate Student**Huy Dang**(University of Virginia), Graduate Student**Jacob Dennerlein**(University of Tennessee Knoxville), Graduate Student**Lian Duan**(University of Massachusetts Amherst), Graduate Student**Taylor Dupuy**(University Of Vermont), Assistant Professor,**Plenary Speaker****Juanita Duque**(Colorado State University), Graduate Student**Kirsten Eisentraeger**(Penn State University), Professor,**Plenary Speaker****Lydia Eldredge**(Florida Sate University), Graduate Student**Kelly Emmrich**(Colorado State University), Graduate Student**Luís Finotti**(University of Tennessee Knoxville), Associate Professor**Nicholas Geis**(The Ohio State University), Graduate Student**David Gill**(Wake Forest University), Graduate Student**Andy Gilliam**(University of Tennessee Knoxville), Graduate Student**Lindsay Grinstead**(University of Tennessee Knoxville), Graduate Student**Justin Groves**(University of Tennessee Knoxville), Graduate Student**Daniel Hast**(Rice University), Postdoc,**Speaker****Peter Humphries**(University of Tennessee Knoxville), Lecturer**Marie Jameson**(University of Tennessee Knoxville), Assistant Professor**Maryam Khaqan**(Emory University), Graduate Student**Debanjana Kundu**(University of Toronto), Graduate Student**Matilde Lalín**(University of Montreal), Professor,**Plenary Speaker****Mat Langford**(University of Tennessee Knoxville), Assistant Professor**Wanlin Li**(University of Wisconsin-Madison), Graduate Student,**Speaker****Delong Li**(University of Tennessee Knoxville), Graduate Student**Adam Logan**(Tutte Institute for Mathematics and Computation), Researcher**Allysa Lumley**(York University), Graduate Student,**Speaker****Travis Morrison**(University of Waterloo), Postdoc,**Speaker****Jackson Morrow**(Emory University), Graduate Student,**Speaker****Shashikant Mulay**(University of Tennessee Knoxville), Professor**Alexis Newton**(Wake Forest University), Graduate Student**Jennifer Park**(Ohio State University), Assistant Professor,**Plenary Speaker****Tuoc Phan**(University of Tennessee Knoxville), Associate Professor**Nar Rawal**(Hampton University), Assistant Professor**Demmas Salim**(University of South Carolina), Graduate Student**Padmavathi Srinivasan**(Georgia Institute of Technology), Postdoc,**Speaker****Jiazhen Tan**(Cornell University), Undergraduate Student**Kalani Thalagoda**(UNC Greensboro), Graduate Student**Morwen Thistlethwaite**(University of Tennessee Knoxville), Professor**Stella Thistlethwaite**(University of Tennessee Knoxville), Lecturer**Tulsi Upadhyay**(Copiah-Lincoln Community College), Instructor-Mathematics**Isabel Vogt**(MIT and Stanford), Graduate Student,**Speaker****José Felipe Voloch**(University of Canterbury), Professor,**Plenary Speaker****Richard Vradenburgh**(University of Virginia), Graduate Student**Biao Wang**(SUNY at Buffalo), Graduate Student**Tom Wright**(Wofford College), Associate Professor**Shaoyun Yi**(University of Oklahoma), Graduate Student