PAlmetto Number Theory Series XXXVII

December 9th and 10th, 2023 at the University of Georgia

Description

The Palmetto Number Theory Series (PANTS) is a series of number theory meetings held in South Carolina, the Palmetto State, and other places in the Southeast. The core members of the PANTS consortium are:

PANTS is funded for the 2023-2024 academic year by a grant from the NSF and NSA.

For links to webpages of all PANTS meetings (PANTS I-XXXVII), see the PANTS home page.

Abstracts/Schedule

All talks will take place on the third floor of the Boyd Research and Education Center at UGA. A check-in desk will be outside Boyd 328.

Schedule.pdf

Abstracts.pdf

Invited Speakers

Chantal David (Concordia)

Quadratic Gauss sums at primes arguments

Abstract: Gauss sums are fundamental objects in number theory. Quadratic Gauss sums were stud- ied by Gauss who gave a simple formula depending only on the argument of the Gauss sums modulo 4. Higher degree Gauss sums behave differently. It was conjectured by Kummer that the cosines of the angles of (normalized) cubic Gauss at prime arguments are not equidis- tributed, and exhibit a bias towards positive values. This was disproved by Heath-Brown and Patterson in 1979, and they showed that (normalized) cubic Gauss at prime arguments are equidistributed. This was later generalized by Patterson to general nth-order Gauss sums. We explain in this talk what is involved in proving those results, and how to improve the results of Patterson for the distribution of quartic Gauss sums at prime arguments. (Joint work with A. Dunn, A. Hamieh and H. Lin.)

Nicole Looper (UIC)

Arakelov-Green functions in arithmetic geometry and dynamics

Abstract: This talk will discuss the role played by Arakelov-Green functions in both Diophantine geometry and dynamical systems. We will focus especially on the non-archimedean setting, as well as the dynamical analogue introduced by Baker and Rumely and its arithmetic applications. The talk will end with a discussion of recent progress in transferring these ideas to dynamical systems on higher-dimensional varieties, and of some key open problems in this direction.

PANTS features two invited faculty speakers, an invited postdoctoral speaker, and an invited graduate student speaker.

Artane Siad (IAS/Princeton)

Anomalous class group statistics and spin structures on number fields

Abstract: The Cohen–Lenstra heuristics describe the distribution of class groups of number fields. After reviewing the philosophy guiding these heuristics, we discuss anomalies observed in class group statistics over certain thin families of number fields and outline an explanation in terms of extra structure on the class group. This structure is the analogue of one enjoyed by spin 2- and 3-manifolds: a choice of spin structure determines a quadratic refinement of the mod 2 intersection pairing and the linking pairing on first homology. Based on joint work with Akshay Venkatesh.

Sun Woo Park (Wisconsin)

A probabilistic and a geometric approach to prime Selmer groups of cyclic prime twist families of elliptic curves over global function fields

Abstract: Let K = Fq(t) be the global function field over the finite field Fq of characteristic coprime to 2 and 3. Let E be a non-isotrivial elliptic curve over K. Fix a prime number l such that ul is contained in Fq. Assuming some mild conditions, we present two distinct methodologies - a probabilistic approach (utilizing Markov chains) and a geometric approach (utilizing Grothendieck-Lefschetz trace formula) - to verify the Bhargava-Lane-Lenstra-Poonen-Rains heuristics on the moments of prime Selmer groups of (l-1) dimensional abelian varieties obtained from the kernel of the norm map from the Weil restriction of E to E with respect to cyclic order l Galois extensions over K. If time allows, we will also explore how one can obtain new geometric insights from incorporating these two distinct approaches.

Local Information

The conference will take place at the Boyd Research and Education Center building on the UGA Campus. Boyd is centrally located, about a 20 minute walk from Downtown, most hotels, and places to eat. Here is a map of the area around UGA.

Accommodations: Participants will need to book their own hotels. We suggest you compare the prices of local hotels before booking, as the prices fluctuate. Some hotels in the area include, but are not limited, to the following:

Holiday Inn Express (we have a limited block of rooms here at a rate $179/night, call and mention "UGA Number Theory" or use this link and select check-in as 12/8 and checkout as 12/10. If you book early enough, you may be able to get a better rate here or elsewhere).
Hyatt Place Athens Downtown
Georgia Center Hotel

We encourage you to share rooms to reduce costs. If you would like to share accomodations but need help finding a roomate, feel free to email the local organizers, and we'll try to connect you.

Travel: The closest airport is the Hartsfield-Jackson Atlanta Airport (ATL), which is roughly 75 miles from Athens. There is a shuttle from ATL to UGA operated by Groome for around $49, these run fairly frequently but should be booked in advance. Outside rush hour, rideshares from ATL to Athens run around $100, but this fluctuates with traffic.

Parking: Most hotels are about a 20-30 minute walk from the math deparment and there are many places to eat close to hotels and within 20-30 minutes walking from the math department. There is paid parking available in lots/decks near the math department. The parking deck closest to campus is the South Deck. More information about parking at UGA can be found here. We cannot provide parking permits.

Food: There are many places to eat Downtown and in the Five Points Neighborhood. Both are about a 20-30 minute walk from the math building. Places include, but are by no means limited to:

Nearest to Boyd: Cali N Titos, The Georgia Center Hotel Savannah Room (dinner), Grindhouse Burgers, Bitty and Beau's (coffee/snacks), Georgia Center Cafe (coffee/snacks)

Downtown: Mother Pho, Thai Spoon, The Place, Taqueria Tsunami, The Globe, Mellow Mushroom Pizza

Five Points: Royal Peasant, ZZ and Simone's (dinner), Pine Bar, Local 706

Contact Local Organizers: Daniel Keliher [keliher (at) uga.edu] , Jiuya Wang [jiuya.wang (at) uga.edu]

Registration

[CLICK HERE TO REGISTER FOR PANTS XXXVII]

Registration is free. To request funding or apply to give a 20 minute contributed talk, please register by November 10th. (You may still register, but we can neither guarantee funding nor a speaking slot if you register after the deadline.)

PANTS has a limited budget from which to provide (partial) support for travel up-to approximately $200. In order to stretch our budget, we encourage car pooling and students to share lodging as much as possible.

Here is the Visitor Expense Form which may be used to request support

For reimbursement related questions, contact Hui Xue [ huixue (at) clemson (dot) edu ]

Organizers

Local Organizers: Daniel Keliher, Jiuya Wang

PANTS Contacts:

At Clemson University: Robert Dicks, Hui Xue.

At University of South Carolina: Matthew Boylan, Michael Filaseta, Frank Thorne, Wei-Lun Tsai.

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