During the 2023-24 academic year the School will have a special program on the p -adic arithmetic geometry, organized by Jacob Lurie and Bhargav Bhatt, who will be the Distinguished Visiting Professor.

The last decade has witnessed some remarkable foundational advances in p-adic arithmetic geometry (e.g., the creation of perfectoid geometry and the ensuing reorganization of p-adic Hodge theory). These advances have already led to breakthroughs in multiple different areas of mathematics (e.g., significant progress in the Langlands program and the resolution of multiple long-standing conjectures in commutative algebra), have uncovered new phenomena that merit further investigation (e.g., the discovery of new structures on algebraic K-theory, new period spaces in p-adic analytic geometry, and new bounds on torsion in singular cohomology), and have made hitherto inaccessible terrains more habitable (e.g., birational geometry in mixed characteristic). This special year intends to bring together a mix of people interested in various facets of the subject, with an eye towards sharing ideas and questions across fields.

The study of rational points on varieties is a field of special interest to arithmetic geometers. Over the past few decades, many techniques have been used to decide whether a variety over a number field has a rational point or not, and even to describe those points completely. In this program, we are mainly interested in the study of rational points on modular curves.

Elliptic curves, modular forms and modular curves are central objects in arithmetic geometry. Modular curves can be thought of as moduli spaces for elliptic curves with extra level structures. The objective of this program is to understand the theoretical and computational aspects of determining K-rational points on modular curves X_H(K) for various fields K and subgroups H of GL_2(ℤ/Nℤ) for any natural number N.

In the mini-courses, we give an advanced introduction to the theory of rational points on modular curves, under both theoretical and computational aspects. These courses would include an introduction to the geometry of modular curves, their ℚ-rational points, classical and non-abelian Chabauty methods, and related computational aspects. We also attempt to strike a balance between the more advanced topics and the down-to-earth examples.

In the discussion meeting, we wish to bring many experts across the world in the area of arithmetic geometry together to share their ideas and the current state of research that will facilitate future research in this direction. We strongly encourage participation of young researchers.

On the occasion of the 60th birthday of Tonghai Yang, we would like to bring together people from modular forms and arithmetic geometry to discuss recent advances in this conference.

The conference is about recent developments in Arithmetic Algebraic Geometry. Its central theme is the geometry of Shimura varieties and related spaces in all its facets. Topics to be covered include: Integral models of Shimura varieties and the geometry of their reductions, p-adic and perfectoid geometry, special cycles on Shimura varieties, moduli spaces of Galois representations and (φ, Γ)-modules.

The goal of the FRNTD is to provide a venue for faculty, graduate students, and undergraduates on the Front Range who are interested in number theory to meet, learn, and collaborate.

For more information and to register, please visit our website: https://sites.google.com/colorado.edu/front-range-number-theory-day/home. If you would like to give a five-minute talk, please include your title on the registration form.

Weekend regional conference in number theory and arithmetic geometry, featuring two mini-courses and additional research lectures.

Fall workshop for the IAS special year on p-adic arithmetic geometry in 2023--2024. Registration form on website.

This is the first meeting of the Arithmetica Transalpina, a joint Number Theory seminar between ETH Zürich, FernUni Schweiz and the universities of Milan, Padova and Genova.

Magma is a world-leading computer algebra system developed by the Computational Algebra Group at the University of Sydney. It supports cutting-edge computations in algebra, number theory, algebraic geometry, and algebraic combinatorics and is used on a daily basis by thousands of research mathematicians in over 70 countries.

The group is led by Professor John Cannon, the founder of Magma, and of its predecessor Cayley. A world expert on the development and implementation of algorithms for mathematics, John's scientific contributions are recognised via many awards including the CSIRO Medal (1993); the ATSE Clunies Ross Award (2001); the Richard D. Jenks Memorial Prize (2006); and his election in 2022 as Fellow of the Australian Academy of Science.

This meeting brings together a group of leading international researchers who have many connections to both John and the broad subject areas. We will celebrate two events: the 30th anniversary of the official launch of Magma, and the outstanding contributions of John as he reaches a personal milestone.

The main conference , 27-29 November 2023
A workshop on Computational Number Theory , 30 November - 1 December 2023

Please register if you would like to attend the conference.

Courses:

 Valentijn Karemaker: TBA
 Ben Moonen: TBA
 Rachel Pries: TBA
 Joe Silverman: TBA

With Clay lecturer Barry Mazur

This workshop, sponsored by AIM and the NSF, will be devoted to the study of degree d points on algebraic surfaces over a number field.

The study of degree d points on algebraic curves over ℚ is a rich and mature area of research, with the Abel-Jacobi map and the Mordell-Lang conjecture providing powerful tools for exploration. However, for higher dimensional varieties there is no such approach that works in general. Because of this, we lack even a conjectural framework for understanding which higher dimensional varieties over ℚ should have "many" degree d points.

The workshop will focus on questions aimed at addressing this dearth, concentrating on the case of algebraic surfaces. For instance, what does it mean for a surface over ℚ to have "many" degree d points? What are some geometric constructions that give rise to abundant degree d points? Are these related to geometric measures of irrationality? If HilbdX has a Zariski dense set of ℚ-points for some small d, does that yield any arithmetic or geometric consequences for X? If X embeds into its Albanese, can we obtain results analogous to that of curves?

Participants will be researchers from a broad array of backgrounds (e.g., arithmetic of surfaces, geometry of Hilbert schemes of surfaces, geometric measures of irrationality, arithmetic of 0-cycles, to name a few), ideally with a curiosity and interest in arithmetic questions.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

In the 1980s, Ceresa exhibited one of the first naturally occurring examples of an algebraic cycle, the Ceresa cycle, that is in general homologically trivial but algebraically nontrivial. In the last few years, there has been a renewed interest in the Ceresa cycle, and other cycle classes associated to curves over arithmetically interesting fields, and their interactions with analytic, combinatorial, and arithmetic properties of those curves. We hope to capitalize on this momentum to bring together different communities of arithmetic geometers to fully explore explicit computations around the arithmetic and geometry of cycles, when these various approaches are systematically combined.

The summer school is dedicated to graduate students and young researchers, and aims to give an introduction to recent techniques and topics of additive combinatorics. The lectures of the summer school will concentrate on recent developments of the polynomial method, some combinatorial methods of additive combinatorics, and the introduction of Fourier analytic techniques connected to them. The main topics will be presented by top researchers of the area.

The lecturers will be Julia Wolf, Christian Elsholtz, Peter Pal Pach, Sean Prendiville.

This conference is devoted to the most recent results of Additive Combinatorics. The topic of the conference is aimed to emphasize the rich interactions between additive combinatorics, harmonic analysis and number theory. The conference will bring together some recognized experts of the field, junior researchers (postdoctoral fellows and graduate students), and senior researchers from various aspects of the main topic. Beside the discussion on the recent progress in the field, it is also aimed to initiate interaction and collaboration among the participants.

See conference website

After the success and impact of Spec(Q⎯⎯⎯⎯), held at the Fields Institute in 2022, Spec(Q¯(2πi)) again aims to celebrate and promote research advances of LGBTQ2I (Lesbian, Gay, Bisexual, Transgender, Queer, 2-spirit , Intersex) mathematicians specialising in algebraic geometry, arithmetic geometry, commutative algebra, and number theory. The first edition of the conference proved to be extremely important to lay the foundations for a fertile, supportive and stimulating scientific queer community in the areas of algebraic geometry, commutative algebra and number theory. Building on the strengths of the first edition, Spec(Q¯(2πi)) will create an empowering and engaging environment which provides LGBTQ2I visibility in algebraic geometry, will support junior LGBTQ2I academics, and will crystallise new collaborative networks for participants.

Algebraic geometry, classically, is the study of the geometry of solutions of polynomial equations; through modern advances it has become an intersectional mathematical field, drawing from various aspects of algebra, number theory, geometry, combinatorics and even mathematical physics. This conference aims to highlight strong mathematical research in a wide array of topics in algebraic geometry, broadly defined. The conference will feature some plenary talks by world-leading researchers from a range of areas of algebraic geometry. To facilitate new connections across the various threads of algebraic geometry, plenary talks at Spec(Q¯(2πi)) will be aimed at a general algebro-geometric audience.

Hilbert’s twelfth problem asks for explicit constructions of the abelian extensions of a given number field, similar to what is known for the rational numbers and for imaginary quadratic fields. These abelian extensions are known as class fields because their Galois groups are identified with certain generalized ideal class groups. In the two known cases, the class fields are obtained via the adjunction of roots of unity and of torsion points on elliptic curves with complex multiplication. These are special values of complex analytic functions – the exponential function and elliptic functions with complex multiplication. Hilbert may have envisioned the use of special values of complex analytic functions to construct class fields of more general base fields.

In the 1970s, Harold Stark proposed a strikingly original approach to the generation of class fields, based on his conjectures on the leading term of Artin L-functions at s = 0 [St75]. In the case of abelian L-functions with a simple zero at s = 0, Stark predicted that the first derivative was the logarithm of a unit in the respective class field [St76], so exponentiating this derivative would give a generator for the abelian extension. In the two known cases, this reduced to the theory of circular and elliptic units, thanks to Dirichlet’s analytic class number formula and Kronecker’s limit formula. Although there is now extensive computational evidence that Stark’s conjecture is correct, there has been little progress on its solution.

In the 1980s Benedict Gross formulated some p-adic [Gr82] and tame [Gr88] analogues of Stark’s conjectures, which gave more information on the p-adic expansions of the conjectural units. Since the p adic L-functions involved in Gross’s conjecture are related to certain Galois modules via the main conjecture in Iwasawa theory, these conjectures have proved more amenable than their complex analogs. Refinements of the Gross-Stark conjecture were proposed in [DD06], and the p-adic conjectures of [Gr82] was proved in [DDP11]. This line of argument has culminated in the recent work of Samit Dasgupta and Mahesh Kakde [DKa], [DKb] which, by proving a large part of the conjectures of [Gr88] (along with the refinement [DD06] of the conjectures of [Gr82] in the broader setting of totally real fields) leads to a p−adic solution to Hilbert’s twelfth problem for this large class of fields.

The goal of this workshop is to take stock of this striking recent development and of other progress around the theme of related approaches to explicit class field theory. The key to much of the progress over the years is the careful study of p-adic and tame deformations of modular forms, most notably, of Hilbert modular Eisenstein series. The p-adic interpolation of classical Eisenstein series was introduced by Jean-Pierre Serre [Se72] to study the congruences of special values of L-functions and the construction of p-adic L-functions for totally real fields, and was further developed by Barry Mazur and Andrew Wiles in their proof of the main conjecture of Iwasawa theory [MW84]. The workshop will focus on the breakthroughs in [DKa] and [DKb], with a lecture series by the two authors forming the cornerstone of the activity.

The biannual ANTS meetings are the premier international forum for the presentation of new research in computational number theory and its applications, including algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography.

The aim of this meeting is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and L-functions and their special values, and modular forms and functions. Considerable attention is paid to computational issues, but the emphasis is on aspects that are of interest to the pure mathematician.

Because of limited space, participation is by invitation only.

In 2019, the organizers created a workshop called Number Theory in the Americas, which brought together junior and senior mathematicians from North, Central, and South America, to work together on research projects. The workshop resulted in at least seven publications, and served as a first collaboration experience for many of the junior researchers. The organizers propose to create a follow-up workshop in order to provide collaboration opportunities for the PhD students and postdocs who were too young to participate the last time. The workshop will be held in Spanish in order to erase the additional obstacle of communicating in English. We will welcome native and non-native speakers alike.

Our workshop is modeled after several other workshops that have been successful at fostering mathematical collaboration. Participants will be divided into small project groups (3-5 participants) containing a mix of junior and senior researchers. Each group will be led by one or two senior mathematicians. Project groups will be assigned based on research area, with care taken to ensure that each group contains researchers from both continents who have not previously worked together. Background reading will be sent to project group members several months in advance so that they are prepared to work on their respective problems together when they arrive in Oaxaca. The bulk of the workshop will be devoted to working in the project groups, but there will be introductory talks on the first day and final reports on the last day. There will also be panel discussions on topics of particular interest to junior researchers. The expectation is not that each project group will write a paper by the end of the week. Rather, it is meant to be an opportunity to exchange ideas and a starting point for potential future collaboration.

The conference "Number Theory, Quantum Chaos and their Interfaces” aims at gathering distinguished researchers working in either of the disciplines to discuss recent research advances in these fields, and serve as a playground for the exchange of ideas between these, rather diverse, research communities. Another purpose of our conference is to provide a solid educational platform for more junior researchers who aspire to conduct research in the relevant fields and expose them to some of the outstanding results and open problems.

The René 25 conference's purpose is to celebrate the research interests of René Schoof.

During the 2025-26 academic year the School will have a special program on Arithmetic Geometry, Hodge Theory, and O-minimality. Jacob Tsimerman, University of Toronto will be the Distinguished Visiting Professor.

The purpose of this special year will focus on recent developments in hodge theory and o-minimality and their applications to arithmetic geometry. There has been much progress over the last 15 years in using transcendental uniformization maps to study arithmetic questions (general shafarevich theorems, results on unlikely intersections, general bounds on rational point counts). It has become increasingly clear that hodge theory (both classical and P-adic) and the resulting period maps form a natural home for these kinds of investigations to arise. In the other direction, O-minimality has been applied with success to make progress on questions in Hodge theory (Griffiths conjecture, definable period maps), and has recently had its own explosion of results (sharply O-minimal sets, the resolution of Wilkie's conjecture).

The goal of this year will be to bring together researchers in these different fields, with the aim of extending the collaboration between areas, share key insights, and investigate how far existing methods can be pushed.

Senior participants: Gal Binaymini, Ben Bakker (to be confirmed), Jonathan Pila and Claire Voisin (STV)