Fourth Number Theory Portuguese meeting

This will be a one-week school as part of the Number Theory Portuguese meetings.

The school will take place at the University of Aveiro, at the "Auditório José Grácio" (department of mechanical engineer). It starts early on September the 1st and ends in the afternoon of September 5th (see the schedule).

The school consists of the following five courses:

• Local Fields, by Rachel Newton (from King's college). Contents

  Valuations, Ostrowski's Theorem, definition of the p-adic numbers as a completion of Q, proof that Qp is a field, p-adic valuation, p-adic integers, p-adic units, power series representation of p-adic numbers, Hensel's lemma, structure of the p-adic units, quadratic extensions of Qp.

  Bibliography

  - Fernando Gouvêa: p-adic Numbers: An Introduction

  - Jean Pierre Serre: Local Fields

  Exercise sheet 1 Exercise sheet 2
• p-adic L functions, by Óscar Rivero (from Universidade de Santiago de Compostela). Contents

  This course will focus on the construction of p-adic L-functions. The first session will review basic concepts of p-adic numbers and discuss some examples of p-adic variable functions and the concept of p-adic interpolation. In the second session, we will present the construction of the p-adic zeta function and the Kubota--Leopoldt p-adic L-functions, introducing Bernoulli distributions for this purpose. Finally, in the third and last talk, we will address the construction of the p-adic L-function of a modular form using the theory of modular symbols.

  Bibliography

  - Fernando Gouvêa: p-adic Numbers: An Introduction

  - Lawrence Washington: Introductionto cyclotomic fields.

  - Chris Williams: An introduction to p-adic L-functions II: modular forms

• Galois Representations by Luis Dieulefait (from Universitat de Barcelona). Bibliography

  - Luis Dieulefait, Ariel Pacetti, Fernando Rodriguez Villegas: Representaciones de Galois

• Arithmetic Statistics by Alex Bartel (from Glasgow University). Contents

  Arithmetic Statistics is concerned with the behaviour of number theoretic objects in natural families. These days there are many variants on this theme: most classically, average behaviour of ideal class groups in families of number fields, but also average behaviour or ranks of elliptic curves, and many other fascinating research directions. In this course we will concentrate on the classical problem of statistics of ideal class groups of number fields. The investigation of these goes back to Gauss, but modern research in the area is driven by the Cohen–Lenstra heuristic, which is what this course will be about.

  Bibliography

  The best preparation for the course is simply knowing what class groups of number fields are. Understanding the statement of Dirichlet's unit theorem, in general and in the special cases of quadratic number fields, will also be helpful. For that, any introductory book on algebraic number theory should work, for example:

  - D. A. Marcus, Number Fields, up to Chapter 5, for those who like a friendly and leisurely style, or

  - the beautiful book by D. Cox, Primes of the Form x^2 + ny^2 for an even slower introduction to the topic, closer in style to how Gauss would have thought about class groups, or

  - J. Neukirch, Algebraic Number Theory, Chapter 1, Sections 1-7 for those who prefer a much terser style or just need a reminder.

  Additional relevant references for the extra-curious:

  - Original paper by Cohen and Lenstra: H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Noordwijkerhout 1983, Lecture Notes in Mathematics 1068 (Springer, Berlin, 1984) 33–62.

  - The earlier paper that provided one of the major motivations for Cohen–Lenstra: H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. Lond. A 322 (1971) 405–420.

  - The much more recent disproof and correction of the original heuristic: A. Bartel and H. W. Lenstra Jr., On class groups of random number fields, Proc. Lond. Math. Soc. (3) 121 (2020), no. 4, 927–953.

  Exercise sheet 1
• Expander graphs, with perspectives to number theory by Harald Helfgott (from CNRS). Contents

  Different definitions of expander graphs: combinatorial, spectral, probabilistic. Equivalences. 2. Existence of (combinatorial) expanders. 3. Spectral strategies: returns to the identity and eigenvalue multiplicity. Bourgain-Gamburd. 4. Spectral strategies: what is a local expander? how do short edges mimic multiplicity? 5. Overview of applications: affine sieve. Weak Chowla in degree 2.

  Bibliography

  - H. Helfgott Expander graphs. Notes by Han Zhicheng

  - Sh. Hoory, N. Linial, A. Wigderson. Expander Graphs and Their Applications.

• Hotel Jardim.
• Hotel Afonso V.
• Hotel das Salinas.
• Hotel Aveiro Center.
• Hotel Imperial.

All interested students should register to the school by sending an email to the address encontros.tn.pt@gmail.com. There will be some limited funding to cover the local expenses (accommodation) of some participants. Those interested in applying for economial assistance should make it explicitly in the registration's email body and register. Also include (as an attachment) a curriculum vitae containing academic information (degrees obtained, ongoing studies, etc) as well as other information considered important together with a few lines explaining your interest for participating in this event.