# Reading course on modular curves

## Organisers: Guido M. Lido, Pietro Mercuri.

• Title: Arithmetic of elliptic curves,
Speaker: Daniele Cozzo,
When: Monday 23/10/2017, hours 18:00-20:00.
Where: Room "D'Antoni", "Tor Vergata" University, Rome.
Abstract: In this lecture I recall some basics of elliptic curves. I begin with the definition of elliptic curves and talk about their group structure and defining equations. Then the concepts of isogeny, torsion subgroups and Tate module are introduced. Following this, I talk about elliptic curves over finite fields (Frobeniuses, Riemann hypothesis), over local fields (singular fibers) and over global fields (minimal model, conductor, Mordell-Weil). Finally, I introduce the p-adic uniformisation.
Main references: "Artihmetic of Elliptic Curves", J.H. Silverman; "Algebraic Curves", W. Fulton; "l-adic representations and elliptic curves", J.P. Serre; "Mathematics of Public Key Criptography", S.D. Galbraith.

• Title: Modular forms, part 1,
Speaker: Dario Antolini,
When: Thursday 27/10/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk I introduce the basic notions of modular forms. Starting with the concept of congruence subgroup, I give the definition of modular form and fundamental domain for such subgroups. Then I focus on the algebras of modular forms and how to decompose them in some particular cases.
Main reference: "A First Course in Modular Forms", F. Diamond, J. Shurman.

• Title: Modular forms, part 2,
Speaker: Dario Antolini,
When: Thursday 02/11/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk I give another decomposition of the cusp forms in the so-called "old" and "new" forms through a description of Hecke operators. If it remains time, I will also give some examples.
Main references: "A First Course in Modular Forms", F. Diamond, J. Shurman; "Twists of newforms and pseudo-eigenvalues of W-operators", A.O.L. Atkin, W. Li.
Notes: pdf.

• Title: Modular curves,
Speaker: Guido M. Lido,
When: Monday 06/11/2017, hours 18:00-20:00.
Where: Room "D'Antoni", "Tor Vergata" University, Rome.
Abstract: In this talk we would like to review the basic theory of modular curves. We will describe explicitly the complex structure of these curves and give "modular" interpretation of some of them. We will then see how basic algebraic geometry applied on these curves helps when dealing with modular forms. Finally we will prove the existence of algebraic models of such curves defined over number fields.
Main reference: "A First Course in Modular Forms", F. Diamond, J. Shurman.

• Title: Complex multiplication,
Speaker: Lorenzo Pagani,
When: Thursday 16/11/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk we discuss the basic properties of complex multiplication on elliptic curves and how the class group of the CM field is involved. Finally we give an explicit description of the Hilbert class field of the CM field.
Main reference: "Advanced Topics in the Artihmetic of Elliptic Curves", J.H. Silverman.

• Title: Serre's uniformity conjecture, part 1,
Speaker: René Schoof,
When: Monday 20/11/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: We describe some of the methods used by Mazur and by Bilu, Parent and Rebolledo in the proofs of their results regarding Serre's uniformity conjecture over Q.

• Title: The Jacobian associated to a modular curve,
Speaker: Daniele Di Tullio,
When: Thursday 30/11/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: I will start the lecture with a review of homology groups, Jacobians associated to Riemann Surfaces and Abelian Varieties. Then I will focus on the Jacobian associated to a modular curve and how they can be described. I will finish describing how is possible to associate Abelian Varieties to newforms and how the Jacobian of the modular curve associated to the congruence group Gamma_1(N) can be described as a direct sum of these Abelian Varieties.
Main references: "A First Course in Modular Forms", F. Diamond, J. Shurman; "Algebraic Curves and Riemann Surfaces", R. Miranda; "Introduction to Topological Manifolds", J.M. Lee; "Basic Algebraic Geometry 1", I.R. Shafarevich.

• Title: Serre's uniformity conjecture, part 2,
Speaker: René Schoof,
When: Monday 04/12/2017, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: We describe some of the methods used by Mazur and by Bilu, Parent and Rebolledo in the proofs of their results regarding Serre's uniformity conjecture over Q.

• Title: Explicit equations of modular curves,
Speaker: Pietro Mercuri,
When: Monday 18/12/2017, hours 18:00-20:00.
Where: Room "D'Antoni", "Tor Vergata" University, Rome.
Abstract: In this talk I explain how to get explicit equations for a model of some kind of modular curves. This is done using the Fourier coefficients of the cusp forms associated to the modular curve and the canonical embedding.
Main reference: "Equations and Rational Points of the Modular Curves X_0^+(p)", P. Mercuri.

• Title: Modular curves over finite fields,
Speaker: Claudio Stirpe,
When: Monday 15/01/2018, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk I will introduce upper and lower bounds on the number of F_q rational points on a (smooth, projective, absolutely irreducible) curve of genus g. In particular I will state, without proof, important results due to Weil, Serre, Oesterlè and Ihara. Later, I will explain how to compute the number of F_q rational points on the modular curve X_0(N). I will also discuss how the modify the algorithm for dealing with X_0^+(N) and X_ns(N). Finally, we will compare our results with the currently best known lower bounds. What is the best? Come and see...

• Title: Automorphisms of Modular Curves,
Speaker: Valerio Dose,
When: Friday 26/01/2018, hours 17:00-19:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: Modular Curves can be seen as the compactification of some quotient of the upper complex half-plane. It is an interesting question to ask if all automorphisms of a modular curve are induced by some automorphism of the overlying upper complex half-plane. When this does not happen, this can be source of exceptional behaviour in the framework of Serre's uniformity. In this talk, we will present results and ideas of proof on determining the automorphism group of modular curves, especially concentrating on some recent work regarding the non-split Cartan case.

• Title: A modular interpretation of the class number one problem,
Speaker: Valerio Dose,
When: Friday 02/02/2018, hours 17:30-19:30.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In his "Disquisitiones arithmeticæ" Gauss conjectured that there are exactly nine imaginary quadratic fields with number class equal to one, namely Q(√-1), Q(√-2), Q(√-3), Q(√-7), Q(√-11), Q(√-19), Q(√-43), Q(√-67) and Q(√-163). It took about one hundred and fifty years before a proof was found thanks to the independent works of Hegneer, Baker and Stark. In this talk I am going to show how such a problem reduces to the Diophantine problem of finding rational points with integral j-invariant on special types of modular curves.
Main references: "Lectures on Mordell-Weil theorem", J.P. Serre; "Normalizers of non-split Cartan subgroups, modular curves and the class number one problem", B. Baran; "Complex multiplication", J.P. Serre, chapter XIII in "Algebraic number theory" by Cassels & Fröhlich.

• Title: Fourier coefficients of cusp forms,
Speaker: Pietro Mercuri,
When: Friday 09/02/2018, hours 17:30-19:30.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk I explain how to get the Fourier coefficients of cusp forms associated to nonsplit Cartan subgroups. This will be done using representation theory of the finite group GL2(Z/pZ).

• Title: Post Quantum Cryptography - Elliptic Curves and More!,
Speaker: Ankan Pal,
When: Monday 12/02/2018, hours 18:00-20:00.
Where: Room "Dal Passo", "Tor Vergata" University, Rome.
Abstract: In this talk, I will be discussing some aspects of quantum computing and I will try to demonstrate that 'why' RSA and DLP will not be able to provide sufficient security in the Post-Quantum era. The central subject of this talk would be Post Quantum (Isogeny-Based) Elliptic Curve Cryptography. But, I would also like to portray some of the other algebraic structures (such as Groups and Lattices) which could be good candidates for providing security in the Post Quantum Paradigm. I would also talk about some methods from Algebraic Geometry which can be used to develop an attack algorithm on EC-DLP. The idea is to have an open discussion on the Applications of Algebraic Geometry and Number Theory in the field of Cryptography and also to see the interconnections between 'Cryptography and the theory of Modular Curves'.

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