|
Recap lectures, Jeroen Sijsling
Program
This course will recall the background theory required to understand the other lectures and research group topics. For example, it will recall elliptic curves, their j-invariants, and their isogenies, as well as other basic material. The goal of this course is to ensure a common background for all participants. Videos will be made available before the school as well; any remaining questions can be posed in this lecture.
Prerequisites: None.
References:
- J. Silvermann, (2009). The arithmetic of elliptic curves, Springer-Verlag
|
Modularity and the LMFDB, Aurel Page
Program
This main series of plenary lectures will explore various themes around modularity, as well the broader Langlands program in the context of the LMFDB. Its guiding example is that of the elliptic curves over the rationals and over the finite fields, where it will study the group law, the group of rational points, the determination of the cardinality of the larger group by means of the trace of the Frobenius and the relation of these results with isogenies which generalises to abelian varieties in higher dimensions. This is followed by the discussion of what it means for an elliptic curve to be modular. In particular, modular curves and modular forms are defined and it is shown that these are related to elliptic curves over the rationals by means of Galois representations and Wiles' seminar results. The theory is illustrated by the concrete implementation of these various cross-connections in the LMFDB, and the course concludes with broad vista of the possible generalisations.
Background: Linear Algebra and basic Geometry.
References:
-
-
|
Geometry of algebraic curves, Jennifer Paulhus
Program
This research group will study algebraic curves over the complex numbers and their Jacobians from a geometric point of view; possible topics include invariants of curves and the study of curves as well as the decompositions of their Jacobians as abelian varieties under group actions or by means of their endomorphism algebra, as well as the geometric and arithmetic consequences of these results.
Background:
References:
-
-
|
Cyclic Galois covers of the projective line (Superelliptic curves and Artin-Scheier curves), Elisa Lorenzo Garcia and Beth Malmskog
Program
In this research group, we will recall some basic facts about covers of curves and we will state Riemann-Hurwitz Theorem. We will focus then on cyclic Galois covers of the projective line and the curves they define. In particular, we will study their ramification and genus. This construction produces some interesting families of curves useful for different reasons. During the exercises session we will see how to "well" parametrize these families and how to study their reduction type.
Background: basics on algebraic curves, morphisms, divisors, like in the notes at Download notes here! .
References:
|
Rational points on elliptic curves and on curves of genus 2, Fabien Pazuki
Program
The first part of the course will be focused on rational points on elliptic curves. We will discuss the Mordell-Weil theorem as well as points over finite fields, and give links with other Diophantine problems. In the second part of the course, we will compare with the situation of rational points over genus 2 curves. We will use several examples to illustrate the main results. Previous knowledge of elliptic curves or genus 2 curves is not necessary. Previous knowledge of basic algebraic geometry and basic number theory is recommended.
Background: Basic algebraic geometry and algebraic number theory.
References
|
Arithmetic statistics 2, David Kohel and Leonardo Colo
Program
Arithmetic statistics can be summarized as the counting or enumeration of arithmetic objects "up to a certain bound". Examples of results in this field are the study of the average rank of elliptic curves. Another main topic, explored in the LMFDB, is the Sato--Tate conjecture and the resulting distribution of traces of Frobenius of algebraic curves modulo various primes of good reduction. This research group introduces the topic, describes its main results, as well as various
open directions that can be pursued further.
Background:
References
|
Modular curves and modular forms, Cecile Armana
Program
This research group will focus on modular curves (which parametrize elliptic curves with extra structures) and modular forms (which are complex analytic functions with special transformation properties) as well as the
connection between them. Using data from the LFMDB, we will also gain a better understanding of them through associated objects such as Hecke operators, modular symbols, and modular parametrizations of elliptic curves.
Prerequisites: linear algebra and elliptic curves.
References:
|
L-functions and Galois representations, Ahmad El-Guindy, Barinder Banwait
Program
L-functions provide a fruitful way of "packaging local information" about elliptic curves (as well as a variety of other mathematical objects) into a function whose analytic properties capture a lot of arithmetic information. In particular, this research group will explore the original computations of Birch and Swinnerton-Dyer (using a precursor of the L-function) which led them to formulate their conjecture, which is currently one of the outstanding open problems in number theory. We will also explore the connections between L-functions and Galois representations, which are another crucial theme in modern number theory which provide a bridge between algebraic and analytic considerations by studying the action of the absolute Galois group on certain points and modules obtained from the elliptic curves. The aim will be to encourage students to explore these themes through concrete examples and computations using the LMFDB and various computer algebra packages.
Prerequisites: Knowledge of elementary number theory as well as a working knowledge of the basics of Galois Theory.
References:
|
Isogeny graphs and cryptography, Sorina Ionica, Sabrina Kunzwieler
Program
Isogenies are special morphisms between elliptic curves, or more generally between the Jacobians of algebraic curves, in the sense that they preserve the group law. In elliptic curve cryptography, isogenies are employed both for constructing protocols and for studying the security of these protocols. In classical cryptography, the security of curve based schemes is based on the discrete logarithm problem. For example, isogenies may be used to reduce the discrete logarithm problem from the Jacobian of a curve where this problem is hard to the Jacobian of a curve where this problem is easy. In general, for elliptic curves the discrete logarithm problem is hard, but for higher genus curves more efficient algorithms are known. The idea of this project is to compute rational isogenies from a product of elliptic curves to the Jacobian of some higher genus curve defined over finite fields and derive more efficient attacks on the discrete logarithm problem.
Background: Basic algebraic geometry and algebraic number theory.
Bibliography:
-
J. Silverman, (2009). The Arithmetic of Elliptic Curves, Springer: Chapters III, V and XI, in particular Sections XI.4, XI.5, XI.6.
- M. Stoll, (). Arithmetic of hyperelliptic curves, Chapters 1, 2 and 4. Download notes here!
|
Abelian varieties over finite fields, Stefano Marsegalia Program
Some of the most extensively studied objects at the intersection of number theory and algebraic geometry are abelian varieties, which are projective varieties whose points form a group. Abelian varieties have a very rich algebraic structure which makes them fundamental tools for understanding the geometry and arithmetic of curves through their Jacobians and they constitute the geometric background of some integrable systems. Moreover, abelian varieties of dimension 1, also known as elliptic curves, defined over finite fields are the basis for some of the most secure and used cryptosystems.
Two abelian varieties are called isogenous if there is a surjective morphism with finite (as a group scheme) kernel from one to the other. Being isogenous turns out to be an equivalence relation. Honda and Tate proved in a series of papers that each isogeny class of abelian abelian varieties over a finite field is uniquely determined by a polynomial with integer coefficients, called Weil polynomial. By exploiting the properties of these Weil polynomial one can enumerate all isogeny classes of abelian varieties of a given dimension over a fixed finite field. For example, the LMFDB contains a list of all Weil polynomials associated to the isogeny classes of abelian varieties up to dimension 5 over several finite fields.
In this project, the students will study such the theory behind such Weil polynomials and experiment with the data contained in the LMFDB.
Background: Basic algebraic geometry and algebraic number theory.
Bibliography:
-
-
|
|
