Tag - Elliptic curves

Ana Caraiani: A glimpse into the Langlands programme

20th May, 2024

The goal of this lecture series is to give you a glimpse into the Langlands programme, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. In the first lecture, we will look at a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. I will try to give you a sense of the different meanings of modularity and of the multitude of ingredients that go into establishing such a result. In the following lectures, I will focus on the more geometric ingredients, first in the special case of the modular curve and then for higher-dimensional Shimura varieties.

David Loeffler: Euler systems and the Bloch-Kato conjecture

27th March, 2024

The Bloch-Kato conjecture, relating special values of L-functions to algebraic data, is one of the most important open problems in number theory; it includes the Birch-Swinnerton-Dyer conjecture for elliptic curves as a special case. I will describe some recent breakthroughs establishing special cases of this conjecture (and related problems such as the Iwasawa
main conjecture) using the method of Euler systems.

Sofia Tirabassi: Lieblich-Olsson deformation technique and applications

12th September, 2023

In this lecture series I will explain how one can use deformation theory to study derived categories in positive characteristic.

I will start by giving an overview on what does it mean to 'lift' something 'to characteristic 0' and when is this possible. Then I will present a baby example: the study of the Fourier-Mukai partners of products of elliptic curves over algebraically closed fields of characteristic at least 5. After that, I will present Lieblich-Olsson deformation technique which allows us to deform derived equivalence. This is a very versatile tools with many applications (not just in positive characteristic!). I will conclude the series by going over some of these applications in greater details.

Noam Elkies: Families of genus-2 curves with 5-torsion

14th July, 2023

We give birational parametrizations of pairs (C,T) where T is a 5-torsion point on the Jacobian of a genus-2 curve C, possibly satisfying one or more of the following additional conditions:

   •  T = (P) - (P0) for some points P0, P in C with P0 Weierstrass;
   •  C has one or more additional rational Weierstrass points;
   •  Jac(C) has real multiplication by (1+√5)/2, and T is a √5-torsion point.

Nina Zubrilina: Root Number Correlation Bias of Fourier Coefficients of Modular Forms

20th April, 2023

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number.

In my talk, I will discuss this root number correlation bias when the average is taken over weight 2 modular newforms of all Galois orbit sizes simultaneously. I will point to a source of this phenomenon in this case and compute the correlation function exactly.

Kenji Iohara: On Elliptic Root Systems

20th February, 2023

In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G ⊂ F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G ⊂ F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.

Wanlin Li: Ordinary and Basic Reductions of Abelian Varieties

16th February, 2023

Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc). In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang. Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves.

Michael Larsen: The Other Galois Representation of an Elliptic Curve

5th December, 2022

Let E be an elliptic curve defined over ℚ. The ℚ̅-points of E form an abelian group on which the Galois group Gℚ=Gal(ℚ̅/ℚ) acts. The usual Galois representation associated to E captures the action of Gℚ on the points of finite order. However, one could also look at the action of Gℚ on the free part of E(ℚ̅). This infinite-dimensional representation encodes a great deal of interesting arithmetic information. I will state a conjecture concerning this other Galois representation and present supporting evidence from probability theory, Ramsey theory, algebraic geometry, and number theory.

Ping Xi: Analytic approaches towards Katz’s problems on Kloosterman sums

8th September, 2022

Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from ℓ-adic cohomology.