Tag - Elliptic curves

Joseph Silverman: More Tips on Keeping Secrets in a Post-Quantum World: Lattice-Based Cryptography

30th July, 2020

What do internet commerce, online banking, and updates to your phone apps have in common? All of them depend on modern public key cryptography for security. For example, there is the RSA cryptosystem that is used by many internet browsers, and there is the elliptic curve based ECDSA digital signature scheme that is used in many applications, including Bitcoin. All of these cryptographic construction are doomed if/when someone (NSA? Russia? China?) builds a full-scale operational quantum computer. It hasn't happened yet, as far as we know, but there are vast resources being thrown at the problem, and slow-but-steady progress is being made. So the search is on for cryptographic algorithms that are secure against quantum computers. The first part of my talk will be a mix of math and history and prognostication centred around the themes of quantum computers and public key cryptography. The second part will discuss cryptographic constructions based on hard lattice problems, which is one of the approaches being proposed to build a post-quantum cryptographic infrastructure.

Jordan Ellenberg: What’s up in arithmetic statistics?

23rd July, 2020

If not for a global pandemic, a bunch of mathematicians would have gathered in Germany to talk about what's going on in the geometry of arithmetic statistics, which I would roughly describe as "methods from arithmetic geometry brought to bear on probabilistic questions about arithmetic objects". What does the maximal unramified extension of a random number field look like? What is the probability that a random elliptic curve has a 2-Selmer group of rank 100? How do you count points on a stack? I’ll give a survey of what’s happening in questions in this area, trying to emphasize open questions.

Joint equidistribution of adelic torus orbits and families of twisted L-functions

31st May, 2020

The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful classical interpretations, such as the equidistribution of integer points on spheres, of Heegner points or packets of closed geodesics on the modular surface, or of supersingular reductions of CM elliptic curves. In the mid 20th century, Linnik and his school established the equidistribution of many of these classical variants through his ergodic method, under a congruence condition on the discriminants modulo a fixed auxiliary prime. More recently, the Waldspurger formula and subconvex estimates on L-functions were used to remove these congruence conditions, and provide effective power-savings rates.

In their 2006 ICM address, Michel and Venkatesh proposed a variant of this problem in which one considers the product of two distinct inner forms of GL2, along with a diagonally embedded torus. One can again specialize the setting to obtain interesting classical reformulations, such as the joint equidistribution of integer points on the sphere, together with the shape of the orthogonal lattice. This hybrid context has received a great deal of attention recently in the dynamics community, where, for instance, the latter problem was solved by Aka, Einsiedler, and Shapira, under supplementary congruence conditions modulo two fixed primes, using as critical input the joinings theorem of Einsiedler and Lindenstrauss.

In joint (ongoing) work with Valentin Blomer, we remove the supplementary congruence conditions in the joint equidistribution problem, conditionally on the Riemann hypothesis, while obtaining a logarithmic rate of convergence. The proof uses Waldsurger’s theorem and estimates of fractional moments of L-functions in the family of class group twists.

Ana Caraiani: Local-global compatibility in the crystalline case

16th April, 2020

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GLn over F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems.

I will discuss joint work with J. Newton, where we establish local-global compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of using P-ordinary parts, and improves on earlier results obtained in joint work with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne in certain Fontaine-Laffaille cases.

Florian Sprung: How does the rank of an elliptic curve grow in towers of number fields?

18th April, 2019

On an elliptic curve y2=x3+ax+b, the points with coordinates (x,y) in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field.

For the simplest example with infinitely many number fields, fix a prime p. Adjoining to ℚ the pth, p2th, p3th,... roots of unity produces a *tower* of number fields

ℚ⊂ℚ(ζp)⊂ℚ(ζp2)⊂....

One may guess that the rank should keep growing in this tower ('more numbers mean more solutions'). However, this guess turns out to be incorrect: the rank is always bounded, as envisioned by the theories of Iwasawa and Mazur in the 1970s.

The above tower started with ℚ, but there are analogous towers that start with an imaginary quadratic field instead. Given the above boundedness result, one would now guess that the rank is bounded in these towers, too. Surprisingly, this is not the case: there are scenarios both for bounded and unbounded rank. So how does the rank grow in those towers in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.

Emmanuel Lecouturier: An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

14th February, 2019

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined L-invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's Eisenstein ideal to study in which power of the Eisenstein ideal L belongs. As a corollary of our study, we give a surprising elementary formula on supersingular j-invariants.

Rong Zhou: Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

25th October, 2018

Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to use techniques from local harmonic analysis to compute the asymptotics of a certain twisted orbital integral which counts the number of 𝔽q-points on the ADLV as q goes to infinity. This is joint work with Yihang Zhu.

Hector Pasten: Shimura curves and new abc bounds

28th November, 2017

Existing unconditional progress on the abc conjecture and Szpiro's conjecture is rather limited and coming from essentially only two approaches: The theory of linear forms in p-adic logarithms, and bounds for the degree of modular parametrizations of elliptic curves by using congruences of modular forms. In this talk I will discuss a new approach as well as some unconditional results that it yields. For a fixed elliptic curve E over the rationals one has several modular parametrizations coming from various Shimura curves X, and our method amounts to using Arakelov theory to bound how these degrees vary as we change the source curve X, keeping E fixed. Unlike linear forms in p-adic logarithms, our method is global and deals with all local contributions at once. Concrete unconditional consequences will be discussed, such as bounding the number of divisors of abc triples polynomially on the radical, bounding the product of the ''fudge factors'' of elliptic curves polynomially on the conductor, and new lower bounds for truncated counting functions in the context of Vojta's arithmetic conjecture.

Ilya Khayutin: Joint equidistribution of CM points

21st November, 2017

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will explain the proof of a recent theorem making progress towards this conjecture.

Currently, this problem does not seem to be amenable to methods of automorphic forms even assuming GRH. Nevertheless, assuming a splitting condition at two primes the joining rigidity theorem of Einsiedler and Lindenstrauss applies. As a result the obstacle to proving equidistribution is the potential concentration of mass on graphs of Hecke correspondences and translates thereof. I will present a method to discard this possibility using a geometric expansion of a relative trace, description of the relative orbital integrals in terms of integral ideals and a sieve argument.