Tag - Elliptic curves

Emmanuel Lecouturier: On the BSD conjecture for certain families of abelian varieties with rational torsion

24th March, 2022

Let N and p at least 5 be primes such that p divides N−1. In his landmark paper on the Eisenstein ideal, Mazur proved the p-part of the BSD conjecture for the p-Eisenstein quotient J(p) of J0(N) over ℚ. Using recent results and techniques of the work of Venkatesh and Sharifi on the Sharifi conjecture, we prove unconditionally a weak form of the BSD conjecture for J(p) over a quadratic field K (which can be real or imaginary). This includes results in positive analytic rank, as the analytic rank of J(p) over K can be greater than or equal to 2 for well-chosen K.

Jef Laga: Arithmetic statistics and graded Lie algebras

17th March, 2022

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves.

Chandrashekhar Khare: A Wiles-Diamond numerical criterion in higher dimensions

17th February, 2022

Wiles's proof of the modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of a modularity lifting method. There were two strands to the method:

   1.  A numerical criterion to for a map of rings to be an isomorphism between complete intersections that are finite flat over ℤp in Wiles's paper on FLT, subsequently generalized by Fred Diamond.
   2.  Patching (in his paper with Taylor)

The patching method has been vastly generalized; in particular Calegari-Geraghty found a way to generalize it in principle to prove (potential) modularity of elliptic curves over imaginary quadratic fields (a situation of "positive defect"). Their method has been made unconditional to prove modularity lifting results over CM fields in the ten author paper. The numerical criterion has yet to be generalized to positive defect.

In joint work with Srikanth Iyengar and Jeff Manning we give a development of the Wiles-Diamond numerical criterion to situations of positive defect (for example to proving modularity results for torsion Galois representations over imaginary quadratic fields). This in principle allows one to prove integral R=T theorems (in minimal and non-minimal situations), for which just the use of patching seems inadequate. One interest of proving such integral versions of modularity lifting is that in these situations, the Betti cohomology groups of 3-dimensional Bianchi manifolds (the analog of the modular curves over imaginary quadratic fields) have a lot of torsion. Our strategy consists of proving a higher dimensional version of the numerical criterion of Wiles-Diamond and applying it to prove integral R=T theorems (in the non-minimal case) after patching.

Congling Qiu: Modularity and Heights of CM cycles on Kuga-Sato varieties

14th October, 2021

We study CM cycles on Kuga-Sato varieties over X(N) via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are associated to higher-weight holomorphic cuspidal automorphic representations of GL2(ℚ). This is proved via theta lifting. Our second result is a higher weight analogue of the general Gross-Zagier formula of Yuan, S. Zhang and W. Zhang.

This is proved via relative trace formula, provided the modularity of CM cycles.

Naomi Sweeting: Kolyvagin’s Conjecture and Higher Congruences of Modular Forms

22nd April, 2021

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The proof builds on work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome.

William Chen: Strong approximation for the Markoff equation via non-abelian level structures on elliptic curves

5th November, 2020

Following Bourgain, Gamburd, and Sarnak, we say that the Markoff equation x2 + y2 + z2 - 3xyz = 0 satisfies strong approximation at a prime p if its integral points surject onto its 𝔽p-points. In 2016, Bourgain, Gamburd, and Sarnak were able to establish strong approximation at all but a sparse (but infinite) set of primes, and conjecture that it holds at all primes. Building on their results, in this talk I will explain how to obtain strong approximation for all but a finite and effectively computable set of primes, thus reducing the conjecture to a finite computation. The key result amounts to establishing a congruence on the degree of a certain line bundle on the moduli stack of elliptic curves with SL2(p)-structures. To make contact with the Markoff equation, we use the fact that the Markoff surface is a level set of the character variety for SL2 representations of the fundamental group of a punctured torus, and that the strong approximation conjecture can be expressed in terms of the mapping class group action on the character variety, which in turn also determines the geometry of the moduli stack of elliptic curves with SL2(p)-structures. As time allows we will also describe a number of applications.

Lue Pan: On the locally analytic vectors of the completed cohomology of modular curves

22nd October, 2020

A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of SL2(ℝ). We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic analogue of this result. As applications, we are able to prove a classicality result for overconvergent eigenforms and give a new proof of Fontaine-Mazur conjecture in the irregular case under some mild hypothesis. One technical tool is relative Sen theory which allows us to relate infinitesimal group action with Hodge(-Tate) structure.

Philippe Michel: Simultaneous reductions of CM elliptic curves

8th October, 2020

Let E be an elliptic curve with CM by the imaginary quadratic order OD of negative discriminant D. Given p a prime, if p is inert or ramified in the quadratic field generated by √D then E has supersingular reduction at a(ny) fixed place above p. By a variant of Duke’s equidistribution theorem, as D grows along such discriminants, the proportion of CM elliptic curves with CM by OD whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes p1, . . . ,ps and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank 2 actions on products of locally homogeneous spaces implies that as D grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual pi (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement — obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.

Richard Hatton: Heegner points and self-points on elliptic curves

21st September, 2020

In the arithmetic of elliptic curves, we are interested in the construction of points on an elliptic curve. In particular, it has been shown that we are able to bound certain Selmer groups using modular points, specifically the use of Heegner points by Kolyvagin and self points by Wuthrich. We will define these points and will show how they can be used to create the bounds and its generalisations.

Umberto Zannier: Torsion in elliptic familes and applications to billiards

1st September, 2020

We shall consider elliptic pencils, of which the best-known example is probably the Legendre family Lt: y2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on Lt depending on t) it is an issue to study the set of complex b such that P(b) is torsion on Lb. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,𝞹), there are only finitely many cases for which both billiard trajectories are periodic.