56-ABKLS seminar in Lille

[Aloys Krieg]

Dear colleagues,
The next ABKLS seminar will take place in Lille, on Wednesday, February 27.
Hope to see you in Lille!

Best regards,
Valery and Aloys


56th Seminar Aachen-Bonn-Köln-Lille-Siegen
on Automorphic Forms
University of Lille, February 27, 2019

Organizers: K. Bringmann, J. Bruinier, V. Gritsenko, A. Krieg, P. Moree,
G. Nebe, N-P. Skoruppa, S. Zwegers

When: Wednesday, February 27, 2019
Where: University of Lille, Cité Scientifique, Villeneuve d’Ascq, Bat.
M2, la salle de réunion


14.00 – 15.00 Haowu Wang (LabEx CEMPI, Lille). The Weyl-invariant weak
Jacobi forms for the root system E_8.
The Chevalley type theorem for affine root systems is equivalent to the
fact that the bigraded ring of the Weyl-invariant weak Jacobi forms for
a classical root system R is a pure polynomial algebra. The
corresponding generators play an important role in the theory of
Frobenius varieties. This subject was developed by E. Looijenga, K.
Saito, J. Bernstein, O. Schwarzman, K. Wirthmüller, B. Dubrovin, M.
Bertola and others. In 1992, Wirthmüller proved that the bigraded ring
of W(R)-invariant weak Jacobi forms is a polynomial algebra over the
ring of SL(2, Z) modular forms except the root system E8. It is still an
open problem how to extend the Wirthmuller's theorem to the case R=E_8.
The Weyl invariant E_8 Jacobi forms have many applications in
mathematics and physics, but very little has been known about its
structure. In this talk, I will present a description of
W(E_8)-invariant Jacobi forms of small indices . As a corollary we give
the negative answer on this old problem: the ring of W(E_8)-invariant
weak Jacobi forms is NOT a polynomial algebra. Thus a Chevalley type
theorem is NOT true for the root system E_8. Then I give a proper
extension of the Chevalley type theorem to the case of the affine root
system E_8.


15.15 – 16.15 Amir-Kian Kashani-Poor (Laboratoire de physique théorique
de l’ENS, Paris)
Curve counting and (extended) Weyl invariant Jacobi forms.

Packaging enumerative invariants into generating functions has proven to
be a powerful strategy to compute them. Symmetries underlying the
geometry act on such functions, and in propitious cases, substantially
simplify their computation. In this talk, I will address curve counting
in elliptically fibered Calabi-Yau manifolds from this vantage point.
After a short excursion into how the problem fits into physics, I will
discuss the structure of the topological string partition function
Z_top, a physically motivated generating function for curve counting
invariants, on a class of such geometries. Weyl invariant Jacobi forms —
Lie algebras make an appearance due to the Kodaira classification of
singularities of elliptic fibers — will play an important role in the
construction of Z_top. To fully exploit the symmetries of the problem, I
will introduce a subring of such forms, invariant also under diagram
symmetries of the associated affine Dynkin diagrams.


16:15—17:15 Coffee Break

17.15 – 18.15 Shoyu Nagaoka (Kindai University)
Theta Operator for Modular Forms.

Theta operator \Theta is a kind of differential operator operating on
modular foms. It is a generalization of the classical Ramanujan's
\theta-operator. For a prime p, the mod p kernel of the theta operator
is defined as the set of modular form F such that \Theta (F) \equiv 0
\mod p. Namely, the element in the set can be interpreted as a mod p
analogue of the singular modular form. In this talk, I will give
examples of modular forms in the mod p kernel of the theta operator.


18:30 Buffet (in the building M2)

For further informations concerning this meeting please send an email to
Valery.G...@math.univ-lille1.fr. For the previous meetings see
http://www.matha.rwth-aachen.de/en/forschung/abkls/