Galois representations and pencils of Calabi-Yau motives

Details

The workshop "Galois representations and pencils of Calabi-Yau motives" will be held at the MPIM Bonn during the week 15-21 December, 2012 as part of the activity "Quantum cohomology, Frobenius manifolds and pencils of Calabi-Yau varieties" in 2012-2013.

Email: galois2012@mpim-bonn.mpg.de [4]

Organisation (in short):

The workshop will consist of three parts. On each of the days 15-17 December the agenda is broken into three consecutively running seminar classes, titled as follows:

1. Congruences (moderator: Neil Dummigan)
2. Crystals (moderator: Richard Crew)
3. Congruences sheaves/Mirror symmetry (moderator: Alessio Corti)

We meet 12PM-6PM. A number of fixed talks will be given on each of these days, however, discussions and ad-lib talks are encouraged.

A short conference will run on 18 December. The Seminar on Algebra, Geometry and Physics will be part of the conference, with Jan Stienstra's talk on connections between number theory and physics. There will be a social event in the afternoon.

The days 19-21 December are dedicated to joint work in groups. However, Thursday 20 December will start with Number Theory Seminar at 10:45. Apart from that, we will start at 11 AM and finish at 5PM on these days.

The lunchtime on workdays is at 13:00, and the institute tea is 16:00, with the exception of Friday, 21 December.

Program and abstracts

December 14, afternoon:

Arrival

On the next three days seminars run continuously 12:00 - late afternoon, normally with one fixed talk in each seminar.

December 15, Seminars:

~12PM: Neil Dummigan: Critical values, congruences, Selmer groups

Since the seminar I am moderating is called congruences, I'll start off by explaining the possibilities for the composition factors of the reduction mod p of the type of 4-dimensional Galois representation coming from a CY 3-fold, and comparing these with the kinds of congruences observed by Anton Mellit, calling on him to report on observations about pairs of congruences and common values of p. This would set the scene. At some point later I could say something about how for at least one of the three kinds of such congruences, it should lead to an element of order p in some Selmer group, how that appears in the Bloch-Kato conjecture, which then predicts the appearance of p in an L-value, comparing and contrasting with the cases of congruences for Saito-Kurokawa lifts, and congruences connected with p-torsion on elliptic curves.

~3PM: Jan Stienstra: Formal groups associated to pencils of Calabi-Yau varieties.

The Cartier-Dieudonne module of the Artin-Mazur formal group (AMFG) equals the the unit root crystal in crystalline cohomology. A Laurent polynomial (LP) with reflexive Newton polytope defines Calabi-Yau hypersurfaces in toric varieties. There is a very concrete formula for a logarithm for a group law for the AMFG in terms of the constant terms  in powers of the LP. The AMFG is a formal group over the ring of coefficients of the LP. This ring has a natural structure of a $\lambda$-ring. For formal groups over $\lambda$-rings Cartier's theory can be reformulated in terms of Dirichlet series. This immediately leads to congruences. The natural geometric context for Laurent polynomials is toric geometry and the natural context for their variations is Gelfand-Kapranov-Zelevinsky's theory of hypergeometric functions. The aforementioned formal group law logarithm is such a GKZ hypergeometric function. This has consequences for the unit roots of L-functions.

~5PM: Piotr Achinger: A characterization of toric varieties in characteristic p

A theorem of J. F. Thomsen states that Frobenius push-forwards of line bundles on smooth toric varieties are direct sums of line bundles. Using characterization of toric varieties in terms of their Cox rings, we show that this property in fact characterizes smooth projective
toric varieties.

December 16, Seminars:

~12PM: Richard Crew: Introduction to F-isocrystals on the line

We will review basic properties of F-isocrystals on a smooth variety, with particular attention to the case of an open subset of the projective line. Topics: convergence conditions, Dwork's trick, the slope filtration, and the local monodromy theorem and its applications. If time permits we will discuss Lauder's work on the explicit computation of Frobenius matrices, and Katz's congruence formulas for the Frobenius of a curve.

~3PM: Francesco Baldassarri: Log-growth and Frobenius slope filtrations.

Convergence polygons, log-growth polygons and Frobenius slope filtrations will be explained.

~5PM: Lucia di Vizio: Galois theory of differential equations with an action of an endomorphisms.

I'll explain how one can construct a Galois theory for differential ­equations that takes into account the action of a difference operator,i.e.,  an endomorphisms, on the solutions.The theory attaches a group scheme to a differential equation, which encodes the algebraic difference relations among the solutions of the differential equation. This is typically the case of p-adic differential equation with a Frobenius structure. This is a joint work with C. Hardouin and M. Wibmer.

December 18, Conference

10:00-10:50 Lucia Di Vizio, Introduction to Frobenius in q-difference equations.
11:00-11:50 Alexei Panchishkin, TBA
12:00-12:50 Guenter Harder Modular construction of mixed motives and congruences

I will briefly describe a construction of mixed (Anderson) motives for the symplectic group $GSp_2$. These mixed motives are labelled by  automorphic cusp forms $f$ on $Sl_2$ and they are extensions of pure Tate motives. I will give a formula for the Betti-de-Rham extension classes. Some rather speculative arguments suggest that these motives "create" congruences between elliptic and Siegel modular forms. These congruences have been checked in experiments by Berstroem, Faber, van der Geer and Ghitza, Ryan and Sulon.

Lunch

14:00-15:00 Seminar on Algebra, Geometry and Physics

Jan Stienstra (Utrecht) An amazing coincidence of formulas for the unit root part of crystalline cohomology and the density of states for physical crystals.

Certain Laurent polynomials can be viewed as discretizations of a Laplace operator with quasi-periodic boundary conditions. The eigenvalue/eigenfunction equation for this situation then describes the level sets of the Laurent polynomial. The density of states is a measure for the size of these level sets as a function of the level. It can be approximated, in yet another discretization, by counting how the points of which the coordinates are N-th roots of 1 are distributed over the various level sets. The limit as N tends to infinity is also known in number theory/algebraic geometry as the Mahler measure of the Laurent polynomial. By a simple transformation which is reminiscent of the Mellin transform the formula for the density of states becomes a formula for the formal group which describes the unit root part of the crystalline cohomology of the level sets which are then viewed as hypersurfaces in a toric variety. For real solid state physics the Laurent polynomial has to be replaced by a matrix (i.e. an endomorphism of a vector bundle) over a toric variety.

16:00 Institute tea

Social event

December 19, Joint work in groups

December 20, Joint work in groups

11.15-12.15 Number theory seminar

December 21, Joint work in groups
Departure
Official closing time of MPIM at around 14:30