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.