tt∗ geometry of modular curves

Elliptic Curves , Arithmetic Geometry , and Modular Forms

Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

We introduce some algebraic geometric models in cosmology related to the “boundaries” of space-time: Big Bang, Mixmaster Universe, Penrose’s crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light c...

tt∗-GEOMETRY IN QUANTUM COHOMOLOGY

We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt-geometry near the large radius limit. We compute an example of P which is pure and polarized over the whole Kähler moduli space H(P,C).

Selections from the Arithmetic Geometry of Shimura Curves I: Modular Curves

This first lecture is on (classical) modular curves. It is a rather “straight up” expository account of the subject, suitable for people who have heard about modular curves before but seen little about them. After reviewing some truly classical results about modular curves over the complex numbers, we will discuss rational and integral canonical models, focusing in on the case of X0(N) for squa...

Modular Curves

H is the upper half plane, a complex manifold. It will be helpful to interpret H in multiple ways. A lattice Λ ⊂ C is a free abelian group of rank 2, for which the map Λ ⊗Z R → C is an isomorphism. In other words, Λ is a subgroup of C of the form Zα⊕Zβ, where {α, β} is basis for C/R. Two lattices Λ and Λ′ are homothetic if Λ′ = θΛ for some θ ∈ C∗. This is an equivalence relation, and the equiva...