# An algebro-geometric theory of vector-valued modular forms of half-integral weight

# An algebro-geometric theory of vector-valued modular forms of half-integral weight

We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves. We discuss extensions to the cusp at infinity and give an algebraic notion of q-expansions of vector-valued modular forms. We give algebraic notions of holomorphic vector-valued modular forms and cusp forms and compute algebraic dimension formulas for these spaces over any algebraically closed field of characteristic not dividing 2,3, by using the Riemann-Roch theorem for DM stacks. A special case of this theory can be used to give an algebro-geometric theory of modular forms of half-integral weight, as defined in the complex-analytic case by Shimura.