Modular forms and arithmetic geometry

Modular forms and arithmetic geometry

The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmetical algebraic geometry. At the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel–Weil formula relating suitable averages of...

Elliptic Curves , Arithmetic Geometry , and Modular Forms

The Arithmetic Subgroups and Their Modular Forms

Arithmetic subgroups are finite index subgroups of the modular group. Classically, congruence arithmetic subgroups, which can be described by congruence relations, are playing important roles in group theory and modular forms. In reality, the majority of arithmetic subgroups are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences be...

Arithmetic of Certain Hypergeometric Modular Forms

In a recent paper, Kaneko and Zagier studied a sequence of modular forms Fk(z) which are solutions of a certain second order differential equation. They studied the polynomials e Fk(j) = Y τ∈H/Γ−{i,ω} (j − j(τ))τ k, where ω = e2πi/3 and H/Γ is the usual fundamental domain of the action of SL2(Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e Fp−1(j) (mod p) is the...

2 9 A ug 2 00 3 Modular forms and arithmetic geometry

The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmetical algebraic geometry. At the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel–Weil formula relating suitable averages of...