Modular Forms of Half Integer Weight

Abstract

Let k be a positive odd integer, and let λ = (k − 1)/2. In this chapter we shall look at modular forms of weight k/2 = λ + 1/2, which is not an integer but rather half way between two integers. Roughly speaking, such a modular form f should satisfy f((az + b)/(cz + d)) = (cz + d)^λ+1/2f(z) for $\left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right)$ in Γ = SL₂(ℤ) or some congruence subgroup Γ′ ⊂= Γ. However, such a simple- minded functional equation leads to inconsistencies (see below), basically because of the possible choice of two branches for the square root. A subtler definition is needed in order to handle the square root properly. One must introduce a quadratic character, corresponding to some quadratic extension of ℚ. Roughly speaking, because of this required “twist” by a quadratic character, the resulting forms turn out to have interesting relationships to the arithmetic of quadratic fields (such as L-series and class numbers). Moreover, recall that the Hasse-Weil L-series for our family of elliptic curves E _n : y² = x³ − n²x in the congruent number problem involved “twists” by quadratic characters as n varies (see Chapter II). It turns out that the critical values L(E _n , 1) for this family of L-series are closely related to certain modular forms of half-integral weight