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 weightKeywordsModular FormEisenstein SeriesCusp FormDouble CosetEuler ProductThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.