Elliptic Modular Forms and Their Applications - Up To

30 D. ZagierFor the weight 1/2 result, the meaning of “essentially” is that the productshould be primitive, i.e., it should have the form ∏ η(n i z) ai where the n iare positive integers with no common factor. (Otherwise one would obtaininfinitely many examples by rescaling, e.g., one would have both θ M (z) =η(z) 2 /η(2z) and θ M (2z) =η(2z) 2 /η(4z) on the list.) The classification is thenas follows:Theorem (Mersmann). There are precisely 14 primitive eta-products whichare holomorphic modular forms of weight 1/2 :η(z) ,η(z) 2 η(6z)η(2z) η(3z) ,η(z) 2η(2z) , η(2z) 2η(z) , η(z) η(4z),η(2z)η(z) η(4z) η(6z) 2η(2z) η(3z) η(12z) ,η(z) η(4z) η(6z) 5η(2z) 2 η(3z) 2 η(12z) 2 .η(2z) 3η(z) η(4z) , η(2z) 5η(z) 2 η(4z) 2 ,η(2z) 2 η(3z)η(z) η(6z) , η(2z) η(3z) 2η(z) η(6z) , η(z) η(6z) 2η(2z) η(3z) ,η(2z) 2 η(3z) η(12z)η(z) η(4z) η(6z),η(2z) 5 η(3z) η(12z)η(z) 2 η(4z) 2 η(6z) 2 ,Finally, we mention that η(z) itself has the theta-series representationη(z) =∞∑χ 12 (n) q n2 /24 = q 1/24 − q 25/24 − q 49/24 + q 121/24 + ···n=1where χ 12 (12m ± 1) = 1, χ 12 (12m ± 5) = −1, andχ 12 (n) =0if n is divisibleby 2 or 3. This identity was discovered numerically by Euler (in the simplerlookingbut less enlightening version ∏ ∞n=1 (1 − qn )= ∑ ∞n=1 (−1)n q (3n2 +n)/2 )and proved by him only after several years of effort. From a modern point ofview, his theorem is no longer surprising because one now knows the followingbeautiful general result, proved by J-P. Serre and H. Stark in 1976:Theorem (Serre–Stark). Every modular form of weight 1/2 is a linearcombination of unary theta series.Explicitly, this means that every modular form of weight 1/2 with respectto any subgroup of finite index of SL(2, Z) is a linear combination of sums ofthe form ∑ n∈Z qa(n+c)2 with a ∈ Q >0 and c ∈ Q. Euler’s formula for η(z) isa typical case of this, and of course each of the other products given in Mersmann’stheorem must also have a representation as a theta series. For instance,the last function on the list, η(z)η(4z)η(6z) 5 /η(2z) 2 η(3z) 2 η(12z) 2 ,hastheexpansion∑ n>0, (n,6)=1 χ 8(n)q n2 /24 ,whereχ 8 (n) equals +1 for n ≡±1(mod8)and −1 for n ≡±3(mod8).We end this subsection by mentioning one more application of the Jacobitheta series.