On singular moduli. (English) Zbl 0545.10015

Gross, Benedict H.; Zagier, Don B.

J. Reine Angew. Math. 355, 191-220 (1985).

The values of the modular function \(j(\tau)\) at imaginary quadratic arguments in the upper half plane are known as singular moduli. They are algebraic integers, and their differences turn out to be highly divisible numbers. We determine the prime factorization of the absolute norm of \(j(\tau_ 1)-j(\tau_ 2)\), where \(\tau_ 1\) and \(\tau_ 2\) are arguments of discriminants \(d_ 1\) and \(d_ 2\), and show that all primes \(\ell\) which divide this norm must divide a positive integer of the form \((D-x^ 2)/4\), where \(D=d_ 1d_ 2\). We also factor values of the modular polynomial \(\Phi_ m(x,y)\) at singular moduli; in this case the primes dividing \(\Phi_ m(j(\tau_ 1),j(\tau_ 2))\) must divide a positive integer of the form \((m^ 2D-x^ 2)/4.\)
Two methods of proof are given. The first is algebraic, and exploits the connection between the arithmetic of maximal orders in quaternion algebras of prime discriminant \(\ell\) over \({\mathbb{Q}}\) and the geometry of supersingular elliptic curves in characteristic \(\ell\). The second is analytic, and is based on the calculation of the Fourier coefficients of the restriction to the diagonal of an Eisenstein series for the Hilbert modular group of \({\mathbb{Q}}(\sqrt{D})\). Both methods may be viewed as the special case \(N=1\) of the theory of local heights for Heegner points on \(X_ 0(N)\).

MSC:

11F03	Modular and automorphic functions
14K10	Algebraic moduli of abelian varieties, classification
14H45	Special algebraic curves and curves of low genus
11F41	Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11R52	Quaternion and other division algebras: arithmetic, zeta functions

Keywords:

modular function \(j(\tau)\); singular moduli; prime factorization of the absolute norm; modular polynomial; arithmetic of maximal orders in quaternion algebras; geometry of supersingular elliptic curves; Fourier coefficients; Eisenstein series; Hilbert modular group; local heights; Heegner points