ahlgrenono2005 - Compositio Math. 141 (2005) 293312 DOI:

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Compositio Math. 141 (2005) 293–312 DOI: 10.1112/S0010437X04001198 Arithmetic of singular moduli and class polynomials Scott Ahlgren and Ken Ono Abstract We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic Felds. We also study generalizations of Lehner’s classical congruences j ( z ) | U p 744 (mod p )(wh e r e p 6 11 and j ( z ) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p . 1. Introduction and statement of results Let j ( z )= q 1 + 744 + 196 884 q + 21 493 760 q 2 + ···∈ 1 q Z [[ q ]] denote the usual elliptic modular function on SL 2 ( Z )( q := e 2 πiz throughout). The values of j ( z ) at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli; two important examples are the evaluations j ( i ) = 1728 and j ± 1+ 3 2 ² =0 . (1.1) Singular moduli are algebraic integers which play many important roles in classical and modern number theory. ±or example, they generate ring class Feld extensions of imaginary quadratic Felds. Also, the work of Deuring [ Deu58 , Deu46 ] highlights their deep connections with the theory of elliptic curves with complex multiplication. In recent work, Borcherds [ Bor95a , Bor95b ] used them to deFne an important class of automorphic forms possessing certain striking inFnite product expansions. There is a vast amount of literature on the computation of singular moduli which dates back to the works of Kronecker; this includes the classical calculations of Berwick [ Ber28 ]andW ebe r [ Web61 ]. In more recent work, Gross and Zagier [ GZ85 ] computed exactly the prime factorization of the absolute norm of suitable di²erences of singular moduli (further work in this direction has been carried out by Dorman [ Dor89 , Dor88 ]). In this paper we investigate the divisibility properties of the traces and Hecke traces of singular moduli in terms of the factorization of primes in imaginary quadratic Felds. We begin by Fxing notation. Throughout, d denotes a positive integer congruent to 0 or 3 modulo 4 (so that d is the discriminant of an order in an imaginary quadratic Feld). Denote by Q d the set of positive deFnite integral binary quadratic forms Q ( x, y ax 2 + bxy + cy 2 Received 29 April 2003, accepted in fnal Form 14 May 2004, published online 10 ±ebruary 2005. 2000 Mathematics Subject Classifcation 11±33, 11±37 (primary). Keywords: singular moduli, class polynomials, modular Forms. The frst author thanks the National Science ±oundation For its support through grant DMS 01-34577. The second author is supported by the National Science ±oundation, a Guggenheim ±ellowship, a Packard Research ±ellowship, and an H. I. Romnes ±ellowship.
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This note was uploaded on 12/03/2011 for the course MATH 8430 taught by Professor Clark during the Fall '11 term at University of Georgia Athens.

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ahlgrenono2005 - Compositio Math. 141 (2005) 293312 DOI:

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