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June 1998 A Conjectural Extension of the Gross-Zagier Formula on Singular Moduli
Tim HUTCHINSON
Tokyo J. Math. 21(1): 255-265 (June 1998). DOI: 10.3836/tjm/1270042000

Abstract

The values of the elliptic modular $j$-invariant at imaginary quadratic arguments are algebraic integers, known as singular moduli of level one. If $d_1$ and $d_2$ are imaginary quadratic discriminants, then we may consider a generalized resultant of the class polynomials of the orders of discriminant $d_1$ and $d_2$; this is the norm of the differences of singular moduli of the corresponding orders, denoted here by $J(d_1,d_2)$. These resultants are highly factorizable; Gross-Zagier established a closed formula for $J(d_1,d_2)^2$ when $d_1$ and $d_2$ are fundamental discriminants, with $(d_1,d_2)=1$. In this paper we present a conjectural extension of the Gross-Zagier formula to the case when $d_1$ and $d_2$ are not necessarily fundamental, and $(d_1,d_2)=l^e$, where 1 is a prime not dividing the product of the conductors of $d_1$ and $d_2$.

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Tim HUTCHINSON. "A Conjectural Extension of the Gross-Zagier Formula on Singular Moduli." Tokyo J. Math. 21 (1) 255 - 265, June 1998. https://doi.org/10.3836/tjm/1270042000

Information

Published: June 1998
First available in Project Euclid: 31 March 2010

zbMATH: 0918.11032
MathSciNet: MR1621530
Digital Object Identifier: 10.3836/tjm/1270042000

Rights: Copyright © 1998 Publication Committee for the Tokyo Journal of Mathematics

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Vol.21 • No. 1 • June 1998
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