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The title says it all. Singular moduli of the j-function satisfy polynomials, but as the class number grows, these polynomial coefficients become very large. Weber functions are modular (not over the full modular group), and their values also satisfy polynomials. But the Weber polynomials tend to have much smaller coefficents. Why?

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The question "why?" sounds too philosophical. One simply tries to determine the corresponding size (for level 4 it is done in [V.D. Mirokov, Math. Notes 86 (2009) 216–233;] while classical level 1 is treated in [P. Tretkoff, Math. Proc. Cambridge Philos. Soc. 95 (1984) 389--402;]) and then compare the results. – Wadim Zudilin Dec 22 '10 at 23:28
Nice references. Thank you, Wadim. – Steven Heston Dec 24 '10 at 21:35
up vote 6 down vote accepted

Simply because they satisfy an equation of the form $P(f)-fj$ for some polynomial $P$. This immediately implies that the height of $f(z)$ will be around $1/deg(P)$ of the height of $j(z)$, or more precisely, asymptotic to it as the discriminant of $z$ goes to infinity.

See A. Enge and F. Morain's "Comparing invariants for class fields of imaginary quadratic fields", ANTS-V 2002.

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