Singular moduli for a distinguished non-holomorphic modular function

Dose, Valerio; Green, Nathan; Griffin, Michael; Mao, Tianyi; Rolen, Larry; Willis, John

doi:10.1090/S0002-9939-2014-12289-1

Singular moduli for a distinguished non-holomorphic modular function
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by Valerio Dose, Nathan Green, Michael Griffin, Tianyi Mao, Larry Rolen and John Willis PDF

Proc. Amer. Math. Soc. 143 (2015), 965-972 Request permission

Abstract:

Here we study the integrality properties of singular moduli of a special non-holomorphic function $\gamma (z)$, which was previously studied by Siegel, Masser, and Bruinier, Sutherland, and Ono. Similar to the modular $j$-invariant, $\gamma$ has algebraic values at any CM-point. We show that primes dividing the denominators of these values must have absolute value less than that of the discriminant and are not split in the corresponding quadratic field. Moreover, we give a bound for the size of the denominator.

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