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Arithmetic of the modular functions \(j_{1,2}\) and \(j_{1, 3}\). (English) Zbl 1196.11062

Some years ago the authors studied a field generator for the field of modular functions on the group \(\Gamma_1(4)\) [Acta Arith. 84, No. 2, 129–143 (1998; Zbl 0907.11014)]. Now they do the same for the groups \(\Gamma_1(2)\), \(\Gamma_1(3)\) which also have genus \(0\). A field generator \(j){1,2}\) for \(\Gamma_1(2)\) is defined in terms of classical Jacobi theta functions. Expressing theta functions by eta products, the definition amounts to \(j_{1,2}(z)= 2^8\Delta(2z)/\Delta(z)\), where \(\Delta\) is the discriminant function.
{Reviewer’s remark: Starting with this definition from the outset would make the proof of Theorem 6 considerably shorter.}
Then they use the Eisenstein series \(E_4\) to define the field generator \(j_{1,3}(z)= E_4(z)/E_4(3z)\) for \(\Gamma_1(3)\).
{Reviewer’s remark: One can as well take \((\Delta(z)/\Delta(3z))^{1/2}\), with similar properties.}
Finally the authors discuss relations of \(j_{1,2}\), \(j_{1,3}\) with Thompson series (“moonshine”), and they (re-)prove that values at imaginary quadratic points are algebraic integers.

MSC:

11F11 Holomorphic modular forms of integral weight
11F06 Structure of modular groups and generalizations; arithmetic groups
11G18 Arithmetic aspects of modular and Shimura varieties

Citations:

Zbl 0907.11014
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