Generation of class fields by using the Weber function. (English) Zbl 1411.11115
Summary: Let \(K\) be an imaginary quadratic field and \(\mathcal{O}_K\) be its ring of integers. Let \(h_E\) be the Weber function on a certain elliptic curve \(E\) with complex multiplication by \(\mathcal{O}_K\). We show that if \(N (>1)\) is an integer prime to 6, then the function \(h_E\) alone generates the ray class field modulo \(N \mathcal{O}_K\) over \(K\) when evaluated at some \(N\)-torsion point of \(E\), which would be a partial answer to the question mentioned in [K. Ramachandra, Ann. Math. (2) 80, 104–148 (1964; Zbl 0142.29804), p. 105].
MSC:
| 11R37 | Class field theory |
| 11G15 | Complex multiplication and moduli of abelian varieties |
| 11G16 | Elliptic and modular units |
Citations:
Zbl 0142.29804References:
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