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Search: MSC category 33E05 ( Elliptic functions and integrals )

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1. CMB 2009 (vol 52 pp. 583)

Konstantinou, Elisavet; Kontogeorgis, Aristides
Computing Polynomials of the Ramanujan $t_n$ Class Invariants
We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have much smaller coefficients than the Hilbert polynomials.

Categories:11R29, 33E05, 11R20

2. CMB 1999 (vol 42 pp. 427)

Berndt, Bruce C.; Chan, Heng Huat
Ramanujan and the Modular $j$-Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants.

Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields
Categories:33C05, 33E05, 11R20, 11R29

3. CMB 1997 (vol 40 pp. 276)

Chouikha, Raouf
Fonctions elliptiques et équations différentielles ordinaires
In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function $\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$ where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case, the bifurcations of $(E)$ depend on one parameter only. Moreover, this global method seems not to apply to the cases $n > 3$.

Categories:33E05, 34A05, 33E20, 33E30, 34A20, 34C23

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