Product formulas for the $5$-division points on the Tate normal form and the Rogers-Ramanujan continued fraction

Abstract

Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5]-⟨(0,0)⟩$ as products of linear fractional quantities in terms of $5$-th roots of unity and a parameter $u$, where the parameter $b$ which defines the curve $E_5$ is given as $b=({\epsilon }^5{u}^5-{\epsilon }^{-5})/(u^5+1)$ and $\epsilon =(-1+\sqrt{5})/2$. If $r(\tau )$ is the Rogers-Ramanujan continued fraction and $b=r^5(\tau )$, then the coordinates of points of order $5$ in $E_5[5]-⟨(0,0)⟩$ are shown to be products of linear fractional expressions in $r(5\tau )$ with coefficients in $\mathbb{Q}({\zeta }_5)$.

19 pages; the paper now discusses the connection with the Rogers-Ramanujan continued fraction; in version 4 several formulas in version 3 have been corrected

References in corpus (4)

• Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, II: The Rogers-Ramanujan continued fraction
• Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)
• The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields
• Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, I

Cited by in corpus (2)

• Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, I
• The Hasse invariant of the Tate normal form $E_5$ and the class number of $\mathbb{Q}(\sqrt{-5l})$