# zbMATH — the first resource for mathematics

Elliptic curves and modular functions. (English) Zbl 0225.14016
Teoria Numeri, 1968, Algebra, 1969, Symp. Math. 4, 27-32 (1970).
The author considers $$E_k^{(D)}: Dy^2 = x^3 + Ax + B$$ $$(A,B,D\in\mathbb Z)$$, the (projective) elliptic curve over the field $$k$$ (with $$k = \mathbb Q$$, $$D=1$$ understood when omitted). The classic question is to find the rank of the solution group (finite by the Mordell-Weil Theorem). The author constructs curves for which the rank is positive using methods based on a paper of K. Heegner [Math. Z. 56, 227–253 (1952; Zbl 0049.16202)]. If the curve $$J_N: F_N(u,v) =0$$ connects $$j(z)$$ and $$j(Nz)$$, this curve has coordinates at some point which are complex conjugates and generate $$K(-D)$$ over $$\mathbb Q(iD^{1/2})$$ where $$S^2-4NT^2 =-D<0$$, and $$K(-D)$$ is the ring class field modulo $$M$$ (with $$D=EM^2$$, $$-E$$ the field-discriminant). For certain values of $$N$$, these curves are (genus one) given by R. Fricke [Lehrbuch der Algebra. Bd. III: Algebraische Zahlen. Braunschweig: Vieweg (1928; JFM 54.0187.20)] in the equivalent form $$C_N: \sigma^2 =f_N(\tau)$$ for $$f_N$$ quartic. Such values of $$N$$ (which avoid rational roots of $$f_N)$$ are $$N=14,15,17,20,21,24,32,36,49$$. Here $$E_{(N)}$$ is the (equivalent) Jacobian of $$C_N$$ and the object is to show $$E_{(N),k}(-D)$$ has solutions (with $$k= \mathbb Q(iD^{1/2}))$$, by showing that $$C_N$$ has points in $$k$$ not in $$\mathbb Q$$. By considering the class field $$K(-D)$$ and its maximum real subfield $$L(-D)$$ whose degree is a power of $$2$$, the author sets conditions which make $$L(-D) = \mathbb Q$$. For example, if $$D$$ is a prime $$=4NT^2-S^2$$, then $$E_{(N)}^{-D}$$ has infinitely many rational points. Modifications are possible for $$h(-D)$$ even. Explicit formulas for $$f_N$$ and $$E_N$$ are listed, e.g., $$f_{36} = (t-1)^4-12t^2$$; $$E_{36}: Y^2 = X^3 +1$$, etc.
For the entire collection see [Zbl 0221.00003].

##### MSC:
 14H52 Elliptic curves 11G05 Elliptic curves over global fields 14G05 Rational points 11F03 Modular and automorphic functions 14H40 Jacobians, Prym varieties