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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