Elliptic modular invariants and numerical ranges. (English) Zbl 1314.14056

For any complex square matrix \(A\) of dimension \(n\) define \[ F_A(t,x,y):=\det\left( tI_n+x\frac{A+A^*}{2}+y\frac{A-A^*}{2i} \right) \] as the associated real ternary form. Such forms are hyperbolic with respect to the vector \((1,0,0)\), i.e., \(F(1,0,0)\neq 0\) and for any \((t_1,x_1,y_1)\in \mathbb{R}^3-\mathbb{R}(1,0,0)\) the equation \(F(s(1,0,0)+ (t_1,x_1,y_1))=0\) has \(n\) real roots. If those roots are distinct the form is called strongly hyperbolic. The paper deals with elliptic curves which can be written projectively as a (transformation of) \(F_A(t,x,y)=0\) for matrices of dimensions 3 and 4.
Using the classifications of cubics arising from \(3\times 3\) matrices provided in [R. Kippenhahn, Math. Nachr. 6, 193–228 (1951; Zbl 0044.16201)] (while also correcting an argument of the original paper) the authors show that real irreducible strongly hyperbolic forms correspond to elliptic curves with positive discriminant and \(j\)-invariant larger than or equal to 1. A similar result holds for forms arising from \(4\times 4\) matrices but with more stringent hypotheses on symmetry of the quartic \(F(t,x,y)\) (i.e., \(F(t,x,y)=F(t,x,-y)\,\)) and on the type of its singular points (their case by case analysis covers the cases of two imaginary cusps, a real tacnode, two imaginary nodes and two real nodes not on the line \(y=0\)).


14H52 Elliptic curves
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory


Zbl 0044.16201
Full Text: DOI


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