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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 $$y=0$$).

##### MSC:
 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
##### Keywords:
ternary forms; elliptic curve; invariants
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