This paper is a continuation of two others by the authors [in The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 437-480 (1990; Zbl 0726.14024); “Variétés de Kritchever de solitons elliptiques”, in Proc. Indo-French Conf. Geometry, Bombay 1989)], which were devoted to the study of the structure of the solutions of the KdV equation called elliptic solitons. To a hyperelliptic curve \(C\) of genus \(g\), equipped with a double point \(p\) of the hyperelliptic involution and a divisor \(D\) of degree \(g-1\), is associated a meromorphic function \(u\) on the complex plane, which is a so-called potential with a finite number of instability zones, and from which the triple \((C,p,D)\) can be recovered (the Schrödinger equation with potential \(u\) on the plane has a family of eigenfunctions parametrized by \(C\) with poles at \(D)\). The function associated with the divisor \((g-1)p\) is called the source potential. This paper is concerned with the case where \(C\) is a “tangential at \(p\)” covering of an elliptic curve \(E\). Here “tangential at \(p\)” means that the images of \(C\) and \(E\) in Jac\(C\) by the morphisms defined by \(p\) are tangent at the origin. The potentials associated to such a covering “live” on \(E\) and are combinations of Weierstrass functions there. To each such tangential covering of an elliptic curve by a hyperelliptic one is associated naturally a four-tuple of (intersection) numbers. And some of these four-tuples determine (for each \(E)\) a unique such tangential hyperelliptic covering. These are the so-called exceptional coverings.
The paper studies the above correspondence between “exceptional” four- tuples and combinations of Weierstrass functions, characterizing in particular those combinations which are reached by source potentials. The formulas describing this picture are full of triangular numbers, whence the title.