JournalslemVol. 59, No. 1/2pp. 73–113

Triangle groups, automorphic forms, and torus knots

  • Valdemar V. Tsanov

    Georg-August Universität Göttingen, Germany
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Abstract

This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, C×\mathbb{C}^\times-equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group PSL2(Z)PSL_2(\mathbb Z): the quotient of the nonzero tangent bundle on the upper-half plane by the action of PSL2(Z)PSL_2(\mathbb Z) is biholomorphic to the complement of the plane curve z327w2=0z^3-27w^2=0. This can be shown using the fact that the algebra of modular forms is doubly generated, by g2,g3g_2,g_3, and the cusp form Δ=g2327g32\Delta=g_2^3-27g_3^2 does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between PSL2(R)/PSL2(Z)PSL_2(\mathbb R)/PSL_2(\mathbb Z) and the complement of the trefoil knot - the local knot of the singular curve. This construction is generalized to include all (p,q,)(p,q,\infty)-triangle groups and, respectively, curves of the form zq+wp=0z^q+w^p=0 and (p,q)(p,q)-torus knots, for p,qp,q co-prime. The general case requires the use of automorphic forms on the simply connected group SL2~(R)\widetilde{SL_2}(\mathbb R). The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case.

Cite this article

Valdemar V. Tsanov, Triangle groups, automorphic forms, and torus knots. Enseign. Math. 59 (2013), no. 1, pp. 73–113

DOI 10.4171/LEM/59-1-3