### Philadelphia Area Number Theory Seminar

Wednesday, April 17, 2024 - 3:30pm

#### Christopher-Lloyd Simon

Penn State University

We study several arithmetic and topological structures on the set of conjugacy classes of the modular group PSL(2; Z), such as equivalence relations or bilinear functions. A) The modular group PSL(2; Z) acts on the hyperbolic plane with quotient the modular orbifold M, whose oriented closed geodesics correspond to the hyperbolic conjugacy classes in PSL(2; Z). For a field K containing Q, two matrices of PSL(2; Z) are said to be K-equivalent if they are conjugated by an element of PSL(2; K). For K = C this amounts to grouping modular geodesics of the same length. For K = Q we obtain a refinement of this equivalence relation which we will relate to genusequivalence of binary quadratic forms, and we will give a geometrical interpretation in terms of the modular geodesics (angles at the intersection points and lengths of the ortho-geodesics). T) The unit tangent bundle U of the modular orbifold M is a 3-dimensional manifold homeomorphic to the complement of trefoil in the sphere. The modular knots in U are the periodic orbits for the geodesic flow, lifts of the closed oriented geodesics in M, and also correspond to the hyperbolic conjugacy classes in PSL(2; Z). Their linking number with the trefoil is well understood as it has been identified by E. Ghys with the Rademacher cocycle. We are interested in the linking numbers between two modular knots. We will show that the linking number with a modular knot minus that with its inverse yields a quasicharacter on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate. We will also associate to a pair of modular knots a function defined on the character variety of PSL(2; Z), whose limit at the boundary recovers their linking number.