Monodromy groups of Belyi Lattes maps (Edray Goins, Pomona College)

An elliptic curve $ E: y^2 + a_1 \, x \, y + a_3 \, y = x^3 + a_2 \, x^2 + a_1 \, x + a_6 $ is a cubic equation which has two curious properties: (1) the curve is nonsingular, so that we can draw tangent lines to every point $ P = (x,y) $ on the curve; and (2) the collection of complex points, namely $ E(\mathbb C) $, forms an abelian group under a certain binary operation $ \bigoplus: E(\mathbb C) \times E(\mathbb C) \to E(\mathbb C) $.   In particular, for every positive integer $N$, the map $ P \mapsto [N] P $ which adds a point $ P \in E(\mathbb C) $ to itself $N$ times is a group homomorphism.   A rational map $\gamma: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) $ from the Riemann Sphere to itself is said to be a Latt\`{e}s Map if there are “well-behaved” maps $ \phi: E(\mathbb C) \to \mathbb P^1(\mathbb C) $ and $\psi: E(\mathbb C) \to E(\mathbb C) $ such that $\gamma \circ \phi = \phi \circ \psi$.  We are interested in those Latt\`{e}s Maps $\gamma$ which are also Bely\u{\i} Maps, that is, the only critical values are $ 0 $, $ 1 $, and $ \infty $.  Work of Zeytin classifies all such maps: For example, if $ E: y^2 = x^3 + 1 $ then $ \phi: (x,y) \mapsto (y+1)/2 $ while $\psi = [N] $ for some positive integer $N$.

We would like to know more about Bely\u{\i} Latt\`{e}s Maps $\gamma$.  What can we say about such maps?  What are their Dessin d’Enfants?  In some cases, this is a bipartite graph with $ 3 \, N^2 $ vertices.  What are their monodromy groups? Sometimes this is a group of size $ 3 \, N^2 $.  In this talk, we explain the complete answers to these questions, exploiting the relationship between fundamental groups of Riemann surfaces and Galois groups of function fields.  This work is conducted as part of the Pomona Research in Mathematics Experience (DMS-2113782).