the genus of a Riemann surface
/ JEE-nus /
Once a Riemann surface is built and sealed up into a closed (compact) surface, the most basic question is: what shape is it? Topologically, every closed orientable surface is a sphere with some number of handles glued on, and that number is its genus. Genus 0 is the sphere (no handles); genus 1 is the torus, the surface of a doughnut (one handle, one hole you can put your finger through); genus 2 is the two-holed pretzel, and so on. The genus is the single integer that captures the surface's essential shape — its number of holes — and it does not change if you bend or stretch the surface without tearing it.
For the Riemann surface of an algebraic function — one defined by a polynomial relation like w^2 = p(z) — the genus can be computed from the data of sheets and branch points. The Riemann-Hurwitz formula relates the genus of the surface to the number of sheets and the total amount of branching: roughly, more branch points (or higher-order ones) build up more handles. As a worked feel for it: the surface of w = sqrt(p(z)) with p a polynomial of degree 2g+1 or 2g+2 (with distinct roots) is a hyperelliptic surface of genus exactly g. So sqrt of a quadratic gives genus 0 (a sphere), sqrt of a cubic or quartic gives genus 1 (a torus) — which is why elliptic functions and the torus are intertwined — and higher-degree polynomials give higher genus.
Genus matters because it is the master invariant controlling almost everything about the surface and its functions. It appears in the Riemann-Roch theorem (counting functions with prescribed poles), it fixes the curvature available by the uniformization theorem (genus 0 surfaces are spherical, genus 1 flat, genus at least 2 hyperbolic), and it ties Riemann surfaces to algebraic curves: a compact Riemann surface is the same thing as a smooth projective algebraic curve, and its genus is the curve's genus. So a single topological count — how many holes — governs the analysis, the geometry, and the algebra all at once.
The surface of w = sqrt((z - e_1)(z - e_2)(z - e_3)(z - e_4)), with four distinct roots, has branch points at the four e_i and two sheets. Gluing them up yields a torus — genus 1. This is exactly the surface on which a Weierstrass elliptic function lives, and it is why a cubic or quartic under a square root is the gateway to elliptic curves.
Square root of a quartic with distinct roots glues into a torus: genus 1, the home of elliptic functions.
Genus is a topological count of holes, not of sheets or branch points directly — you compute it from those via Riemann-Hurwitz. And genus only applies cleanly to a CLOSED surface; you must compactify (add the point at infinity) before the count is well defined.