Ernesto Girondo

Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

It is known that the largest disc that a compact hyperbolic surface of genusg may contain has radiusR=cosh−1(1/2sin(π/(12g−6))). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially withg. Elsewhere the authors have shown that for genusg>3 extremal surfaces contain only one extremal disc.Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries.

On extremal discs inside compact hyperbolic surfaces

Abstract It has been proved by C. Bavard that the radius of a disc isometrically embedded in a compact hyperbolic surface of genus g is bounded by R g = cosh ⁡ − 1 ( 1 2 sin ⁡ π 12 g - 6 ) , and that surfaces containing discs of such extremal radius are found in every genus. By constructing explicit surfaces of genus 2, he has also shown that extremal discs may or may not be unique. Here we show that a compact surface of genus g > 3 has at most one embedded extremal disc.

Compact non-orientable hyperbolic surfaces with an extremal metric disc

The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus g ≥ 3. We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case g = 3, we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface y 2 = Π 2g+2 i=1 (x-a i ) as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values a i . Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittakers conjecture cannot be true in its full generality. Recently, numerical computations have shown that Whittakers prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms. The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittakers conjecture.

On extremal Riemann surfaces and their uniformizing fuchsian groups

Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g = 2, 3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g ≥ 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms. 1991 Mathematics Subject Classification. 30F35, 30F10

On the structure of the Whittaker sublocus of the moduli space of algebraic curves

We prove the compactness of Whittaker sublocus of moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves which satisfy Whittaker’s conjecture on the uniformization of hyperelliptic curves via monodromy of Fuchsian differential equations. In the last part of the article we drive our attention to the statement made by R.A. Rankin more than forty years ago to the effect that the conjecture “has not been proved for any algebraic equation containing irremovable arbitrary constants”. We combine our compactness result with other facts coming from Teichmuller theory to show that in the most natural interpretations of this sentence we can think of, this is, in fact, impossible.

RIGHT TRIANGLES WITH ALGEBRAIC SIDES AND ELLIPTIC CURVES OVER NUMBER FIELDS

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point Pλ of infinite order in the Mordell-Weil group of the elliptic curve Y2 = X3 − n2X over ℚ(λ).