arXiv subjects

Ronen E. Mukamel

Publications and source records attributed to Ronen E. Mukamel.

4 recordsLinked to original sources

arXiv2016

Algebraic models and arithmetic geometry of Teichmüller curves in genus two

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichmüller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.

math.AG↗

arXiv2016

Orbifold points on Teichmüller curves and Jacobians with complex multiplication

For each integer $D \geq 5$ with $D \equiv 0$ or $1 \bmod 4$, the Weierstrass curve $W_D$ is an algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichmüller curves in genus two. The primary goal of this paper is to determine the number and type of orbifold points on each component of $W_D$. Our enumeration of the orbifold points, together with work of Bainbridge and McMullen, completes the determination of the homeomorphism type of $W_D$ and gives a formula for the genus of its components. We use our formula to give bounds on the genus of $W_D$ and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on $W_D$.

math.GT↗

arXiv2016

Visualizing the unit ball for the Teichmüller metric

We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated to the Finsler Teichmüller metric. Our method involves computing the periods of abelian double covers and is easy to implement for Riemann surfaces presented as algebraic curves using existing tools for approximating period matrices of plane algebraic curves. We illustrate our method by depicting the unit sphere in the cotangent space to moduli space at a particular surface of genus zero with five punctures and by corroborating the proof of a theorem of Royden's for our example.

math.CV↗

arXiv2016

Real multiplication through explicit correspondences

We compute equations for real multiplication on the divisor classes of genus two curves via algebraic correspondences. We do so by implementing van Wamelen's method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant 5 and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

math.AG↗