Lecture

Modular Curves: Genus and Mapping Theorems

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In course

MATH-511: Number theory II.b - Modular forms

In this course we will introduce core concepts of the theory of modular forms and consider several applications of this theory to combinatorics, harmonic analysis, and geometric optimization. ...

Description

This lecture covers the concept of holomorphic maps between Riemann surfaces, including the definition of holomorphic maps, isomorphisms, and ramification points. It also explores the Open Mapping Theorem, the Degree of a holomorphic map, and the Riemann-Hurwitz formula. The lecture concludes with a detailed proof regarding the genus of a modular curve.

Instructor

Maryna Viazovska

Maryna Viazovska did her bachelor studies at the Kyiv National Taras ShevchenkoUniversity and completed her MSc at the Technical University Kaiserslautern.She obtained her PhD in 2013 in Bonn.She was a postdoctoral researcher at the Institut des Hautes Etudes Scientifiquesand at the Humboldt University of Berlin, and in 2017 was a Minerva DistinguishedVisitor at Princeton University. She joined EPFL in 2017 as Tenure-Track AssistantProfessor and was promoted Full Professor in 2018.

Official source

Related concepts (32)

Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from onto the open unit disk This mapping is known as a Riemann mapping. Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that is biholomorphic implies that it is a conformal map and therefore angle-preserving.

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

Modular curve

In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane).

Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

Classical modular curve

In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant. The curve is sometimes called X0(n), though often that notation is used for the abstract algebraic curve for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x).

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