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Everything about Modular curve

Algebraic variety

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 to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers Q or a cyclotomic field Q(ζn). The latter fact and its generalizations are of fundamental importance in number theory. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Modular curveA 5-minute orientation
  2. Skim a recent paper: The least degree of a CM point on a modular curveInfluential work since 2022
  3. Pick up a classic book: Geometric Modular Forms and Elliptic CurvesThe most-reprinted book on the subject
  4. Read the landmark paper: Area-Time Efficient Hardware Implementation of Modular Multiplication for Elliptic Curve CryptographyCited 68 times
  5. Go sideways: Modular formA closely related topic
  6. Go sideways: Modular groupA closely related topic

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🗂️ The subject at a glance

  1. Analytic definition
  2. Examples
  3. Genus
  4. Relation with the Monster group

🏆 Landmark papers (most cited)

🆕 Influential since 2022

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