Geometry of Elliptic Curves

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Monday 09.00-11.00 / Dr Elena Denisova (University of Glasgow)

Introduction to the Geometry of Elliptic Curves

Elliptic curves lie at the crossroads of algebraic geometry, number theory, and complex analysis, making them a central object of mathematical study. This course provides a graduate-level introduction to algebraic curves, with a primary focus on the geometry of elliptic curves over complex numbers.

We begin with the basic theory of algebraic curves, covering plane curves, singularities, and projective embeddings. The focus then shifts to elliptic curves - smooth projective curves of genus one with a specified point. We will study their properties, including the group law, isomorphisms, and etc. Key geometric topics include divisors, line bundles, morphisms to projective space, and the Riemann-Roch theorem with its applications.

The course is intended for graduate students with prior exposure to algebraic geometry (such as schemes or classical varieties) and abstract algebra. It is particularly suited for those interested in further study in algebraic geometry, arithmetic geometry, or related areas. The central reference for the course will be J.H. Silverman’s "The Arithmetic of Elliptic Curves". Depending on the interests of the audience, we may also study elliptic curves over fields other than C.

ASSESSMENT

One individual assignment in the form of a short presentation.

Students will prepare and deliver a 15-20-minute presentation on a related topic of their choice, to be agreed upon with the lecturer. This format allows students to deepen their understanding of an aspect of the theory, develop independent research skills, and practice mathematical communication.

Below is a list of suggested topics for the final presentations. Each topic is intended to extend beyond the material directly covered in the lectures, giving students the chance to explore related ideas in greater depth. Presentations should be 15-20 minutes long, focusing on conceptual clarity rather than technical computations.

Suggested Presentation Topics

Geometry-focused (G)

  1. Elliptic surfaces and K3 surfaces - Show how elliptic curves appear in families on algebraic surfaces, and explain their role in the geometry of K3 surfaces.
  2. Elliptic curves in cubic surfaces and plane cubics - Show how smooth hyperplane sections of cubic surfaces are elliptic curves, and explain the special projective geometry of plane cubics, including the fact that every smooth cubic has nine flex points.
  3. Degenerations of elliptic curves - From smooth genus-one curves to nodal and cuspidal cubics when the discriminant vanishes.
  4. Embeddings into projective space - Elliptic curve embeddings beyond plane cubics, via higher linear systems such as |nO| where O is the origin.
  5. Elliptic curves and del Pezzo surfaces - The appearance of elliptic curves as anticanonical curves on del Pezzo surfaces.

Arithmetic and Modern Applications (A)

  1. Mazur’s torsion theorem (overview) - A conceptual overview of Mazur’s classification of torsion subgroups of elliptic curves over Q; statement and significance, but without proof details.
  2. Elliptic curves in cryptography - Why elliptic curves are used in modern cryptographic systems.
  3. Elliptic curves and modular forms - Conceptual connections without analytic detail.
  4. Elliptic curves in Diophantine equations - How classical Diophantine problems lead to elliptic curves.
  5. The Birch and Swinnerton-Dyer conjecture (overview) - A conceptual overview of the conjecture; statement and significance as one of the central open problems in mathematics, without technical proofs.

Historical (G/A)

  1. Historical development of elliptic curves - From the 18th century to their modern role in geometry.

Note to Students

The above list is only a set of suggestions. The students are welcome to propose their own topic, provided it is related to elliptic curves and approved by the lecturer by Week 7.

Students must inform the lecturer of their chosen presentation topic by Week 7 for approval. Each topic can be chosen once, and allocation will be on a first-come, first-served basis.