**Time and place:** Thursday, 10:00-12:00, room: HG03.085, HG03.054, HG02.028, or HG00.308 (**see the individual talk**)

In this seminar we want to understand some aspects of elliptic curves. The theory of elliptic curves is a vast subject with applications in many areas (e.g., algebraic topology, number theory, cryptography) and the seminar will provide an introduction to the subject. Although we will only scratch the surface of the theory, this seminar is hopefully a good preparation for both a more thorough independent study and a course in algebraic geometry.

In the first part of the seminar (Talk 01 - Talk 04) we lay some foundations of basic algebraic geometry. In order to minimize the prerequisites we recall some central notions from field theory (Talk 01), algebraic geometry (affine varieties, projective varieties, dimension, morphisms; Talks 02), and 'commutative algebra of dimension one' (Talk 03). In the last talk of this part we introduce the category of algebraic curves. We will mention that the category of smooth curves over a fixed ground field K and non-constant morphisms is equivalent to a certain category of 'one-dimensional field-extensions of K'.

In the second part of the seminar (Talk 05 - Talk 09) we want to improve our general mathematical education and give sketch proofs of three important theorems in the theory of algebraic curves. The first one, Bezout's theorem (Talks 05-06), tells us how many intersection points two algebraic curves over an algebraically closed field have - at least if we count properly. The second one (Talk 07) states that every birational equivalence class of irreducible, projective curves contains a smooth representative and that this representative is unique up to isomorphism. The third theorem (Talks 08-09) is the celebrated Riemann-Roch theorem whose statement needs some preparation and whose importance can hardly be overestimated.

In the third part of the seminar (Talk 10 - Talk 17) we turn to elliptic curves. We begin by defining them using the Weierstrass equation. The Riemann-Roch theorem will allow us to show that elliptic curves are precisely the pointed, smooth curves of genus one. At the beginning of this part we focus on the geometry of elliptic curves, i.e., we work over an algebraically closed field. The special case of the field of complex numbers allows us to see the relation to the classical theory of elliptic functions and elliptic integrals. At the end of this part we will introduce the notion of a formal group law and see how elliptic curves provide some examples.

**Prerequisites:** Some standard field theory and some basics from commutative algebra are needed. An expertise in algebraic geometry is not assumed but some elementary knowledge (as treated in Reid, Undergraduate Algebraic Geometry) is helpful.

**References:** The main reference for us will be the book by Silverman but we also mention some further references.

Fulton: Algebraic Curves

Husemöller: Elliptic Curves

Knapp: Advanced Algebra

Miranda: Algebraic Curves and Riemann Surfaces (treats the picture over the complex numbers)

Lang: Introduction to Algebraic and Abelian Functions

Perrin: Algebraic Geometry, An Introduction

Reid: Undergraduate Algebraic Geometry (very nice introduction to elementary algebraic geometry)

Silverman: The Arithmetic of Elliptic Curves (main reference for talk 05 and talks 12-20)

In the first part of the seminar (Talk 01 - Talk 04) we lay some foundations of basic algebraic geometry. In order to minimize the prerequisites we recall some central notions from field theory (Talk 01), algebraic geometry (affine varieties, projective varieties, dimension, morphisms; Talks 02), and 'commutative algebra of dimension one' (Talk 03). In the last talk of this part we introduce the category of algebraic curves. We will mention that the category of smooth curves over a fixed ground field K and non-constant morphisms is equivalent to a certain category of 'one-dimensional field-extensions of K'.

In the second part of the seminar (Talk 05 - Talk 09) we want to improve our general mathematical education and give sketch proofs of three important theorems in the theory of algebraic curves. The first one, Bezout's theorem (Talks 05-06), tells us how many intersection points two algebraic curves over an algebraically closed field have - at least if we count properly. The second one (Talk 07) states that every birational equivalence class of irreducible, projective curves contains a smooth representative and that this representative is unique up to isomorphism. The third theorem (Talks 08-09) is the celebrated Riemann-Roch theorem whose statement needs some preparation and whose importance can hardly be overestimated.

In the third part of the seminar (Talk 10 - Talk 17) we turn to elliptic curves. We begin by defining them using the Weierstrass equation. The Riemann-Roch theorem will allow us to show that elliptic curves are precisely the pointed, smooth curves of genus one. At the beginning of this part we focus on the geometry of elliptic curves, i.e., we work over an algebraically closed field. The special case of the field of complex numbers allows us to see the relation to the classical theory of elliptic functions and elliptic integrals. At the end of this part we will introduce the notion of a formal group law and see how elliptic curves provide some examples.

**Schedule of the seminar:**

Talk 00, 04.10.12: Overview (HG03.054, Moritz Groth)

Talk 01, 11.10.12: Recap of field theory (HG03.054, Giovanni Caviglia)

Talk 02, 18.10.12: Affine varieties, projective varieties, and maps between varieties (HG03.054, Dimitri Ara)

Talk 03, 25.10.12: Discrete valuation rings, Dedekind domains (HG03.054, Frank Rouman)

Talk 04, 01.11.12: The category of algebraic curves (HG03.054, Sander Rieken)

Talk 05, 08.11.12: Bezout's Theorem I (HG02.028, Giovanni Caviglia)

Talk 06, 22.11.12: Bezout's Theorem II (HG02.028, Matan Prezma)

Talk 07, 29.11.12: Smooth representative of birational equivalence class of irreducible, projective curves (HG02.028, Dimitri Ara)

Talk 08, 13.12.12: Cohomology of sheaves (HG02.028, Matan Prezma)

Talk 09, 18.12.12: Riemann-Roch Theorem (HG00.308, Moritz Groth)

Talk 10: 10.01.13: The Weierstrass equation and the tangent-chord group law I (HG03.085, Frank Rouman)

Talk 11: 24.01.13: The Weierstrass equation and the tangent-chord group law II (HG03.085, Sander Rieken)

Talk 12: 31.01.13: Isogenies (HG03.085, Giovanni Caviglia)

Talk 13: 11.02.13: The invariant differential of an elliptic curve and the dual isogeny (HG03.085, Roberta Iseppi)

Talk 14: 21.02.13: Elliptic curves over the complex numbers (HG03.085, Sander Rieken)

Talk 15: 28.02.13: The Mordell-Weil theorem (HG03.085, Dimitri Ara)

Talk 16: 14.03.13: The formal group law associated to an elliptic curve (HG03.085, Matan Prezma)

Talk 17: 21.03.13: A glimpse at elliptic cohomology (HG03.085, Moritz Groth)

Fulton: Algebraic Curves

Husemöller: Elliptic Curves

Knapp: Advanced Algebra

Miranda: Algebraic Curves and Riemann Surfaces (treats the picture over the complex numbers)

Lang: Introduction to Algebraic and Abelian Functions

Perrin: Algebraic Geometry, An Introduction

Reid: Undergraduate Algebraic Geometry (very nice introduction to elementary algebraic geometry)

Silverman: The Arithmetic of Elliptic Curves (main reference for talk 05 and talks 12-20)