Arithmetic of elliptic curves

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Type of exam

Oral exam


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Master and PhD students with a basic knowledge of classic algebraic geometry. Prior knowledge in algebraic number theory is suggested, although not strictly necessary.


- Recaps of algebraic geometry. Riemann-Roch theorem.

- Weierstrass equations. Invariants. Legendre form. Group law geometrically, algebraically and via divisors.

- Isogenies and endomorphism ring. Complex multiplication. Weil pairing.

- Recaps of Galois theory. Profinite groups. Tate module. Faltings' and Serre's theorems.

- Elliptic curves over finite fields. The zeta function and the Weil conjectures.

- Formal groups.

- Recaps of number theory. Local and global fields.

- Elliptic curves over local fields. Good and bad reduction. Neron-Ogg-Shafarevich criterion.

- Galois cohomology and the Kummer sequence.

- Weak Mordell-Weil theorem.

- Heights on projective spaces. The descent procedure. Proof of Mordell-Weil theorem.

- Integral points and Nutz-Nagell theorem.

- Hints of advanced topics.

Educational aims

Introducing the students to basic arithmetic and geometric aspects of elliptic curves theory over arbitrary fields. We will focus on elliptic curves over number fields and we will prove Mordell-Weil theorem, one of the most important results in arithmetic geometry.

Bibliographical references

- J. Silverman, "The arithmetic of elliptic curves"

- D. Husemöller, "Elliptic curves"