AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the teaching is to provide advanced contents of number theory on elliptic curves which are considered fundamental for students who intend to continue their studies in a PhD.

AIMS AND LEARNING OUTCOMES

The aim of this course is to provide an introduction to the arithmetic theory of elliptic curves (with some discussion of Abelian varieties of arbitrary dimension), which is one of the central topics of modern arithmetic geometry and algebraic number theory. More specifically, the ultimate aim of the course is to prove the Mordell–Weil theorem, which states that the abelian group of rational points on an elliptic curve defined over a number field (i.e. a finite-degree extension of the field of rational numbers) is finitely generated. The course can therefore be seen as an introduction to the study of Diophantine problems (particular arithmetic problems arising in a geometric context) using techniques from algebraic number theory and commutative algebra.

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PREREQUISITES

The content of the compulsory algebra and geometry modules on the Mathematics degree programme. It is certainly preferable (though not strictly necessary) to have taken Number Theory 2.

TEACHING METHODS

Traditional format.

Students with a certified learning disability, disability or other special educational needs are advised to contact the lecturer at the start of the course to agree on teaching and assessment arrangements which, whilst adhering to the course objectives, take into account individual learning styles and provide appropriate compensatory measures.

SYLLABUS/CONTENT

- A refresher on algebraic geometry (if necessary).
- Elliptic curves and Weierstrass equations.
- The group law on an elliptic curve.
- Isogenies between elliptic curves.
- Tate modules of an elliptic curve.
- Elliptic curves over finite fields: the Artin–Hasse theorem.
- Reduction types of an elliptic curve.
- The semi-stable reduction theorem.
- Heights on projective spaces.
- Selmer and Shafarevich–Tate groups of an elliptic curve over a number field.
- Elliptic curves over number fields: proof of the weak Mordell–Weil theorem.
- Heights on elliptic curves.
- Elliptic curves over number fields: proof of the Mordell–Weil theorem.

RECOMMENDED READING/BIBLIOGRAPHY

- J. H. Silverman, "The arithmetic of elliptic curves", Springer-Verlag, 1986.
- D. Husemöller, "Elliptic Curves", Springer-Verlag, 1987.
- A. W. Knapp, "Elliptic curves", Princeton University Press, 1992.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

See the link https://corsi.unige.it/corsi/11907/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy