### MS-E1996 - Elliptic curves, 01.06.2020-31.08.2020

Credits: 5

Schedule: 01.06.2020 - 31.08.2020

Contact information for the course (applies in this implementation):

Prof. Guillermo Mantilla-Soler: gmantelia@gmail.com, guillermo.mantillasoler@aalto.fi

Details on the course content (applies in this implementation):

This course is an introduction the arithmetic study of elliptic curves. An elliptic curve E over a field K is a genus 1 curve, defined over K, together with a K-rational point on it. For practical purposes, our working definition of elliptic curve will be a cubic polynomial f(x, y) ∈ K[x, y] such that the equation f(x, y) = 0 has solutions in K, and such that f has no singular points in an algebraic closure of K. Elliptic curves are at the center of modern research in number theory, not only in terms of pure arithmetic but also in applied mathematics. Elliptic curves over the rationals are protagonist in Wiles’ proof of Fermat’s last theorem, moreover over finite fields elliptic curves appear quite often in modern public key cryptography. In this course we will study elliptic curves as interesting pure mathematical objects. The class is aimed at advanced undergraduate, and graduate, students interested in an introduction to a very active field of research in algebra and number theory.

In terms of prerequisites for the class a second course in abstract algebra and mathematical maturity are assumed. Desirable, although not necessary to follow the course, are Galois Theory and a bit of algebraic number theory.

Elaboration of the evaluation criteria and methods, and acquainting students with the evaluation (applies in this implementation):

This will be a pass-fail class, with mandatory assistance for those who want the pass grade.  I will assign two or three homework sets and based on their completeness, and attendance, the fail or pass grade will be determined. There are no exams

Details on the course materials (applies in this implementation):

The following topics will be discussed:

• Introduction: How geometry determines arithmetic; classification of rational points of rational elliptic curves according to their genus, the trichotomy  $g=0,1$ and $g \ge 2$.
• Projective geometry, quick review of algebraic curves,  and properties of intersection in projective spaces. Singular points, rational parametrizations.
• Group structure on points of an elliptic curve.
• Weierstrass form of an elliptic curve. Isogenies.
•  Complex and real elliptic curves. Uniformization theorem. The Weil Pairing over complex elliptic curves
•  Mordell-Weil theorem. Rank, Mazur's theorem on torsion.

Course Homepage (valid 01.08.2018-31.07.2020):

https://mycourses.aalto.fi/course/search.php?search=MS-E1996

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Additional information for the course (applies in this implementation):

Although I will not follow strictly any text book, the following three sources are good companions for the class:

•  J. Silverman, J. Tate. Rational points on elliptic curves.
•  J. Silverman. The arithmetic of elliptic curves}.
•  N. Koblitz.  Introduction to Elliptic curves and Modular forms.}

Details on the schedule (applies in this implementation):

The class will consist of  a total of six lectures to take place via Zoom Mondays and Wednesdays  from 12:15 to 14:00, starting on Monday 1.06 and finishing on Wednesday 18.06.