Elliptische Kurven II, Winter 2016-2017

This is the Homepage for the Vertiefungsvorlesung Elliptische Kurven II, given in Mainz during the Wintersemester of 2016-2017. If you're taking this course, please enroll on Jogustine. This course is the follow-up of the course on elliptic curves given the preceding semester; click here.

Practical information

Teacher: Ariyan Javanpeykar

Course: Tuesday 14h-16h and Tuesday 16h-18h.

Room: 04-516

Prerequisites: Elliptische Kurven I  

Course material and description

The spoken language of the course will be German. On the other hand, the references and the homework sheets will be in English.

We will follow Chapter I of Hartshorne's Algebraic Geometry. We will also use Silverman's The Arithmetic  of Elliptic Curves. We will also sometimes use Silverman's sequel to the previous book Advanced Topics in the Arithmetic of Elliptic CurvesWe will follow the proof of Mordell-Weil given by James Milne in Chapter VI.1, VI.2, VI.3 and VI.4 of his notes on elliptic curves. We will also use Neukirch's Algebraic Number Theory.

Examination: Everybody has to hand in the  homework sheets. These won't be graded, but rather awarded with a minus or plus. If you have enough +'s, you can take the exam.


We recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under this link.

I prefer you e-mail me your homework solutions. You can of course also just give them to me.


  1. To be handed in 2016 Nov 8th. Homework 1
  2. To be handed in 2016 Nov. 29th. Homework 2
  3. To be handed in 2016 Dec. 13th. Homework 3
  4. To be handed in 2017 Jan. 13th. Homework 4
  5. To be handed in 2017 February 3rd. Homework 5
  6. To be handed in 2017 Feb. 17th. Homework 6

TAKE HOME EXAM (click), to be handed in on March 3rd (or before).

ORAL EXAMS: Monday March 13th


  1. Tu. Oct. 25th. Finitely generated abelian groups. Elliptic curves over number fields. Statement of Mordell-Weil theorem. Examples. Hermite's theorem for monic polynomials with integer coefficients.
  2. Tu. Nov. 8th. Hand in HW1 Trace, norm, and discriminant of a finite separable field extension. Minkowski's geometry of numbers.  Basic theorems of algebraic number theory: finiteness of class number, Dirichlet's unit theorem, and Hermite's finiteness theorem.
  3. Tu. Nov. 15th. Lattices. Central symmetric convex subsets. Volumes. Minkowski's lattice point theorem. Norm of an ideal. Multiplicativity of the norm. Existence of small elements in a given ideal (application of Minkowski's theorem). Proof of finiteness of class number. Dirichlet's unit theorem (sketch of proof).
  4. Tu. Nov 22nd. Independence of characters. Hilbert 90. P^n(L)^{Gal(L/K)} = P^n(K). The group H^1(G,M) for finite groups G acting on an abelian group M. Infinite Galois groups Gal(L/k). Galois correspondence. H^1(Gal(kbar/k),M) for discrete Gal(kbar/k)-modules M. The exact sequence 0->E(k)/nE(k) -> H^1(G, E(kbar)[n]) -> H^1(G, E(kbar))[n] ->0 with G = Gal(kbar/k) and E an elliptic curve over k
  5. Tu. Nov 29th. Hand in HW2 The field of p-adic numbers and Hensel's lemma. Computing torsion groups via reduction modulo a prime p of good reduction.  Examples.
  6. Tu. Dec. 6th.  Torsion over Q injects into torsion over F_p; idea of proof. n-Selmer groups. Proof of finiteness of 2-Selmer group for E/Q with rational 2-torsion.
  7. Tu. Dec 13th. Hand in HW3 Finiteness of n-Selmer groups.
  8. Tu. Dec 20th. Explicit computations.
  9. Tu. Jan. 10th
  10. Tu. Jan. 17th Heights. Neron-Tate height on an elliptic curves. Completion of proof of Mordell-Weil via  descent Hand in HW4
  11. Tu. Jan. 24th. Nagell-Lutz theorem + example. Computing the rank of an elliptic curve using upper bound for rank + example. Discussion of Homework 5.
  12. Tu. Jan. 31st. NO LECTURE
  13. Tu. Feb. 7th. Roth's theorem, unit equation, integral points on elliptic curves, Shafarevich's finiteness theorem, Neron-Ogg-Shafarevich, isogeny theorem, and Serre's irreducibility theorem on Galois representations. Hand in HW5