Class Times: Mondays  Wednesdays  Fridays 12:00
 12:50
No classes on February 1822, March 21 and 24
Office:
CAB 621
Office hours:
Mondays 1:002:00, Wednesdays 10:00 12:00, and by appointment.
Phone:
(780) 4923613
email:
lalin at ualberta . ca or mlalin at math . ualberta . ca
Text: J. S. Milne, Elliptic
Curves, BookSurge Publishers, 2006, available here, here, and
here
Important links:
Homework:
Special Announcements:
 3/10: There won't be classes on Friday, March 14. There will be an
additonal class on Monday March 17 at 3 PM in CAB 563
 3/5: We are starting with Mordell's theorem. This is really cool
stuff! It you left the class becuase you hate complex analysis, this is a
good time to come back!
 2/13: We are having two additional classes on February 25 and
March 17 at 3PM in CAB 563
 1/30: Pari is in the system! To use it, just go to a unix machine
and ssh to numbers.math.ualberta.ca
Open a terminal and type: gp
Page 7678 from the book tell you a little bit of the applications to
elliptic curves. You can find more info
here. If you feel
inclined to this, the best place to start is the tutorial.
 1/28: Midterm has been canceled!!!
 1/25: We're having an additional class on February 4 at 3PM in
CAB 563. This
is besides the additional class on January 28.
 1/16: The deadline for the first homework has been extended until
January 28 at 1PM. We have agreed to have an extra lecture on January 28
at 2:30 PM in CAB 365. If we
manage to have this lecture the midterm will be canceled.
 1/14: I have extended the official time for office hours to Monday
12, Wednesdays 1012. You're always welcome to come to my office is you
see me there (if I'm busy, I'll let you know). The official office
hours are the times when it's guarantee
that I'll be in my office and I'll have time to discuss math with you.
Important dates:
 March 14: no class. (recovered January
28)
 March 28: no class. (recovered February
4)
 April 7: no class. (recovered March 17)
 April 9: no class. (recovered February
25)
Topics covered in
Class:
 4/4: finite basis theorem. Zeta functions from Riemann
Zeta to Zeta of varieties
 4/2: Survey. the canonical height is a quadratic form
 3/28: No class I'll be here.
 3/26: Heights on E, canonical height
 3/19: More on Heights on P^1
 3/17: More applications of weak finite
basis theorem, Heights
 3/14: No class I'll be here.
 3/12: Applications of weak finite basis
 3/10: weak finite basis continued
 3/7:weak finite basis (a la Cassels,
chap 14)
 3/5: end of endomorphisms E/C. Introduction to
Mordell's theorem
and strategy.
 3/3: C/Lambda and E(Lambda) are isomorphic groups.
classification of
elliptic curves over C (satement of uniformization, integrals), torsion
points, beginning of endomorphisms of E/C
 2/29(!!!):
C/Lambda and E(Lambda) are isomorphic Riemann surfaces
 2/27: morphisms of C/Lambda,
 2/25: Laurent series and differential equation for
Pfunctions, Field of elliptic functions, definition of Riemann surfaces
with the example of the Riemann sphere, C/Lambda is a Riemann surface
 2/15:Weierstrass
Pfunction, Einsestein series, (definition and convergence)
 2/13: doubly periodic (aka elliptic) functions (the fact that
their residues in a fundamental domain sum up to zero and similar
results, and that they have to have more than just a simple pole),
 2/11: invariance of the holomorphic differential (dx/y) , lattices
 2/8: Statement of Mazur's thm, examples of
computation of torsion, why elliptic integrals are natural functions from
elliptic curves, construction of periods, why we get a lattice
 2/6:End of proof of LutzNagell: E^1(Q_p) is
torsionfree, torsion injects in the reduced curve mod p for p of
good reduction.
 2/4: Q_p (filtration and such), statement of LutzNagell, partial
proof
 1/32: reduction modulo p (additive, multiplicative, minimal
model, semistable reduction)
 1/30: more on isogenies (statement of the structure of End(E) and
Aut(E), what is the dual isogeny and the main properties, what is complex
multiplication, multiplication by m as an isogeny).
 1/28: (morning) the Weierstrass form of an elliptic
curve (isomorphisms, discriminant, jinvariant, look also at page 49 of
Silverman's book), (afternoon) addition formulas, isogenies,
 1/25: A more serious defintion of an ellitpic curve
(at
last!!!!!!!) We did the equivalence of definitions, but we didn't cover
how to transform O into an inflection point. You may read that from the
book.
 1/23: maps (morphisms) between curves,
 1/21: divisors, statement of RiemannRoch, the
group in cubics again,
 1/18:The group in cubics works, rational and regular functions in
curves
 1/16: More padic numbers (Z_p, Hensel's lemma),
Localglobal principle revisited. Construction of the group law in cubics
(we haven't proved that it works yet).
 1/14: Projective curves, statement of Bezout's theorem,
introduction to padic
numbers (padic absolute values, definition of Q_p)
 1/11: Uniqueness of intersecction numbers, construction
of projective
plane
 1/9: Affine plane curves, definition of intersection numbers
 1/7: Intro to class: (Rough definition. Elliptic curves as
Diophantine equations. Connection with elliptic integrals. (Partial)
connection with congruent numbers. Main problems: elliptic curves over Q,
number of solutions, structure, BSD, Fermat.)
Other Resources:
In no particular order! I'll keep updating this, so let me know of any
suggestions!
 Other interesting books:

J. Silverman, The Arithmetic of Elliptic Curves
 N. Koblitz,
Introduction to Elliptic Cuves and Modular Forms
 A. Knapp,
Elliptic Curves
 J. Cassels, Lectures on Elliptic Curves
 J.
Silverman and J. Tate, Rational Points on Elliptic Curves
Last update: April 12, 2008