Diophantine Approximation
and Abelian
Varieties
Mircea Mustata
Disclaimer:
my notes do not
always faithfully reflect what was actually said during the
corresponding lectures.
- Lecture
1 in PDF
and DJVU
(01/04/2012). The lecture was devoted to an overview of the course and
an introduction to
abelian varieties, including the Rigidity Lemma and some applications
as well as an introduction to the base change theorem.
- Lecture
2 in PDF
and DJVU
(01/06/2012). The lecture was mostly
devoted to the statement of the base change theorem
and of the semicontinuity theorem; the seesaw theorem; the statement of
the theorem of the cube (which will be proved
next time) and some applications of the theorem of the cube to abelian
varieties.
- I
missed Lecture 3
- Lecture
4 in PDF
and DJVU
(01/11/2012). We finished the proof of
the proposition giving various characterizations
of ample line bundles on abelian varieties and then proved the
following corollaries: if A is an abelian variety, then
A is projective as a variety, A is divisible as an abelian group, and
if n is any integer, then the degree of the
endomorphism of A given by multiplication by n is equal to n^{2g},
where g=dim(A).
- Lecture
5 in PDF
and DJVU
(01/13/2012). We discussed the structure
of the group of n-torsion points on an
abelian variety for each integer n>0 and the Picard scheme of a
smooth projective variety over an algebraically
closed field. We began talking about the construction of the dual of an
abelian variety over a field of characteristic 0.
- Lecture
6 in PDF
and DJVU
(01/18/2012). We proved a number of
lemmas on line bundles in Pic^0(X) for an
abelian variety X, e.g., the fact that if L is such a line bundle, then
L is "multiplicative," and if L is also nontrivial,
then H^j(X,L)=0 for all j (including j=0). We also showed that if a
line bundle on X is algebraically equivalent to 0,
then it belongs to Pic^0(X). Finally, we proved that if L is an ample
line bundle on X, then the induced map phi_L
from X to Pic^0(X) is surjective. (It is given by
phi_L(x)=T_x^*(L)\otimes L^{-1}, where T_x is translation by x.)
- Lecture
7 in PDF
and DJVU
(01/20/2012). The lecture was mainly
devoted to a discussion of quotients of
algebraic varieties by finite group actions and to isogenies of
algebraic varieties.
- Lecture
8 in PDF
and DJVU
(01/23/2012). We discussed G-equivariant
(quasi-)coherent sheaves on a variety
X equipped with an action of a finite group G. T we used the
formalism of equivariant sheaves to construct
the dual of an abelian variety together with the Poincare line bundle
(in the characteristic 0 setting).
- Lecture
9 in PDF
and DJVU
(01/25/2012). We finished the proof of
the universal property of the dual of an
abelian variety. We also discussed some properties of the dual and
constructed a canonical isomorphism between
an abelian variety and its double dual. The lecture ended with some
remarks about principal polarizations and
the Albanese variety of a smooth projective variety X (we saw that it
can be identified with the dual of Pic^0(X)).
- Lecture
10 in PDF
and DJVU
(01/27/2012). We proved that if X is an
abelian variety and L is a line bundle on X,
then the degree of the corresponding map from X to the dual of X is the
square of the Euler characteristic of L.
As a consequence, the principal polarizations on X correspond to those
ample line bundles on X whose space of
global sections is 1-dimensional. We then began the discussion of
Jacobians of smooth projective curves.
- Lecture
11 in PDF
and DJVU
(01/30/2012). Let X be a smooth
connected projective curve of genus g>0,
let J be the Jacobian of X and let J' be the dual abelian variety of J.
We recalled the construction of the
Abel-Jacobi map i from X to J and of the Albanese map a from X to J'.
By the universal property of the
Albanese variety, i factors as the composition of a and a unique
morphism f from J' to J. We stated and
mostly finished proving the theorem that f is an isomorphism and
-f^{-1} corresponds to the principal
polarization on J=Pic^0(X)
that comes from a translate of the canonical theta divisor on Pic^{g-1}(X).
- Lecture
12 in PDF
and DJVU
(02/01/2012). We finished the proof of
the theorem about principal polarizations
of Jacobians of smooth projective
curves from the previous lecture. We
then began the second part of the
course, devoted to arithmetic and the theory of heights. We had a
discussion of rational points of schemes
defined over (not necessarily algebraically closed) fields and defined
heights for points of P^n(Q).
- Lecture
13 in PDF
and DJVU
(02/03/2012). The lecture was devoted
to an overview of absolute values on
number fields. We discussed the relation between non-Archimedean places
and valuation rings as well as the
two standard normalizations of absolute values corresponding to the
places of a number field. We then proved
the product formula for number fields and defined heights for points of
projective spaces over number fields.
- Lecture
14 in PDF
and DJVU
(02/06/2012). We defined the absolute
and logarithmic heights for points of
projective space over an algebraic closure of Q
and stated Northcott's theorem.
- Lecture
15 in PDF
and DJVU
(02/08/2012). We proved Northcott's
theorem and began the proof the result
stating that if X is a projective variety over an algebraic closure of Q,
then given a morphism from X to
projective space, the induced logarithmic height on points of X depends
only on the pullback of O(1), up
to adding bounded functions. In particular, we showed that the usual
logarithmic height function on projective
space is invariant under linear changes of coordinates, modulo bounded
functions.
- Lecture
16 in PDF
and DJVU
(02/10/2012). We finished the proof of
the result from last time and deduced
Weil's theorem, which for any projective variety X over an algebraic
closure of Q
gives a
construction of a
canonical group homomorphism from Pic(X) to H(X). Here H(X) is the
quotient of the space of all real-valued
functions on the set of closed points of X by the subspace of bounded
functions. This homomorphism is denoted
by: L goes to h_L. We also discussed some properties of h_L for certain
types of line bundles L on X.
- Lecture
17 in PDF
and DJVU
(02/13/2012). We began the discussion
of heights on abelian varieties over an
algebraic closure of Q.
We
recalled some basic facts about linear, bilinear and quadratic
functions on an abelian
group A with values in R
and
discussed their analogues in the "quasi" setting, where each of the
properties
(linearity etc.) is only required to hold up to a bounded function. We
also proved that every quasi-linear (resp.
quasi-bilinear, resp. quasi-quadratic) function is equivalent to a
unique linear (resp. bilinear, resp. quadratic)
function, where "equivalent" again means that the difference between
the two functions is bounded.
- Lecture
18 in PDF
and DJVU
(02/15/2012). We defined the Neron-Tate
height associated to a line bundle on
an abelian variety and discussed the properties of these heights. We
then stated the Mordell-Weil theorem
and the weak Mordell-Weil theorem and proved that the latter implies
the former.
- Lecture
19 in PDF
and DJVU
(02/17/2012). We gave a complete proof
of the weak Mordell-Weil theorem,
modulo a certain finiteness result from global class field theory.
- Lecture
20 in PDF
and DJVU
(02/20/2012). We discussed some
properties of the Neron-Tate height defined
by the Poincare line bundle for an abelian variety. We then constructed
the Neron-Tate height on the Jacobian
of a smooth projective curve corresponding to a certain principal
polarization of this Jacobian.
- Lecture
21 in PDF
and DJVU
(02/22/2012). The lecture was devoted
to various preparations for the proof
of Mumford's theorem from 1965 (see below).
- Lecture
22 in PDF
and DJVU
(02/24/2012). We finished the proof of
Mumford's theorem, which states
(roughly speaking) that if C is a smooth projective curve of genus
g>1 over a number field K and C(K) is
infinite (though now we know that this assumption could never hold),
then the heights of points of C(K)
grow exponentially. The ideas Mumford introduced in his proof were
influential in the later proof of the
fact that C(K) is actually always finite when g>1.
- Lecture
23 in PDF
and DJVU
(03/05/2012). The lecture was devoted
to the modern viewpoint on heights,
namely, the one related to Arakelov theory. In particular, if R is the
ring of integers in a number field K, we
defined the degree of a metrized line bundle on Spec(R) and used this
notion to define the height function
associated to a metrized line bundle on a reduced projective scheme
over R.
- Lecture
24 in PDF
and DJVU
(03/12/2012). We finished the
discussion of the Arakelov theory approach
to heights on projective schemes over rings of integers in number
fields.
- Lecture
25 in PDF
and DJVU
(03/14/2012). We started a new section
of the course, devoted to Diophantine
approximation. We proved Liouville's and Dirichlet's theorems about
approximating algebraic numbers
with rational numbers, stated Roth's theorem and discussed the first
reduction step in its proof.
- Lecture
26 in PDF
and DJVU
(03/16/2012). The lecture was mostly
devoted to some heuristic computations
related to the proof of Roth's theorem. The goal was to explain why
certain naive ways of trying to extend
the method of the proof of Liouville's theorem by using polynomials of
several variables do not work.
- Lecture
27 in PDF
and DJVU
(03/19/2012). Today we began working
toward a proof of Roth's theorem.
We stated a result (Theorem A) that corresponds to the first step of
the proof of Liouville's theorem and
proved some auxiliary lemmas. The proof of Theorem A will be completed
next time.
- Lecture
28 in PDF
and DJVU
(03/21/2012). We finished the proof of
Theorem A and gave most of the
proof of Theorem B (which corresponds to the second step in the proof
of Liouville's theorem).
- Lecture
29 in PDF
and DJVU
(03/23/2012). We finished the proof of
Theorem B and began discussing
the idea of Theorem C (Roth's Lemma), which is the last step needed to
complete the proof of Roth's
theorem. The second half of the lecture was devoted to generalized
Wronskians.
- Lecture
30 in PDF
and DJVU
(03/26/2012). We proved an auxiliary
result about generalized
Wronskians and began an inductive proof of Roth's Lemma (it will be
finished in the next lecture).
- Lecture
31 in PDF
and DJVU
(03/28/2012). We finished the proof of
Roth's Lemma (Theorem C), and
then combined Theorems A,B,C from the previous classes to give a proof
of Roth's Theorem.
- Lecture
32 in PDF
and DJVU
(03/30/2012). We reformulated Roth's
theorem as a statement about pairs
of linear forms in 2 variables and explained why the naive
generalization to the case of n linear forms in
n>2 variables does not work. We stated the correct
generalization,
which is Schmidt's Subspace Theorem.
A related result -- Faltings's Product Theorem -- will be stated and
proved next week. The rest of today's
lecture was devoted to some preliminaries on numerical intersections on
complete varieties.
- Lecture
33 in PDF
and DJVU
(04/02/2012). The lecture was devoted
to an overview of properties of
intersection numbers for r-tuples of line bundles on an r-dimensional
proper scheme over a field.
- Lecture
34 in PDF
and DJVU
(04/04/2012). Today we began the last
portion of the course, devoted
to Faltings's second proof of the Mordell conjecture. We stated
Faltings's Product Theorem and
discussed some background material on differential operators (on smooth
varieties in characteristic 0).
- Lecture
35 in PDF
and DJVU
(04/06/2012). We explained the first
key step in the proof of Faltings's
Product Theorem (it accounts for about one half of the proof). The
proof will be finished next time.
- Lecture
36 in PDF
and DJVU
(04/09/2012). We finished the proof of
Faltings's Product Theorem.
- Lecture
37 in PDF
and DJVU
(04/11/2012). We began discussing the
geometric ingredients in
Faltings's second proof of (the generalization of) the Mordell
conjecture.
- Lecture
38 in PDF
and DJVU
(04/13/2012). We continued discussing
the geometric ingredients.
- Lecture
39 in PDF
and DJVU
(04/16/2012).
- Lecture
40 in PDF
and DJVU
(04/17/2012).
- Lecture
41 in PDF
and DJVU
(04/20/2012). This was the last lecture
of the course, in which the
arithmetic part of Faltings's proof of the main theorem was sketched.