95%

* page 2: Why do you write "#S := |S|" on page 2?  You're not defining
anything.  You're just saying "I can't make up my mind between two
different notations."

* page 2: In definition of S are you taking *sequences*?  If so, they
are *not* subsets of QQ.  A subset of QQ does not have multiplicities.
You shouldn't use \{ \}'s notation, and shouldn't write \subset QQ.

* page 2: "under this metric;"  That's the wrong use of semicolon, imho.  Maybe a colon?  

* page 2: E(K) is not an elliptic curve.  It's the group of *points*
on an elliptic curve.  You want to write E/K "for E over K".  Also, an
elliptic curve is not a set of points.  It is an algebraic variety
over some field K.  There are sets of points associated to an elliptic
curve, for every field extension F of K.  But the cure is *not* the
points -- it's the object defined by the equation that has points
associated to it.  From a more formal perspective it is a locally
ringed space whose points are prime ideals and whose structure sheaf
involves the ring K[x,y]/(y^2 + ... a_6). 

* page 2: "minimal" for a Weierstrass equation only makes sense when K
has class number 1, e.g., K=Q.  In general no minimal Weierstrass
model need exist.

* page 3: in fig 1, I would draw the lines in a different color than
the plot of the curve itself.  Also, make the points big.

* page 3: writing "E = E(QQ)" would make most number theorists
cringe. See remark about about E(K).  I think you mean to say that
we assume E is defined over Q?

* page 3: "elements of E" --> "elements of E(K)"; You *really* need to
read a basic book on algebraic geometry from a modern (=post 1950)
perspective (i.e. schemes).

* page 4: it is very weird to define the torsion subgroup as the group of points
that generate a finite group.  just say it's the subgroup of points of finite order.

* page 4: You confuse the variables x,y with the specific point (x,y) at the beginning
of section 3.1.  I.e., x,y in K[x,y], then suddenly a second later, x,y in K.

* page 4: "parametrize a Weierstrass equation" -- that makes no sense
at all.  You're defining an invertible algebraic transformation from
the curve to another curve.  Parametrizing?  The u,r,s,t do parameterize
possible transformations, though you _forgot_ the condition that u=/=0.

* page 4: near the bottom -- "integral" only makes sense for certain fields K, e.g., QQ.
Are you assuming K=QQ?  You have to be since you can't define Omega as you do on page 5 unless
you're assuming K=QQ.

* page 5: "itegers" --> "integers"

* page 5: You confuse the curve and the set of points on the curve when defining reduction modulo p.  It
makes no sense to say "E(F_p)" is nonsingular, etc., but "E_{F_p} or E/F_p is nonsingular" makes sense.

* page 5: given the intended audience, define "multiplicative series"

* are you sure about how you define a_p when p | Delta?  What is #E(F_p) in that case? (Hint, I think you're wrong...)

* page 6: "highly nontrivial to prove L is entire" -- you could mention this is the main result that grew
out of Wiles's proof of FLT.

* page 6: An algebraic variety is *NOT* the solution set of a system of polynomials.  It is a mathematical
object defined by polynomials which has pointsets, which are solution sets, associated to it.

* page 6: in definition of C_1(K), you also need X(R) != {}.

* page 6: I wonder if your discussion of p-adics should be closer to
  3.5, since section 3.5 is the first time you use p-adics at all, I
  think.   

* page 6: "One can prove that if E has good reduction then c_p = 1".  That's totally obvious from your definition.

* page 7: "The Neron-Tate canonical height is finite." --> "The Neron-Tate canonical height is well defined."

* page 8: Maybe the group law as you defined it work fine in char 2,3.

* page 8: There are always infinitely many minimal weierstrass equations.   (see your question 2)

* page 8: changing the basis for E(Q)/tor merely changes the regulator by conjugating the matrix you're det'ing by
an invertible matrix in GL_r(Z), which has determinant +/-1. 
 
* page 8: squareness of sha follows from Cassels defining a nondegenerate alternating pairing on Sha.

* page 8: "ord_1 L(E,s)" --> "ord_{s=1} L(E,s)"

* page 8, etc.: Replace E_tor by E(Q)_tor everywhere. 

* page 8: "however it is not even clear that this is a rational
  number".  You could see that when r_an=0 it is a theorem, when
  r_an=1 it is a very deep theorem, and when r_an >= 2 it is a major
  open problem.