address note 08: wording

Comment:

    Maybe "sketch the notions" is more appropriate, since we're not
    really giving a definition - at least for the normalization.
address note 09: what happens upstairs

Comment:

    This is not really true, after desingularizing many things could
    happen. The point is that the number of rational points on the
    desingularization stays finite and can be easily bounded above and
    below (see
    http://math.stackexchange.com/questions/176844/rational-points-on-singular-curves-and-their-normalization)

I ended up removing the remark, and modified the part that used the
remark. Fortunately, it still works.
address note 10 and 11: references

Comment 10:

    You haven't said what O_X is, you need to give a reference for that.

Comment 11:

    Same here: reference for omega and cohomology group… also I think
    that you should use capital Omega since you mean the sheaf of
    differentials.
address note 14: remove sentence

Comment:

    Let's get rid of this sentence...
address note 15: change the notation for a field...

from K to k.

Comment:

    Formally everything is fine, but you may want to use a different
    letter for the field, since K is usually a number field, while often
    we will use it when K is some transcendent extension.
address note 16: typo

Comment:

    period N, not periodic
refilled the paragraph - nothing changed
address note 17: wording

Comment:

    more like, a refinement appropriately cooked for this specific curve.
address note 18: add reference for BSD

Comment:

    Maybe give a reference for the BSD conjecture?
address note 23: typo

Comment:

    "tried" missing