Attempted "analytic model" of universal cover of a Manin kernel:

Let A be an abelian variety over a differential field K which is the
differential function field of a complex algebraic curve S with a rational
vector field (rational/C section of TS -> S).

Deprecated (due to falsity)
    Now consider an arbitrary omega-chain of non-empty simply connected open
    subsets U_i of S(\C) such that \cap_i U_i = {}. Let L_i be the ring of the
    restrictions to U_i of holomorphic functions on neighbourhoods of cl(U_i),
    considered as a differential ring with respect to our chosen vector field. Let
    L be the direct limit.

    Claim: L is an algebraically closed field
    Proof:
	Any non-zero f\in L_n has discrete zero set in U_n, so its restriction to
	some U_n' is invertible. So L is a field.

	Algebraic closure can be seen similarly: given a polynomial over \Z
	    f(X,Z_1,Z_n)
	such that the zero set V projects dominantly to the last n co-ordinates,
	and given and g_1,...,g_n \in L algebraically independent, we want to find
	a solution to f(X,g_1,...,g_n)=0 in L.

	Let R (/=_cl A^n be the ramification locus of the projection. By genericity
	of g_1,...,g_n and \Cap U_i = {}, for some i, (g_1,...,g_n)(U_i) does
	not intersect R. So by picking a solution at a point and analytically
	continuing to U_i, there is a solution in L_i.

	Wait. But why would it extend to a neighbourhood of the *closed* disk?
	What if the standard part of our point is in the ramification locus?

	Fuck.

	This isn't going to work at all. Obviously.

	So... should instead work in the traditional way, taking meromorphic
	functions on inverse limit of finite covers?

    (We can think of L as the field of germs of holomorphic functions at a
    non-standard complex point)

Gropings:
    Let ^S be the inverse limit of finite covers of S, and let L be the field of
    meromorphic functions on ^S, i.e. the direct limit of the fields of
    meromorphic functions on the finite covers.

    No, that doesn't work either - say f has infinite discrete zero set; then to
    find a square root of f we need to take a cover which branches at each of
    those points. So that isn't an algebraic cover, and surely things go crazy if
    we start allowing non-algebraic covers...

    Ah no - we should look at the algebraic closure of the Seidenberg field as
    being a matter of looking at germs on branched covers. I think.

    Yes. See Khovanskii. It's "convergent Puiseux series".

-----

Let A be an abelian variety over the differential field L of germs at 0 of
meromorphic functions on \C. (By Seidenberg, this is a rather general
situation.)

TODO: should actually be able to work with field L of meromorphic functions on
a small enough disc.

Points f \in A(L_k) can be interpreted as sections.

G:=E(A)

We have relative exponential maps, yielding a commutative diagram of complex
analytic groups over a disc around 0:

 H (-> LG --->> LA
 =     |        |
 =     | exp_G  | exp_A
 =     v        v
 H (-> G --->>  A

(Note that we can't expect exp_G to be surjective when we restrict to
L-points - e.g. when G is over \C, we have an analytic isomorphism G ~= G_m^2g
with corresponding exponential map; so e.g. (z,1,...,1) is an L-point without
a logarithm in L.)

G has its canonical D-group structure, which induces one on LG. This agrees
(see [BP Appendix]) with the Gauss-Manin connection. Taking horizontal
L-points and defining LA^# as the image of LG^\d, we have:

H^\d (-> LG^\d -->> LA^#
=         |         |
=         |         | exp_{A^#} := exp_A\rstr^#
=         v         v
=         v         v
H^\d (-> G^\d  -->> A^#

TODO: proof that LG^\d -->> G^\d. One approach is to use that we have local
analytic splittings of D-groups, see below. I suspect an alternative proof can
be given along the following lines:
    (i) any point of G^\d can be lifted to LG^\d after shrinking;
    (ii) but then by the description of LG^\d below,
	the map must be surjective over any domain for which
	the periods are meromorphic.

We may view H, LG, and LA as L-vector spaces. The kernel of exp_G is a local
system of rank 2g, which we may view as a rank 2g free abelian subgroup of LG,
and LG^\d is the \C-subspace of LG generated by it.
g <= dim_\C LA^# <= 2g
[TODO: Hodgeless proof of these statements].

Claim: A^#(L) = A^#(L^diff)
Proof:
    First, note that A^#(L) = A^#(L^alg).
    
    Indeed, L^alg is the direct limit of the fields L_k of germs at 0 of
    meromorphic functions of the branched covers z|->z^k of C
    (so \delta z^q = q z^{q-1})
    (ref Khovanskii).

    But as above, LG^\d(L_k) is the \C-subspace generated by ker, so
    LG^\d(L_k)=LG^\d(L). Hence A^#(L_k)=A^#(L).

    Now A^#(L^diff) = A^#(L^alg). Indeed, A^# is the smallest type-definable
    subgroup containing Tor(A) [TODO: ref BP or Marker], and so any
    model-theoretically algebraically closed subset is an elementary submodel
    [Wagner FieldsFRM].

TODO: I think we saw that Seidenberg gives a direct proof here?
    Like this, I assume: by (the proof of) Seidenberg, we can find a domain
    over which we have the full 2g dimensional \C-VS LG^\d, i.e. an L such
    that LG^\d(L) = LG^\d(L^diff); then as above, A^#(L)=A^#(L^diff).

Hammification:
    By Hamm ([BuiumDiffAlgDiophGeom p.143]), restricting to a disc S', G is
    analytically isomorphic to a product G_0 x S' such that the D-structure on
    G is the obvious one on G_0 x S'.

    Then LG ~= LG_0 x S', again respecting the D-structure, and the
    exponential, and everything is very clear, in particular the surjectivity
    above and the periods being horizontal.
    
    (Note, crucially, that in general H will *not* be horizontal, i.e. there
    won't be an H_0 <= LG_0 such that H ~= H_0 x S' <= LG, so we don't get a
    corresponding representation of LA (or A).)

    Explicitly: let L be the differential field of meromorphic functions on
    this disc. Then LG^\d(L) ~= LG_0(C). Also LG^\d(L^diff) = LG^\d(L);
    indeed, let X be a \C-basis of LG^\d, so for any disc S'' (= S', we have
    LG^\d(S'') = <X>_\C. So by [MarkerDCF Lemma A.1], for any
    y \in LG^\d(L^diff),
    y \in <X>_{\C(L^diff)} = <X>_\C = LG^\d(L).

Clearly, L(A^n)^# = (LA^#)^n.

Since ker exp_G (= LG^\d,
    (*) ker exp_{A^#} = ker exp_A

If B is a connected algebraic subgroup of A, by [BBP 4.9] we have
    (**) A^# \cap B = B^# .
Now LB^# (= LA^#, and combining (*) with (**), we have
    (***) LA^# \cap LB = LB^# .

Now if H is a connected definable subgroup of A^#, then H = B^# where B is the
Zariski closure of H.
Indeed, by (**), H <= A^# \cap B = B^#,
and meanwhile B^# <= H since B^# is the smallest Zariski-dense definable
subgroup of B.

So we make LA^# into a structure in the language of ^T by interpreting ^(B^#)
as LB^#.

Proposition: LA^# |= ^T

Proof:
    (A1)-(A6) are clear from the setup and (***).

    (A7) and (A8) follow from (*).

    (A9)(I): by (***), the exact sequence
	    0 --> L(K^o) --> LG --> LH --> 0
	remains exact on applying #.

    (A9)(II) is by (*).