08 := {P ^ p(P) :=
| * G #°(X, L ^
) } .
The data (Xs), (0S) define a (5—ringed set X$ with underlying set X(Rp).
Let now X
= X = (X, a) (X, X, 7l5 cr2) be a correspondence in the category
of schemes over Rp. Assume X and X are smooth of finite type over Rp and have
irreducible geometric fibers. Then by the discussion above one can attach to the
schemes X and X two S—ringed sets X$ and Xs- Assume moreover that o\ and
&2 are etale. Then the morphisms G\ , a2 : X -^ X induce morphisms between the
corresponding 5—ringed sets. So we end up with a correspondence X«5 = (X$, o~s) in
the category C5. One can easily describe the categorical quotient Xs/crs in C&. The
underlying set of the categorical quotient Xs/as is the set of equivalence classes of
points in X(Rp) with respect to the smallest equivalence relation containing the
image of the map
ax x a2 : X{Rp) - X{Rp) x X(Rp).
Let L be the anticanonical bundle on X. Let us say that a 5—section s G H°(X, Lw)
is a—invariant if its pull-backs to H°(X, Lw) via cr^ and a^ coincide; a— invariant
S—sections will simply be called S— invariants. Then the monoid parameterizing
the structure rings of Xsjos is the monoid of all 5—invariants s G H°(X, Lw) which
are not divisible by p. For s of degree WQ the corresponding structure ring of Xs/crs
{[P] ^ p(P) :=
I t G H°(X,
a J-invariant}.
So morally Xs/cra is non-trivial if one can find at least two Rp—linearly independent
S—invariants in one of the spaces H°(X,
where 0 ^ w G W+. This agrees, of
course, with the ideology of classical invariant theory.
0.2.4. Conjectural picture. At this point we consider a global situation.
Let 5 be a finite set of finite places of the number field F, containing all the places
ramified over Q, assume (for simplicity) that S consists of all the places containing
some integer m G Z , and let O := CV[l/ra] be the ring of S—integers of F. Consider
a correspondence
X = X0 = (X,a),
a = (X,ai,(72), in the category Co of schemes over O. We will assume X and
X are of finite type and smooth over 0 , have irreducible geometric fibers of di-
mension one (so they are "curves" over 0) , and o\,oi are etale. We have induced
= (Xp,crp), X
= ( I c ^ c )
in the category of schemes over Rp and C respectively. We will always assume
that the smallest equivalence relation in X c x Xc containing the image of Xc
X c x X c is Zariski dense in X c x X c so that the categorical quotient X c / ^ c is
trivial in the category of schemes over C. As we already saw, for all p 0 5, we may
consider a correspondence
X-s = (Xs,as)
in the category C$. Our general guiding conjecture will (roughly speaking) assert
The categorical quotient Xs/crs is non-trivial for almost all places p if and
(essentially) only if the correspondence X c admits an "analytic uniformization".
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