The Grothendieck Conjecture on the Fundamental Groups of ...

4 HIROAKI NAKAMURA, AKIO TAMAGAWA, SHINICHI MOCHIZUKI
(GC2) <strong>The</strong> “Hom <strong>Conjecture</strong>.” For hyperbolic algebraic curves X, Y over a field K which
is finitely generated over the rationals, the natural map
Hom K (X, Y ) → Hom Gal(K) (π 1 (X),π 1 (Y ))/ ∼
defines a bijective correspondence between dominant K-morphisms and equivalence classes of
Gal(K)-compatible open homomorphisms (modulo composition with an inner automorphism
induced by an element of π 1 (Y K
)). (In other words, open homomorphisms of the fundamental
group always arise from algebro-geometric morphisms.)
As <strong>Grothendieck</strong> himself observes, the above conjecture bears some resemblance to the Tate
<strong>Conjecture</strong> (proved by G. Faltings [F1]) concerning the 1-dimensional étale homology groups
of abelian varieties:
Hom K (A, B) ⊗ Ẑ ∼ = Hom Gal(K) (H 1 (A K
, Ẑ),H 1(B K
, Ẑ))
(Here, A and B are abelian varieties defined over a global field K, and Ẑ is the profinite
completion of Z.) Moreover, if one applies the Tate <strong>Conjecture</strong> together with the “isogeny
theorem” (as well as the Shafarevich <strong>Conjecture</strong>, etc., which were proven by Faltings along with
the Tate <strong>Conjecture</strong>) to the Jacobian variety of the curves in question, it follows immediately
that there are only finitely many curves with homology group H 1 isomorphic (as a Galois
module) to the H 1 of a given proper algebraic curve of genus ≥ 2. If one observes that H 1
is just the abelianization of π 1 , then one may regard the Fundamental <strong>Conjecture</strong> (GC1) as
the assertion that, if one increases the data that one is given from just the homology group
to the entire fundamental group, then the number of possibilities for a curve possessing the
same invariant (i.e., the same π 1 ) is narrowed down from some unknown finite number to “just
one.” In fact, even effective versions of this sort of finiteness theorem (i.e., the Shafarevich
conjecture, etc.) tend (with few exceptions 5) ) to give only inordinately large estimates for
the number of such possibilities. Thus, from this point of view, there is quite a substantial
gap between <strong>Grothendieck</strong>’s conjectures (GC1), (GC2) and the Tate <strong>Conjecture</strong> applied to the
Jacobian varieties of the curves in question. <strong>Grothendieck</strong> argued, in support of his conjecture,
that the arithmetic fundamental group π 1 (X) possesses an “extraordinary rigidity,” i.e., that
the outer action (1.2) of its “arithmetic quotient” Gal(K) on its “geometric portion” π 1 (X K
)
should be “extraordinarily rigid,” citing by way of comparison the nontriviality of the Galois
representations arising from cohomology theory which were studied by A. Weil and P. Deligne
([G3]).
Finally, among (unsolved) conjectures which may be rigorously formulated, one interesting
conjecture is the following “Section <strong>Conjecture</strong>.” A K-rational point x ∈ X(K) of an algebraic
variety X over K may be regarded as a section x :Spec(K) → X of the structure morphism
X → Spec (K). Thus, a K-rational point x induces a (π 1 (X K
)-conjugacy class of) section
homomorphism(s) α x : Gal(K) → π 1 (X) which splits the fundamental exact sequence (1.1)
discussed above.
(GC3) <strong>The</strong> Section <strong>Conjecture</strong>. For an X/K as in (GC2), every section homomorphism
α :Gal(K) → π 1 (X) of the projection pr X : π 1 (X) → Gal(K) arises either from a K-rational
point of X (in the usual sense), or from the K-rational points “at infinity” 6) of X.