p-adic Representations of the K-rational
Geometric Fundamental Group
1. Introduction
Let K be a number field (or any fin. gen. field)
C/K a (smooth. . .) curve of genus g
F = κ(C) its function field (⇒ F/K regular)
P ∈ C(K) a K-rational point
Definition Let Fnr,P be the field generated by thefinite unramified Galois extensions F 0/F such thatP splits completely in F 0. Then its Galois group
π1(C, P ) = Gal(Fnr,P /F )F
is called the K-rational
geometric fundamentalFnr group of C with basepoint P .
Fnr,P K  π1(CK̄ , P )GK 
→Y. IharaF K(Fnr,Pπ1(C, P )

F






K

GKK

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2. Some Results about π1(C, P )–joint work with G. Frey and H. Völklein
Note: g = 0 ⇒ π1(C, P ) = π1(CK̄ , P ) = {1}.
Theorem 1 (Merel) There is cK such that for all elliptic curves E/K and P ∈ E(K) we have|π1(E, P )| ≤ cK .
Mazur: cQ = 12.
Proposition 1: π1(C, P )ab is always finite.
Theorem 2: Let K ⊃ Q(i) (or K ⊃ Fp(i)). Thenfor every g ≥ 3 there exist (many!) curves C/K ofgenus g with a point P ∈ C(K) such that π1(C, P )is infinite.
Remark: The above situation forπ1(C, P )is very similar to that of the fundamental group π1(K) of anumber field K:
π1(K) = {1} for some K’s (K = Q, Q(i), etc.)|π1(K)ab| = h(K) is always finite.
π1(K) is√often infinite (→ Class field towers: e.g.
K = Q( −30030).)

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3. p-adic Representations
So far, the theory for π1(C, P ) and for π1(K) seemto be very similar. (→ M. Rosen (Hilbert class fields).)
However, this picture changes if we look at p-adic representations, particularly in view of the Fontaine-Mazur
Conjecture:
Fontaine-Mazur Conjecture (1993): Any p-adic
representationρ : π1(K) → GLn(Qp)factors through a finite quotient group.
In particular:
Any quotient group of π1(K)(p), which is a p-adic
analytic group, is finite.
Remark: The above conjecture is actually only a special case of a more general conjecture (also due toFontaine and Mazur):
The Main F-M Conjecture: Every irreducible padic representation on GK which is potentially semistable (at all v|p) comes from algebraic geometry,i.e. is isomorphic to a subquotient of an étale cohomology group H q (XK , Qp(r)), for some projective
smooth variety X/K.

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The analogues of these conjectures for π1(C, P ) arefalse, as the following theorem and its corollary show1:
Theorem 20: Let b ∈ K ×, b4 6= ±1, and put c = 1+b4√22band a = c . (As before, −1 ∈ K). Let C/K bethe curve defined by the equation
s4 = ct(t2 − 1)(t − a)g(t),where g(t) ∈ K[t] is any polynomial withg(a) = 1 and g(0)g(1)g(−1) 6= 0,and put P = (a, 0) ∈ C(K). Then the K-rational
geometric fundamental group π1(C, P ) is infinite;
more precisely, for every prime p ≡ 5 (mod 12)(with p 6= char(K)), the group PSL3(Zp) is a factorof π1(C, P ), i.e. there is a surjectionρ : π1(C, P ) → PSL3(Zp).
Corollary. In the above situation, let Cp denote thefinite cover of C corresponding to a pro-p-Sylow subgroup Up of PSL3(Zp). Then for any point P 0 overP , the fundametal group π1(Cp, P 0) has a quotientwhich is isomorphic to the p-adic analytic group Up.
This also shows that J. Holden’s generalization of the Fontaine-Mazur Conjecture tocurves over finite fields is false as well.
1

54. The Basic Construction: Motivation
Basic Idea: Construct unramified extensions of F via(towers of) torsion points of abelian varieties A/F =κ(C), i.e. look at the p-adic Galois representation
ρA,p : GF = Gal(F /F ) → GL(Tp(A)) ' GL2g (Zp).
Remark: In the language of Fontaine-Mazur this meansthat we are looking at p-adic representations that aresubquotients of H 1(XF , Qp(1)), where X is somecurve (or abelian variety) over F .
Want: A to have good reduction everywhere over C.
Criterion of Neron-Ogg-Shafarevich:
A/F has good reduction everywhere⇔ F (A[m]) is unramified over F, ∀m ≥ 1S⇔ F (Tp(A)) = F (A[pn]) is unramified over F, ∀p.
Assume this from now on.

Unfortunately: F (A[m]) 6⊂ Fnr,P for m >> 0, soin particular F (Tp(A)) 6⊂ Fnr,P , fo all p.
For: ζm ∈ F (A[m]), ∀m but ζm ∈/ K (hence ζm ∈/Fnr,P ), for m >> 0.

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1st Modification: In place of ρA,p, consider instead
its associated projective representation:ρ̃A,p : GF → PGL(Tp(A)) = Aut(P(Tp(A))),i.e. consider the subfield
F (P(Tp(A))) = F (Tp(A))Z(GL(Tp(A))),of F (Tp(A)) which is fixed by the centre Z of thegroup GL(Tp(A)).
Then we have: F (P(Tp(A))) ⊂ Fnr,Pdef⇔ P ∈ C(K) splits completely in F (P(Tp(A)))⇔ GK operates centrally (diagonally) on Tp(AP ),Tate⇔ EndK (AP ) ⊗ Qp = M2g (Qp),where AP denotes the reduction of A at P .
Note: Here we have used the Tate Conjecture for endomorphisms of abelian varieties (which was provedby G. Faltings).

However: The theory of abelian varieties shows that
this is impossible (in characteristic 0); i.e. there is
no abelian variety of dimension g ≥ 1 whose endomorphism ring is a full 2g × 2g matrix algebra.

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2nd Modification: Look for Zp[GF ]-decompositions:rM(1)
Tp(A) =Si ,i=1

and let S i = image of Si in Tp(AP ).
Then: F (P(Si)) ⊂ Fnr,P , for all i⇔ GK operates centrally on each S iTate⇒ AP is of CM-Type.

Remark: If we assume the existence of a decomposition (1) and require the CM-type of AP to be compatible with the S i, then the converse to the last
implication is also true.
Proposition: Let A/F be an abelian variety withgood reduction everywhere. If p is a prime suchthat we have a decomposition (1) such that GK acts
centrally on each S i ⊂ Tp(AP ), then each projective
p-adic subrepresentation
ρ̃Si : GF → PGL(Si) = Aut(P(Si))
of ρ̃A,p factors over π1(C, P ), i.e. induces a homomorphismρ̃Si : π1(C, P ) → PGL(Si) = Aut(P(Si)).

85. The Basic Construction: Some Details
Aim: For F = K(t, s) and P as in Theorem 20, construct an abelian variety A/F satisfying the hypotheses of the previous proposition.
Consider: the cyclic covering φ : X → P1F definedby the equationy 4 = x(x2 − 1)(x − a)3(x − t)2.
Then: 0) X has genus 4, ∃σ ∈ Aut(X) of order 4, andφ factors over the elliptic curve E = X/hσ 2i.
1) The Jacobian JX ∼ E × A, where A = J new isan abelian subvariety of JX of dimension 3.
2) σ acts on A and hence on Tp(A), and if p ≡1 (mod 4), then we have the GF -decomposition into
σ-eigenspacesTp(A) = S1 ⊕ S2, where dim Si = 3.
3) A/F has good reduction everywhere.
4) ρ̃Si : GF → PGL3(Zp) is surjective if p ≡ 5 (12).
5) AP ∼ E1 × E1 × E1, where E1/K is an elliptic
curve with CM by Q(i), so ρ̃Si factors over π1(C, P ).

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Proof Sketch: 0) - 2) Easy.
3) Note first that X, φ, A etc. are defined over F0 :=K(t) ⊂ F . By Völklein’s theory of Thompson tuples, the ramification structure of F0(P(Si[p]))/F0
can be described precisely (for all p ≡ 1 (4)), and so
it follows from the Serre–Tate criterion that A has
potentially good reduction. By analyzing the Neronmodel of JX more closely, it follows that A alreadyhas good reduction over F .
4) Völklein’s theory of Thompson tuples shows that
Gal(F (P(Si[p]))/F ) ' PGL3(p). By an argument
due to Serre, it follows that ρ̃Si is surjective.
5) Here we work out the structure of the fibre CP
at P of the minimal model of C in some detail. Itis here that the judicious choice of c and a becomeimportant.
Remark: Most of the above program (i.e. steps 0)4)) can be generalized to (almost arbitrary) cycliccoverings φ : X → P1K(t). In this case one works
new of the Jacobianwith what we call the new part JX
JX of X, i.e. the part of JX that is orthogonal tothe Jacobians of proper subcovers of φ.

10Proof Sketch of 3):
I. Völklein’s theory of Thompson tuples + choice of F⇒ 1) F (P(Si))/F is unramified
2) eP (F (A[p])/F ) ≤ N r , for p ≡ 5(12) andsome r (indep. of p)
II. The Serre-Tate Criterion:
A/F has potentially good reduction at P
⇔ ∃c : eP (F (A[m])/F ) ≤ c, for all m⇔ ∃c : eP (F (A[p])/F ) ≤ c, for ∞’ly many pr.’s p.
III. A Good reduction criterion:
Show: F (A[p])/F is unramified (Ne-O-Sh).
Enough: F (Si[p])/F (P(Si[p])) is unramified (by I.)
Criterion: SiI 6= {0} ⇒ F (Si[p])/F (P(Si[p])) is unramified. (I = inertia group.)
oI
Recall (Grothendieck, SGA 7I ): Tp(A) ' Tp(AP ),owhere AP = connected component of the identity of

the reduction of the Neron model at P .
oI(Thus: Tp(A) = {0} ⇔ AP is unipotent, i.e. anextension of additive groups.)
Note: The above criterion applies to A = J new foroAP has a large abelian part which meets each S i.

11Reference: G. Frey, E. Kani, H. Völklein, Curves withinfinite K-rational geometric fundamental group. In:
Aspects of Galois Theory (H. Völklein et al., eds.),
LMS Lecture Notes 256 (1999), 85–118.