On a conjecture of Fontaine and Mazur

Richard Taylor ^{1} Department of Mathematics,

Harvard University, Cambridge, MA 02138,

U.S.A.

June 17, 2001

1Partially supported by NSF Grant DMS-9702885

In this talk we discuss the following theorem, which provides some evidence for a very influential conjecture of Fontaine and Mazur (which for many years circulated only as the abstract of Fontaine’s talk at the 1988 Arbeitstagung - MPIM preprints 1988 number 30).

Theorem A Let l > 3 be a prime and fix i : ℚ_{l}^{ac} →^{∼} ℂ. Let ρ : G_{ℚ} → GL_{2}(ℚ_{l}^{ac}) be a continuous irreducible representation such that

• ρ is ramifies at only finitely many primes,

• ρ|_{G_{l}} is crystalline with Hodge-Tate numbers n_{1}, n_{2} where 1 ≤ |n_{1} − n_{2}| ≤ (l − 1)/2,

• det ρ(c) = −1.

Then the following assertions hold.

1. There is a Galois totally real field F/ℚ, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ of the field of rationality of π into ℚ_{l}^{ac} such that ρ|_{G_{F}} ∼ ρ_{π,λ}, the usual l-adic representation associated to π (and λ).

2. If ρ is unramified at a prime p and if α is an eigenvalue of ρ(Frob_{p}) then α ∈ ℚ^{ac} and for any embedding ℚ^{ac} ↪ ℂ we have

|α|^{2} = p^{n_{1}+n_{2}}.

3. There is a rational function L_{l}(X) ∈ ℂ(X) such that the product

L(iρ, s) = L_{l}(l^{−s})^{−1} ∏_{p≠l} i det(1 − ρ^{I_{p}}(Frob_{p})p^{−s})^{−1}

converges in Re s > 1 + (n_{1} + n_{2})/2 and extends to a meromorphic function on the entire complex plane which satisfies a functional equation

(2π)^{n_{2}−s}Γ(s − n_{2})L(iρ, s) = ϵ(iρ, s)(2π)^{s−n_{1}−1}Γ(1 + n_{1} − s)L(iρ^{∨}, 1 − s),

where ϵ(iρ, s) denotes the usual product of local ϵ-factors and so has the form N^{s}W for some constants N and W.

4. If k = 2 further assume that for some prime p ≠ l we have

ρ|_{G_{p}} ∼ ( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} ) .

Then ρ occurs in the l-adic cohomology (with coefficients in some Tate twist of the constant sheaf) of some smooth projective variety over ℚ.

As a consequence we see that the L-function of any regular (i.e. distinct Hodge numbers), rank two motive over ℚ has meromorphic continuation to the entire complex plane and satisfies the expected functional equation. This applies for instance to the L-function of H^{1} of any abelian variety of GL_{2}-type, and to the zeta function of any rigid (i.e. h^{2,1} = 0) Calabi-Yau 3-fold.

Because it is not entirely clear what is meant by motive we can make this statement precise as follows. By a regular, rank 2 weakly compatible system of l-adic representations R over ℚ we shall mean a 5-tuple

(M, S, {Q_{p}(X)}, {ρ_{λ}}, {n_{1}, n_{2}})

where • M is a number field;

• S is a finite set of rational primes;

• for each prime p ∉ S of ℚ, Q_{p}(X) is a monic degree 2 polynomial in M[X];

• for each prime λ of M (with residue characteristic l say)

ρ_{λ} : G_{ℚ} ⟶ GL_{2}(M_{λ})

is a continuous representation such that det ρ_{λ}(c) = −1 and such that for each rational prime p ∉ S ∪ {l}, ρ_{λ} is unramified at p and ρ_{λ}(Frob_{p}) has characteristic polynomial Q_{p}(X); and

• n_{1}, n_{2} are distinct integers such that for all primes λ of M lying above a rational prime l ∉ S the representation ρ_{λ}|_{G_{l}} is crystalline with Hodge-Tate numbers n_{1} and n_{2}.

(We remark that if R arises from l-adic realisations of a regular motive one could use the Hodge realisation to check that det ρ_{λ}(c) = −1 for all λ. Thus we consider this oddness condition part of regularity.) It is easy to see that either all the ρ_{λ} are absolutely reducible or all but finitely many are absolutely irreducible. In the former (resp. latter) case we call R reducible (resp. irreducible).

We will call R strongly compatible if for each rational prime p there is a representation WD_{p}(R) of the Weil-Deligne group of ℚ_{p} such that for primes λ of M not dividing p, WD_{p}(R) is equivalent to the Frobenius semi-simplification of the representation of the Weil-Deligne group associated to ρ_{λ}|_{G_{p}}. If R is strongly compatible and if i : M ↪ ℂ then we define an L-function L(iR, s) as the infinite product

L(iR, s) = ∏_{p} L_{p}(iWD_{p}(R), s)

which may or may not converge. Fix an additive character Ψ = ∏ Ψ_{p} of 𝔸/ℚ with Ψ_{∞}(x) = e^{2πix}, and a Haar measure dx = ∏ dx_{p} on 𝔸 with dx_{∞} the usual measure on ℝ and with dx(𝔸/ℚ) = 1. Then we can also also define an ϵ-factor ϵ(iR, s) by the formula

ϵ(iR, s) = i^{1+n_{1}−n_{2}} ∏_{p} ϵ(iWD_{p}(RS), Ψ_{p}, dx_{p}, s).

Theorem B Suppose that R = (M, S, {Q_{x}(X)}, {ρ_{λ}}, {n_{1}, n_{2}})/ℚ is a regular, irreducible, rank 2 weakly compatible system of l-adic representations with n_{1} > n_{2} and suppose i : M ↪ ℂ. Then the following assertions hold.

1. For all rational primes p ∉ S and for all i : M ↪ ℂ the roots of i(Q_{p}(X)) have absolute value p^{(n_{1}+n_{2})/2}.

2. R is strongly compatible.

3. The L-function L(iR, s) converges to a meromorphic function in Re s > 1+(n_{1}+n_{2})/2, has meromorphic continuation to the entire complex plane, and satisfies a functional equation

(2π)^{n_{2}−s}Γ(s − n_{2})L(iR, s) = ϵ(iR, s)(2π)^{s−n_{1}−1}Γ(1 + n_{1} − s)L(iR^{∨}, 1 − s).