Chapter 10 Comparing Galois representations: the Faltings–Serre method
Original lecture date: October 30, 2019.
In the last lecture, we formulated Proposition 9.4.5, which asserts that two continuous \(\ell\)-adic representations of \(G_{K,S}\) (for \(K\) a number field and \(S\) a finite set of primes) are equal up to semisimplification if and only if their Frobenius traces coincide for all but finitely many primes. In this lecture, we discuss the Faltings–Serre method which makes it psosible to verify this sort of equality by a finite computation. This turns out to have numerous applications, which do not depend on computing with étale cohomology in any direct way.
Section 10.1 Example: modularity of elliptic curves
Before proceeding, let us discuss an example where such a comparison of representations comes up naturally: the modularity of elliptic curves, which had been conjectured in various forms by Taniyama, Shimura, and Weil.
Theorem 10.1.1. Modularity of rational elliptic curves.
Let \(E\) be an elliptic curve over \(\QQ\text{.}\) Then there exists a modular form \(f\) (more precisely, a cuspidal weight \(2\) newform for the group \(\Gamma_0(N)\) where \(N\) is the conductor of \(E\)) such that for every prime \(p\) at which \(E\) has good reduction (i.e., every prime not dividing \(N\)), the trace of Frobenius on \(E/\FF_p\) equals the Fourier coefficient \(a_p(f)\text{.}\)
Proof.
In the case where \(E\) is semistable (i.e., the conductor \(N\) is squarefree), this is part of the work of Wiles [133] and Taylor–Wiles [123] that completed the proof of Fermat’s last theorem. The general case was resolved (based on the aforementioned papers and several intermediate results) by Breuil–Conrad–Diamond–Taylor [13].
Remark 10.1.2.
Before Theorem 10.1.1 was proved in the 1990s, it had been verified in numerous examples, most conclusively by Cremona [27]. This relied on the older Eichler–Shimura theorem, which implies that for any \(f\) as in the theorem, there exists an elliptic curve \(E_f\) for which the desired conclusion holds.
Now suppose in that context, one had in mind a particular elliptic curve \(E\) for which one wanted to confirm the statement of the theorem. Since the conjecture includes a prediction for the level of the newform in terms of \(E\text{,}\) and the newforms for a given level form a computable (via Manin’s method of modular symbols) finite-dimensional vector space, one could then do the finite computation to find all candidates for \(f\text{,}\) and quickly isolate a unique candidate with the first few Fourier coefficients correct.
However, this would not suffice to prove that \(E\) and \(E_f\) are isogenous: applying Corollary 9.4.6 would require an infinite number of equalities, which cannot a priori be established via a finite computation. Instead, one applies some form of the Faltings–Serre method as described below to conclude.
Remark 10.1.3.
It should be emphasized that the Faltings–Serre method cannot be used to prove that an “abstract” infinite list of numbers matches the list of Frobenius traces of a given Galois representation; one must know that the first list itself comes from a Galois representation with some control on the ramified primes. Besides the control on ramification, we need no other information other than the ability to compute entries of both lists on demand.
For example, in the case of elliptic curves, it would not have been possible to rigorously establish modularity of a given elliptic curve \(E\) without knowledge of the existence of the other elliptic curve \(E_f\) produced from \(f\) via Eichler–Shimura. However, it is not necessary to know anything more about \(E_f\) beyond its existence, its ramified primes (or even just an upper bound on this set), and its Frobenius traces.
Section 10.2 The method
Remark 10.2.1.
Let us now break down how to reduce a comparison of infinitely many Frobenius traces, as in Proposition 9.4.5, to a finite computation. This splits into two main steps.
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This is the hard step. We know that these representations can be factored through \(\GL_n(\ZZ_\ell)\) (they are representations of compact groups into Hausdorff targets, so they have closed image), so we would like to see these two representations with images in \(\GL_n(\FF_\ell)\) have the same semisimplification (and in particular, the Frobenius traces are pairwise congruent modulo \(\ell\)). To check that, we try to identify the two kernels and comparing them; this involves enumerating number fields with prescribed ramification.
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Use a group-theoretic argument to promote the mod-\(\ell\) equality to an \(\ell\)-adic equality. This turns out to be much easier since it doesn’t depend on the set \(S\text{;}\) it is a matter of pure group theory.
The difficulty of both steps scales badly with \(\ell\text{,}\) so generally one takes \(\ell = 2\) in practice. (Fortunately, there is no requirement that the representations in question have good reduction at primes above 2.)
To settle the first step, we use the following basic theorem of algebraic number theory.
Theorem 10.2.2. Hermite–Minkowski.
Given a number field \(K\text{,}\) an integer \(d\in \ZZ^+\text{,}\) and a finite set of finite primes \(S\) of \(\calO_K\text{,}\) there are only finitely many isomorphism classes of number fields \(L/K\) which are unramified away from \(S\) and satisfy \([L:K]=d\text{.}\) Moreover, this list is effectively computable.
Proof.
One may reduce to the case \(K = \QQ\text{.}\) In this case, one can bound the contribution of each prime in \(S\) to the discriminant to get a bound on the absolute discriminant of \(L\text{,}\) and then use a geometry of numbers argument (Minkowski’s theorem) to limit the possibilities for \(L\) to a finite set.
In particular, there are finitely many homomorphisms \(G_{K,S} \to \GL_n(\FF_\ell)\) and (in principle!) one can compute them all, and then group them according to semisimplification. By Chebotarëv density, it takes only finitely many Frobenius traces to rule out all but the right candidate.
To achieve the second step (for \(\ell=2, n=2\)), we use the following theorem. See also [16], section 2.3, for a generalization valid for arbitrary \(\ell, n\) but under the additional assumption that the residual representations are absolutely irreducible. (The use of this hypothesis here is closely related to its use in the theory of Galois deformation rings.)
Definition 10.2.3.
A subset \(T\) of a vector space \(V\) over a field (here \(\FF_2\)) is noncubic if every homogeneous polynomial of degree \(3\) which vanishes on \(T\) also vanishes on all of \(V\text{.}\)
Theorem 10.2.4. Livné, after Serre.
Suppose that \(\ell=2, n=2\text{.}\) With notation as in Proposition 9.4.5, if the Frobenius traces of \(\rho_1\) and \(\rho_2\) agree modulo \(2\) for all \(h\in G_K\) (as evidenced by the first step) and agree “on the nose” for some finite set \(T\subseteq G_K\) whose image in \(G_{K,S}^{\ab}/2G_{K,S}^{\ab}\) is noncubic, then the traces agree for all \(h\in G_K\text{.}\)
Remark 10.2.5.
Note that if \(G_{K,S}^{\ab}/2G_{K,S}^{\ab}\) has dimension \(\leq 2\) over \(\FF_2\text{,}\) then it is impossible for \(T\) to be noncubic. This is easily fixed by adding extra primes to \(S\text{.}\)
Remark 10.2.6.
The main practical difficulty in applying this method in practice is to make the enumeration of homomorphisms \(G_{K,S} \to \GL_n(\FF_\ell)\) efficient. There is a lot of work to be done in this direction; see for example [16].
Section 10.3 Other uses of the method
Remark 10.3.1.
To conclude this lecture, we justify our claim that the Faltings–Serre method is useful in practice by citing a few additional examples of its use in the literature.
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There is a conjectural analogue of the modularity theorem for elliptic curves over imaginary quadratic fields, in which the role of classical modular forms is played by Bianchi forms. However, one must consider not only elliptic curves, but also abelian surfaces with quaternionic multiplication (QM); instances of modularity in this context have been exhibited by Dieulefait–Guerberoff–Pacetti [35] for elliptic curves and Schembri [109] for QM abelian surfaces.
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Inspired by some examples arising in mirror symmetry in mathematical physics, numerous authors have identified examples of Calabi–Yau threefolds whose Frobenius traces can be computed in terms of modular forms. See [134] for a survey.
