15 Elliptic curves and Fermat s last theorem

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1 15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine equation a q + b q = c q. We may assume that a, b, c are relatively prime. Let A = a q, B = b q and C = c q. Consider the homogeneous cubic equation E = (y 2 z = x(x A)(x + B)). (1) Solutions to the equation (1) is called the Frey curve. In fact, we are not going to use explicitly A = a p, B = b p and C = c p, but just that the integers A, B, C satisfy the assumption of Lemma 15.2 and 15.4 below. The Frey curve is considered as in a twodimensional projective space P 2. That is, for any field, say C, P 2 (C) is the set of all not-allzero triples of complex numbers modulo C -scaling, and E(C) is set of the complex solutions of (1) modulo C -scaling. One has that E(C) is smooth; it is naturally endowed the structure of a 1-dimensional compact complex manifold (a.k.a. compact Riemann surface). Using merely that E(C) has genus 1 (as an orientable closed surface), it is possible to prove that E(C) is biholomorphic to C/Λ where Λ C is a (rank 2) lattice. Up to translation by C/Λ we may and do assume that (0, 1, 0) E(C) corresponds to C/Λ. The additive group law on C/Λ then imposes a commutative group law on E(C). This group law has the following algebraic description: o := (0, 1, 0) is the identity, and x + y + z = o in E(C) iff there exists a line (given by a linear equation, like x + 2y + 3z = 0) on P 2 (C) such that the line intersects 1 at x, y and z. It also has the analytic description that x + y + z = 0 iff there exists a meromorphic function on E(C) whose divisor (i.e. zeroes and poles) is (x) + (y) + (z) 3(o). With the group law, one may consider for any positive integer m the subgroup E[m] := {x E(C) mx = 0}. Here mx is m of x s added together. The analytic description E(C) = C/Λ implies E[m] = 1 Λ/Λ m = (Z/mZ) 2. On the other hand, by the algebraic description, we have E[m] E( Q). The algebraic description of the group law furthermore implies that G Q acts on E[m] by automorphisms (i.e. preserving E[m] and respecting the group law on it). Now fix a prime l. For any positive integer n, we have a natural map E[l n+1 ] E[l n ] by x lx. As abstract groups we have E[l n ] = (Z/l n Z) 2 and the map above is the 1 When the line is tangent to E(C) P 2 (C) at x, we should count x twice, e.g. y = x. The exception happens when the line only meets E(C) has one point x. In this case it is tangent to E(C) to a higher order and we put y = z = x. 1

2 natural projection (Z/l n+1 Z) 2 (Z/l n Z) 2. Equip each E[l n ] with the discrete topology and consider the inverse limit T l E := lim E[l n ] called the l-adic Tate module. As above, we see that as an topological group we have T l E = lim (Z/l n Z) 2 = Z 2 l. By construction, we have G Q acting on T l E by continuous homomorphisms, which are necessarily Z l -linear. Up to isomorphism, this action gives a Galois representation ρ E,l : G Q GL 2 (Z l ). Let Z l (1) be the topological group Z l equipped with the G Q action by the cyclotomic character. Then it is a general fact (for any elliptic curve) that there is a symplectic G Q - equivariant pairing map T l E T l E Z l (1), call the Tate pairing. In particular this shows that det ρ E,l is the cyclotomic character. Now arithmetic geometry gives us: Lemma The Galois representation ρ E,l is irreducible, and unramified at p iff p ABC (we exclude the case p = l in this statement). For those p, the characteristic polynomial of ρ E,l (Frob p ) has coefficients in Z Q l and does not depends on the choice of l. The Galois representation ρ E,l is de Rham at l, and the corresponding automorphic representation (if exists) is of the form V ( det) where V is a cuspidal automorphic representation with trivial central character and archimedean component V corresponding to modular forms of weight 2, i.e. there exists v V such that R X v = 0 and k θ v = e 2iθ v. Here det is the character GL 2 (A Q ) det A Q R + C. In fact, the more common convention (when dealing with global Langlands) is to normalize the Langlands correspondence by det so that this extra factor disappears. This however will produce an extra factor for local Langlands (as required by local-global compatibility), which is why we didn t do so in the first place. A priori the cuspidal automorphic representation V might depends on l. However, by the second statement of the lemma we have that if we have V and V given by different l s, then V p = V p for almost every p, which by the multiplicity one theorem gives that V = V, and thus we have the independence of l result. We need another result about ρ E,l from arithmetic geometry. Lemma Suppose we had arranged that A 1(4) and 32 B. Then for those p ABC, the restriction ρ E,l GQp has the property that the inertia acts unipotently, that is, elements in ρ E,l (I Qp ) have the only eigenvalue 1. This implies the same is true for the Weil-representation part in the corresponding Weil- Deligne representation. As the inertia has finite image via the Weil representation, this implies that the Weil representation is in fact trivial on inertia, i.e. unramified. As the Weil-Deligne representation is not unramified, this says the N-part of the Weil-Deligne 2

3 representation is non-trivial. As V p has the trivial central character, this says V p is the Steinberg representation St, which appears in 0 St Ind G B( B ) triv 0 Now let I p GL 2 (Z p ) be the Iwahori subgroup at p, that is, the subgroup of matrices whose reduction modulo p is upper triangular. We claim that the I p -fixed vectors St Ip is 1-dimensional. Suppose this is the case. Write N = rad(abc) := p ABC p. Then we have K 0 (N) = p ABC I p p ABC GL 2 (Z p ). In particular, the space of vectors in V satisfying R X v = 0, k θ v = e 2iθ v, and invariant by K 0 (N) is the tensor product of Vp Ip for p ABC and V GL 2(Z p) p for p ABC, which is then one-dimensional. This 1-dimensional space gives us a particular cusp form, which can be normalized to a Hecke eigenform f = q + n 2 a nq n, since Hecke operators (for those p ABC) necessarily preserve the 1-dimensional space. We have seen that the coefficients a p for p ABC reflects the eigenvalues of the Hecke operator. By our normalization one may check that a p = Tr(ρ E,l (Frob p )), which is an integer by the second statement in Lemma Theorem (Wiles, Taylor-Wiles) The modularity theorem holds for semisstable elliptic curves over Q, in the sense that for any elliptic curve over Q satisfying Lemma 15.2, there exists a normalized Hecke eigenform with the property described above. In fact, the modularity theorem holds for elliptic curves over Q in general (with the same statement that they are captured in the above way by Hecke eigenforms) and there are generalizations to elliptic curves over real and imaginary quadratic extensions of Q. For the global function field case, much more is known in the geometric Langlands program. By Theorem 15.3, we see that from a solution a q + b q = c q of the Fermat last theorem we have a Hecke eigenform of weight 2 and level N = rad(abc). To derive a contradiction, we need to use one more arithmetic geometric result from (1). Let us now suppose thatt l = q. Lemma Suppose p q is odd such that val p (ABC) is divisible by q. Then the reduction ρ E,l of ρ E,l modulo q is unramified at p. When p = q it is instead finite flat 2. Apparently the assumption holds when A = a q, B = b q and C = c q. This unramified modulo q property allows Ribet to prove: 2 One may think of this as an analogue of unramifiedness. We will not go into detail of this. 3

4 Theorem (Ribet) Suppose ρ E,l satisfy the conclusion of Lemma 15.4 (for a particular p) and corresponds via the global Langlands correspondence to a Hecke eigenform f E of weight 2 and level N where N is divisible by p but not p 2. Then there exists another Hecke eigenform f E of weight 2 and level N/p which corresponds to ρ E,l modulo q. That is f corresponds to another elliptic curve E with ρ E,l isomorphic to ρ E,l modulo q = l. As N is square-free, by repeatedly applying Ribet s theorem to remove all odd prime factors of N, we arrive from the Hecke eigenform f E which is of weight 2 and level N, to a Hecke eigenform f which is of weight 2 and level 2. However, direct calculation will show that there is no non-zero cusp form of weight 2 and level 2, and thus there can t be any non-trivial solution to the Fermat equation Serre s Conjecture In the rest of the time, we elaborate a bit more on the idea of taking ρ E,l modulo l, which we will now denote by ρ E,l. Let again A, B, C be non-zero integers with A + B = C and E = (y 2 z = x(x Az)(x + Bz)). While the reduction of ρ E,l makes sense in general, it is in fact given by the Galois action of G Q on E[l]. We also have a stronger irreducibility result: Lemma If l 5, then ρ E,l is irreducible. The proof of this lemma makes use of a deep, celebrated theorem of Mazur on the torsion points of elliptic curves over Q. We now assume l 5 and assume this lemma. Let us note that as det ρ E,l is the cyclotomic character, the same is true for det ρ E,l : G Q F l, i.e. det ρ E,l is given by the action of G Q on Gal(Q(ζ l )/Q). In particular, complex conjugation have determinant 1; we call such 2-dimensional, mod-l Galois representation odd. Let us say a 2-dimensional, mod-l Galois representation is modular if it comes from the reduction modulo l of a Hecke eigenform. Using what we have seen before, this has the following definition without any use of global Langlands correspondence: Definition An 2-dimensional, mod-l Galois representation ρ : G Q GL 2 ( F l ) is modular of weight k if there exists a normalized Hecke eigenform f = q + n=2 a nq n of weight k such that a n Q, and for a place v l of Q we have the reduction a p Tr( ρ(frob p )) for almost every p. The condition that a n Q is in fact automatic from the theory of modular curves (the fact that modular curves are defined over Q). We note when ρ = ρ E,l, the modularity of ρ is compatible with that given in Theorem 15.3, but only after we normalize our global Langlands correspondence so that the det factor in Lemma 15.1 disappear. 4

5 Now Serre gave 3 in 1975 (and refined in 1987) the following conjecture, proved later by Khare and Wintenberger. Theorem Let ρ : G Q GL 2 ( F l ) be an odd irreducible 2-dimensional Galois representation. Then ρ is modular. Serre gave a more precise conjecture which we describe further in the following remarks. Firstly, note that as long as ρ : G Q GL 2 (F l ) is irreducible over F l (or similar for any finite extension of F l ), then the oddness of ρ will ensure that it is irreducible over F l. This is because that as the complex conjugation as determinant 1, it has two eigenvalues 1 and 1 and if ρ is reducible, a subrepresentation must be given by one of the eigenspaces (here we use 1 1 in F l ). Such a subspace will then be defined over the ground field F l (or other finite extension). Secondly, in the Serre s conjecture, the conjectural modular form might not be invariant under some Γ 0 (N), but some possibly smaller subgroups. This subgroup depends on how det( ρ) is ramified at places outside l. If we assume that det( ρ) is the cyclotomic character, then it is in particular unramified at every p l, and in this case the conjectured asserts that one may take the Hecke eigenform to be invariant under some Γ 0 (N), i.e. of level N. Serre also conjectured that the level of the modular form can be taken to be the conductor of ρ away from l (i.e. not considering the l-factor). In particular, if ρ is as in Lemma 15.2 and Lemma 15.4, then the Artin conductor is 2; it has no odd factor as it s unramified at all odd places away from l, and at 2 the tame inertia acts with a 1-dimensional fixed subspace while the wild inertia acts trivially. Thirdly, Serre also conjectured that ρ is modular of a weight k which is determined by the restriction ρ GQl. We will only say that when det ρ is the mod-l cyclotomic character and ρ is finite flat as in Lemma 15.4, Serre s recipe gives k = 2. As there is no cusp form of weight 2 and level 2, Serre s conjecture itself implies the Fermat Last Theorem. Serre s conjecture motivated the proof of the Fermat Last Theorem. In fact, the theorem of Ribet was stated in the way that if ρ satisfies the conclusion of Lemma 15.4 and was modular of weight 2 and level N, then it is modular of weight 2 and level N/p. Nevertheless, Serre s conjectured was proved more than a decade after the work of Wiles and Taylor-Wiles Two extra interpretations of the modularity theorem We would like to give two interpretation of the modularity theorem. The first comes from the following theorem: 3 In fact, it was in the 1987 paper that Serre summarize how Wiles later work will imply the Fermat last theorem assuming Ribet s theorem which was proved right after Serre s work. 5

6 Theorem Let X be a proper variety over F p. Then for any l p, the number of F p -points can be given by #X(F p ) = 2 dim X i=0 ( 1) i Tr(Frob : Hét i ( X, Q l )) where X is the base change of X to F p. Similar statement is true when F p is replaced by F q for any p-power q. The philosophy (and that of the proof) of the theorem is that the fixed point of the p-power map (x 1,..., x n ) (x p 1,..., x p n) on coordinates is precisely #X(F p ) (a coordinate in F p is fixed if it is in F p ). The rest then follows as Grothendieck proved a étale cohomology version of the Lefschetz fixed-point theorem, which states that the fixed point of a continuous map from X (if X was a compact toplogical space) is equal to the above alternating sum of its trace on the (co)homology. When X is a curve, e.g. the reduction E p modulo p of our Frey curve E = (y 2 z = x(x Az)(x + Bz)) at those p ABC, one has that Hét 0 ( X, Q l ) = Q l is 1-dimension with trivial G Q -action and Hét 2 ( X, Q l ) = Q l (1) has the cyclotomic action. This implies #X(F p ) = 1 + p Tr(Frob : Hét 1 ( X, Q l )). When X = E p as above and E is modular from a Hecke eigenform f = n=1 a nq n (normalized as a 1 = 1), we had seen that Tr(Frob : Hét 1 (E p, Q l )) = Tr(Frob p : T l E) = a p! In other words, we have a p = p + 1 #E p (F p ). In other words, the modularity theorem could have been stated in a way without referring to the Tate module and Galois representations at all. For example, the modularity of the elliptic curve E = (y 2 z = x(x Az)(x Bz)) simply says that there exists a normalized Hecke eigenform f = n=1 a nq n of weight 2 such that for p AB(A + B), the number of non-trivial solutions to the cubic equation modulo p, up to F p -scaling, is equal to p + 1 a p. This gives the modularity theorem as an elementary statement. There is another way to write the modularity theorem as a statement that requires only complex analysis. The quotient Γ 0 (N)\H is a Riemann surface, which is a compact Riemann surface X 0 (N) punctured a few points. One may prove that a cusp form of weight 2 is equivalent to a holomorphic 1-form on X 0 (N). Using H 0 (X 0 (N), Ω 1 ) H 1 (X 0 (N), Ω 0 ) = H 1 (X 0 (N), C) and that a choice of isomorphism C = Q l gives us, from a 1-dimensional subspace in H 0 (X 0 (N), Ω 1 ) = H 1 (X 0 (N), Ω 0 ) to a 2-dimensional subspace of Hét 1 (X 0(N), Q l ) Q l. Now arithmetic geometric result (the Tate conjecture for abelian varieties proved by Faltings, to be precise) will then imply that if this two-dimensional space is related to the Hét 1 of an 6

7 elliptic curve E, then there exists a non-trivial algebraic map from X 0 (N) to the elliptic curve E. However one may show that the existence of such a map is equivalent to an existence of a holomorphic map, and thus the modularity theorem is equivalent to the following statement: Theorem Every elliptic curve (over Q) receives a holomorphic surjection from X 0 (N) for some N. 7