GENERALIZED NON-ABELIAN RECIPROCITY LAWS: A CONTEXT FOR WILES S PROOF

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1 GENERALIZED NON-ABELIAN RECIPROCITY LAWS: A CONTEXT FOR WILES S PROOF AVNER ASH AND ROBERT GROSS Abstract. In as elementary a way as ossible, we lace Wiles s roof of Fermat s Last Theorem into the context of a general descrition of recirocity conjectured to obtain between algebraic varieties defined over Q and Hecke eigenvectors in the homology of the saces of lattices in R n. We shall find the Cube of the Rainbow, Of that, there is no doubt. But the Arc of a Lover s conjecture Eludes the finding out. Emily Dickinson During the last few decades, the field of number theory has been increasingly ermeated by the theory of automorhic forms and automorhic reresentations. This henomenon often goes under the rubric of the Langlands rogram, although it involves the work of many mathematicians, including R. Langlands, J.-P. Serre, and G. Shimura, to name only three of many. Recently, this rogram became esecially rominent because it forms a background to Wiles s roof of Fermat s last theorem. As exlained by Langlands in [14], arts of this rogram can be viewed as a vast generalization of recirocity laws familiar in number theory, such as quadratic recirocity and Artin recirocity. Some of the ultimate conjectures along these lines are selled out in Clozel s article [4], linking u motives and automorhic reresentations; see also Gelbart [11]. Both of the italicized terms in the revious sentence are rather technical objects. Our goal in this article is to describe a version of these generalized nonabelian recirocity conjectures: a version accessible to a reader who knows nothing beyond basic algebra and the definition of homology grous. Because of these self-imosed limitations, we shall be unable to state the conjectures in their full strength or with total recision. To comensate for these necessary shortcomings, we have included three basic examles that should give the flavor of the conjectures. The first involves quadratic recirocity, the second a modular ellitic curve, and the third a (robably automorhic algebraic surface. Moreover, the conjecture we do state is strong enough to allow its alication to Fermat s last theorem. At the aroriate lace, we shall discuss the term recirocity and what may be thought of as being recirocated. By reducing the rerequisites, we hoe we have increased the number of readers who can obtain some glimses of this beautiful landscae of generalized recirocity. We also believe the formulation of the conjectures in terms of saces of lattices in R n has a certain surrising beauty of its own. We should oint out, however, that it is very unlikely that rogress can be made in roving the conjectures on this elementary level. We begin with a quick review of Wiles s roof of Fermat s last theorem. For a good survey of this together with references to the work of Wiles and his redecessors necessary to the roof, see [15]. Let n be a rime greater than 3, and suose that (a, b, c Z 3 is a nontrivial solution of the Fermat equation a n +b n = c n. It was noticed that one could take these three integers and use them Date: Aril 7, Mathematics Subject Classification

2 2 AVNER ASH AND ROBERT GROSS to define a articular ellitic curve E, y 2 = x(x a n (x + b n, where we think of E as lying in the xy-lane. Ellitic curves have been the object of intense study for much of this century, though many things about them remain unknown. One articular roerty of ellitic curves is that they can be modular. We shall give below a definition of modularity from our oint of view that is equivalent to the standard one. Wiles roved that this articular ellitic curve is modular; Ribet had earlier roved that E is not modular. Therefore, it cannot exist, and so a, b, and c cannot exist either. The work of Wiles [23], Taylor and Wiles [21], Diamond [7], and Conrad, Diamond, and Taylor [5] shows that a rather large class of ellitic curves is modular. Just recently, Christohe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have announced a roof of the modularity conjecture [6], which asserts that every ellitic curve defined over Q is modular. This conjecture is art of a vast rogram which sets u a conjectural corresondence between two large sets. On one side of the corresondence, roughly seaking, is the collection of systems of simultaneous olynomial equations with rational coefficients. A member of this collection is called a variety defined over Q, and is relatively simle to comrehend. (More recisely, this side of the corresondence contains motives, which we shall discuss below. For examle, ellitic curves defined over Q are varieties, because they can be described in general by an equation y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, where the a i are integers, and the discriminant of the ellitic curve, which is a olynomial function of the numbers a i, is non-zero. Points on the curve over the ring R are simly solutions (x, y R 2 to the given equation. The formula for the discriminant is quite comlex, and may be found in [19,. 46]; if we restrict to the simler family of curves given by the equation y 2 = x 3 + Ax + B, then = 16(4A B 2. Excet for a constant factor, this is the same as the discriminant of the cubic olynomial x 3 +Ax+B. For future reference, we mention a more subtle number N called the conductor, which consists of a roduct of the rimes dividing raised to certain owers. The other side of the corresondence is harder to gras; it consists of certain automorhic reresentations of reductive grous defined over Q. (Linking these two collections is something even harder to get an elementary handle on: reresentations of Gal(Q/Q, which we shall discuss further in the Aendix. To simlify our exosition, in this article we shall consider only a certain subset of automorhic reresentations: those which are geometric for GL(n with constant coefficients. This articular subset is still rich enough to allow us to define the concet of modularity for ellitic curves. This narrowing of scoe allows us to relace automorhic reresentations by simler equivalent objects, called H-eigenelements, to be exlained below. We denote by α an element of this collection. The first side of the corresondence will now consist of a more narrowly defined collection of varieties that will still include ellitic curves. We shall denote such a variety by V. Our subrogram of the general recirocity rogram is the following corresondence: ( certain varieties (including ellitic ( H-eigenelements curves defined over Q V α We need to exlain more recisely the terms and nature of this corresondence. First, the lefthand side: by multilying through by the common denominator of all fractions used to define the variety V, we may assume that in fact the equations used to define V have coefficients in Z rather

3 GENERALIZED NON-ABELIAN RECIPROCITY LAWS 3 than in Q. If R is any ring, we can then use V (R to mean the solutions to the equations which are in R. In other words, if V is defined by olynomials {g i (x 1,..., x m = 0, i = 1,..., n}, then V (R = {(a 1,..., a m R m g i (a 1,..., a m = 0 for i = 1,..., n}. In articular, we can take R to be the field F m, the field of m elements, where is any rime and m is any ositive integer. Because the field is finite, we get a set of integers #V (F m for rimes and ositive integers m. These integers are encoded in the Hasse-Weil zeta function of V as described below. Unfortunately, making recise which equivalence classes of varieties we consider on the left-hand side of this corresondence is quite comlex. In fact, we actually need ieces of varieties, or, more recisely, ieces of their cohomology, called motives. When the cohomology of a variety V breaks u into a number of motives, all of which excet one are easily understood, then V as a whole can stand in for its nontrivial motivic iece in the formulas that give a descrition of the corresondence V α. This is what haens in the three examles discussed in this aer. Therefore, the first-time reader might wish to skim this ad hoc introduction to motives. Advanced readers may consult [12] for more information about motives. A motive is a iece of the cohomology of a variety defined over Q. Just like the comlex cohomology of a comlex variety, motives have Hodge tyes, to which we shall need to refer for the sake of accurately stating the conjectures below. One rojective variety V can contain many motives, and one motive can aear in various different varieties. A motive M gives rise to an L-function L(M, t with an Euler roduct L(M, t = L (M, t L (M, t, where runs over all rime numbers. If the variety V yields the set of motives {M i }, then L(M i, t ε i = Z(V, t = Z (V, t Z (V, t i where Z(V, t is the Hasse-Weil zeta-function of V, and ε i is 1 or 1 deending on whether M i aears in an odd- or even-degree cohomology grou of V. The local factor Z (V, t is defined by ( #V (F m Z (V, t = ex t m. m In fact, we even have the local factorization m=1 Z (V, t = i L (M i, t ε i for each rime. An examle could not but hel to clarify these concets. Consider the variety V = P 2, defined over Q. We can think of P 2 as the disjoint union A 0 A 1 A 2. Therefore, #P 2 (F m = 1+ m + 2m. We have 1 + m + 2m log Z (V, t = t m = m=1 m [ t m m + (tm m m=1 + (2 t m ] m = ( log(1 t + log(1 t + log(1 2 t,

4 4 AVNER ASH AND ROBERT GROSS so Z (V, t = 1 (1 t(1 t(1 2 t. It is traditional to substitute t = s, and view s as a comlex variable. We then have and therefore Z (V, s = ( (1 s (1 1 s (1 2 s 1, Z (V, s = ζ(sζ(s 1ζ(s 2. In this case, the three motives corresonding to V are exactly H i (V for i = 0, 2, 4, and L(M i, s = ζ(s i/2. In general, for any smooth rojective variety V defined by equations with integral coefficients, and for almost all rimes, Z (V, t is a rational function with with the factorization Z (V, t = P 1,(tP 3, (t P 0, (tp 2, (t P i, (t = r (1 β ( r,i t. Let V = V (F. Then the Frobenius element φ acts on the étale cohomology of V, with β ( r,i recirocal eigenvalues of φ. We have β ( r,i = i/2. Finally, #V (F m = r,i ( 1 i (β ( r,i m. the Now, a motive M corresonds first to a choice of a single index i and a ositive integer n, and then for each rime a choice of a subset S of {β ( r,i } of cardinality n such that S is the set of recirocal eigenvalues of φ acting on a geometrically defined iece of the étale cohomology of V. For examle, the iece might be all of H i (V. See the discussion of [10] later in this aer for an examle of a motive that is not all of H i (V. We can now define the L-function of a motive M by setting L (M, s = (1 β s 1, for almost all. β S Note the customary change of variable from t to s. We omit the definition for bad rimes and for L, but we mention that L (M, s deends on the Hodge tye of M. Then L(M, s = L (M, s L (M, s. Now we can say: the left-hand side of the corresondence consists of motives of dimension n with Hodge tye (n 1, 0 (n 2, 1 (0, n 1. On the right-hand side of the corresondence, H will denote a ring of oerators acting on certain homology grous, and this action has eigenelements. The eigenvalues of the oerators in H acting on α will also be a set of numbers, and the recirocity conjecture secifies a relationshi between these eigenvalues and the numbers #V (F m. These relationshis are recisely given by an equality of L-functions; but first we have to define the α and their L-functions. What are these mysterious H-eigenelements? We begin with a review of a common definition: Definition. A lattice Λ R n is a free abelian grou generated by a basis of R n. Our next definition is less common, but also simle to comrehend:

5 GENERALIZED NON-ABELIAN RECIPROCITY LAWS 5 Definition. A level N structure on an n-dimensional lattice Λ is a grou isomorhism φ : Λ/NΛ (Z/NZ n. We could consider the collection of all lattices in R n with level N structure, but that turns out to be too large a class. Instead, we consider two such lattices to be equivalent if one can be obtained from the other by roer Euclidean motion (that is, an orthogonal transformation of determinant 1, and ositive homothety (change of scale, where we always view R n as endowed with its usual Euclidean structure. Definition. L n (N = {(lattice Λ R n, level N structure φ} / roer Euclidean motion, ositive homotheties. Before we consider a few examles, we ut a toology on our set L n (N. Given (Λ, φ L n (N, fix a basis for Λ, and then consider nearby oints to be those lattices gotten by small erturbations of the basis in R n, keeing the level N structure φ the same. These saces of lattices come into the game because they are locally symmetric saces on which automorhic forms naturally live. The simlest examle is L 1 (1. Because our lattices are equivalent u to homothety, we can take a basis of our lattice Λ to be 1. A level 1 structure would be an isomorhism from Λ/Λ Z/Z, and since each grou contains only the identity element, there can be only one such isomorhism. Therefore, L 1 (1 consists of a single oint. A more enlightening examle is given by L 1 (N. Again, there is only one lattice Λ u to homothety, so we can take Λ = Z. Our level N structure is an isomorhism φ : Z/NZ Z/NZ. Such a ma is defined by φ(1, and since φ must be invertible, φ(1 must be invertible, and hence in (Z/NZ. Therefore, L 1 (N = (Z/NZ. In order to see how L n (N can contain more interesting geometric information, we consider one more examle: L 2 (1. We start with a two-dimensional lattice Λ. We take a vector of minimal length in Λ, and by means of rotations and stretching, we can take that vector to be (1, 0. We are free to take any linearly indeendent vector in Λ as the second basis vector. Take any such vector (x, y, where we may suose that x 0 and y > 0. By adding multiles of (1, 0 to this vector, we may suose that 1 2 x 1 2. Since this vector must have length at least as large as (1, 0, we also know that x 2 + y 2 1. Furthermore, there are two identifications we can make. Clearly, ( 1 2, y is equivalent to ( 1 2, y. Less obviously, there is another identification. If the second chosen basis vector (x, y has length 1, we can rotate this vector to (1, 0, and then the vector formerly at (1, 0 will rotate to (x, y. After we multily by 1 to get a ositive second coordinate, we see that (x, y is equivalent to ( x, y when x 2 + y 2 = 1. After making these identifications, our sace of lattices is toologically equivalent to a shere with one oint removed. In fact, taking the stri { (x, y R x 1 2, x2 + y 2 1, y > 0 }, and identifying the left and right vertical edges, gives a cylinder. Then glueing (x, y to ( x, y for those oints (x, y with x 2 + y 2 = 1 collases the circle at the bottom of the cylinder to an arc. Toologically, we now have an oen cu, which is homeomorhic to a shere minus a oint. (A more detailed discussion can be found in many laces in the literature; see, for examle, [16, Ch. VII.1], for a different exlanation and a icture. As before, the level 1 structure does not add any further detail, since φ will be a ma from the trivial grou to the trivial grou. Although the general icture is obviously much more comlex, certain features of this situation will remain true. For all n 1, N 1, L n (N will be a nice toological sace with #(Z/NZ comonents, classified by det φ. The situation for n = 2 is esecially favorable: L 2 (N is the disjoint union of Riemann surfaces, and can be defined by equations with Q-coefficients. There are formulas for the genus of L 2 (N; see [18, Ch. 1]. If n 3, then L n (N is not an algebro-geometric sace, but only a V -manifold (a manifold if N 3.

6 6 AVNER ASH AND ROBERT GROSS We next consider the homology of these saces: H d (L n (N, C, which is the grou of d-dimensional cycles modulo d-dimensional boundaries. This is always a finite-dimensional vector sace. There is extra structure, given by the action of the Hecke algebra H, defined as follows. Let be any rime not dividing N, and let k be any integer between 0 and n, inclusive. We can then define T (k : L n (N L n (N (Λ, φ {(Λ, ψ Λ Λ, Λ/Λ = (Z/Z k, ψ = φ Λ } There are several observations to be made about these oerators T (k. (1 Because N, ψ is again a level N structure. (2 T (k is not a function, but rather a one-to-many ma. In fact, the image of (Λ, φ is finite, and the cardinality is given by the number of k-lanes in (Z/Z n. (3 The action on the homology grous is in fact a function. In fact, T (k mas a cycle to the union of all of its images, but that union is again a cycle. Therefore, we do have a well-defined function T (k : H d (L n (N, C H d (L n (N, C. (4 These functions T (k all commute. Therefore, there are simultaneous eigenclasses α. If we write T (k (α = a (k α, then one can rove that these numbers a (k are algebraic integers. For a fixed α, they in fact generate a finite-degree extension of Q. (5 If N N, and α H d (L n (N, C is an H-eigenelement, there always exists α H d (L n (N, C, an H-eigenelement with eigenvalues equal to those of α for those rimes not dividing N. (6 In rincile, these eigenclasses α and numbers a (k are comutable. For n = 1 and any N, the comutation is easy. For n = 2 and relatively small N, the comutation is not imossible; there is much hel coming from the theory of modular forms and ellitic curves. For n = 3 and relatively small N, a comuter can rovide the answers [1]. For n = 4 and very small N, there is some research in rogress by the first author, Paul Gunnells, and Mark McConnell. In theory, whenever one can get a corresondence between a variety V (really a motive M and such an α, there is information to be gained. Such corresondences are generalized recirocity laws, so called because quadratic recirocity can be interreted as such a corresondence, as we shall see shortly. The term recirocity seems to go back to Legendre, as quoted on. 328 of [22]. Originally, the term referred to recirocity between two rimes and q: whether or not was a square modulo q being determined according to a simle rule deending on whether or not q was a square modulo. Eventually, this was interreted as a roerty of φ, the Frobenius element at, restricted to the Galois grou of Q( ±q over Q and vice versa. Later, the term recirocity was extended to a variety of rules that told how φ acted in various situations; see [24] for a good introduction. An early examle of a non-abelian recirocity law was given by Shimura [17]. Its modern use is exlained by Langlands [14, ] as including assertions that an L-function defined by diohantine data, that is, by an algebraic variety over a number field, is equal to an L-function defined by analytic data, that is, by an automorhic form. See also Tate s article [20] in the same volume, and the Aendix below, for some examles of what kind of concrete information a recirocity law can rovide. We can now offer a recise conjecture. An H-eigenvector corresonds to a motive as follows. We can define an L-function corresonding to the H-eigenelement α by defining L (α, s = 1 a (1 s + a (2 1 s a (3 3 s + + ( 1 n a (n 1 2 n(n 1 s.

7 GENERALIZED NON-ABELIAN RECIPROCITY LAWS 7 for almost all (that is, for those finite rimes not dividing the level N, with other factors for rimes dividing N and for, and then define L(α, s = L (α, s L (α, s. Conjecturally, α corresonds to a motive M in the sense that L(α, s = L(M, s. More recisely: Conjecture. (i Given an absolutely irreducible motive M of dimension n with Hodge tye (n 1, 0 (n 2, 1 (0, n 1 with conductor N, there is a cusidal H-eigenclass α H (L n (N, C such that L(α, s = L(M, s. (ii Given a cusidal H-eigenclass α H (L n (N, C, then there is a motive M of dimension n and conductor N and Hodge tye (n 1, 0 (n 2, 1 (0, n 1, such that L(α, s = L(M, s. We make some observations. (1 Part (i is Question 4.16 in [4], and art (ii is Conjecture 4.5 in [4]. We have restricted each of Clozel s formulations to the case of a trivial coefficient module for the homology of L n (N. (2 Other Hodge tyes occur if we allow the cohomology of L n (N with non-trivial coefficient modules. (3 The conductor N of a motive M is a well-defined attribute of M, indeendent of the Conjecture. (4 Cusidal is a technical term whose definition is beyond the scoe of this aer. Roughly seaking, an H-eigenclass α H (L n (N, C is cusidal if it cannot be induced from any sace of lattices in R m for m < n. If n = 2 or 3, then the concet simlifies: α is non-cusidal if for any comact subset K of L n (N, α is homologous to a cycle suorted outside of K. If α is cusidal, then it is known that L(α, s has an analytic continuation to the entire s-lane, and therefore the Conjecture imlies that L(M, s also has an analytic continuation. (5 The equality of L-functions is equivalent to the equality of the local factors: L (α, s = L (M, s for all, and L (α, s = L (M, s. (6 In art (ii, M need not be absolutely irreducible; for examle, α H 1 (L 2 (N, C could be associated to an ellitic curve with comlex multilication. Let us return to our examle of L 1 (N, and show how we can use the conjecture to deduce quadratic recirocity. The grou H 0 (L 1 (N, C is isomorhic to H 0 ((Z/NZ, C, and this homology grou consists of cycles ζ f = f(xx for functions f : (Z/NZ C. Let us work out how the oerator T (1 behaves. Because we are free to scale our lattice by a scalar, we can take Λ to be Z, with distinguished generator g = 1 (although, for clarity, we shall continue to write g. The level N structure φ is determined by φ(g = a (Z/NZ. We know that T (1 (Λ, φ must be the air (Λ, φ Λ = (Λ, ψ. Since Λ has distinguished generator g, we have ψ(g = ψ(g = a, so T (1 ζ f = f(xx = f( 1 xx, where 1 is the inverse of modulo N. Therefore (T (1 f(x = f( 1 x. Suose that α is a simultaneous eigenelement for T (1 for all not dividing N. We may scale α so that α(1 = 1. We know that (T (1 α(x = α( 1 x, and we also have (T (1 α(x = a α(x (we need not write a (1, since this examle is one-dimensional. Therefore, α( 1 x = a α(x. Set x = 1, and we have α( 1 = a. Now set x =, and we have α( = a 1. We can now use induction to conclude that α( k = a k. This formula holds for the image of any rime in

8 8 AVNER ASH AND ROBERT GROSS (Z/NZ, which imlies that α(xy = α(xα(y. In other words, the eigenelement α is a character of (Z/NZ, and the eigenvalue a = α( 1. Notice that the eigenfunction α also determines the eigenvalue a. Let us move now to the other side of the corresondence. Take any non-zero integer W, and let V be the variety defined by the equation x 2 W = 0. We ( comute the Hasse-Weil L-function Z (V, t for rimes W : If the quadratic residue symbol W = 1, then #V (F m = 2 for all ositive integers m. Therefore, ( 2 Z (V, t = ex m tm If ( W m=1 = ex(2 ( log(1 t 1 = (1 t 2 1 = ( (. (1 t 1 W t = 1, then #V (F 2m = 2, and #V (F 2m+1 = 0. Therefore, We now have ( 2 Z (V, t = ex 2m t2m m=1 ( (t 2 m = ex m m=1 = ex( log(1 t 2 = 1 1 t 2 = 1 ( ( (1 t 1 W. t Z (V, s = ζ(sl(χ, s, ( where L(χ, s is the Dirichlet L-series defined by the function χ( = W. The interesting art of the cohomology of V corresonds to the L-series L(χ, s, and this is the finite art of the L-series of the motive M{ that ( we } study. Precisely, for each {( } of good reduction, we must make a choice S of a subset of 1, W, and we select S = W. The conductor N of the variety M is a divisor of 4W (in fact, it might be considerably less than 4 W, since we have not as yet asked that W be square-free; in addition, if W 1 (mod 4, then the factor of 4 is unnecessary. ( The conjecture tells us that there is an eigenelement α H 0 ((Z/4W Z, C so that α( = W. (This is where the fact that the eigenelement α also determines the eigenvalue a comes into lay, along with the helful observation that we can ignore the exonent of 1 in our formulas for α( because 1 1 = 1. We have also used the fifth observation following the definition of T (k, with N = 4 W.

9 GENERALIZED NON-ABELIAN RECIPROCITY LAWS 9 Suose to ( start that W = 1. Since α can be thought of as a character on (Z/4Z, we can conclude that 1 is defined by the residue class of (mod 4. Since ( ( 1 3 = 1, and 1 5 = 1, we have deduced the usual formula for ( 1 Next, take W = 2, and we now have that α is defined ( 5, 7, and 17 give the usual formula for 2.. ( (mod 8. Comutation of 2 for = 3, Next, suose that q (mod ( 4, with ( > q, and let W = ( q/4. Then q (mod 4W, which means that α( = α(q, or W = W q. We have ( ( ( ( ( ( ( ( ( ( 4W + q 4W W W 4W q q 1 q = = = = = = = =, q q q q which imlies most of the usual formula for quadratic recirocity. To derive the remaining case, observe that the congruence x 2 W 0 (mod 4W 1 always has the solution x 2W. This tells us that if W is ositive and is any rime dividing 4W 1, then α( = 1, so α(4w 1 = 1. Now, suose that + q 0 (mod 4, and let W = ( + q/4. Our receding observation imlies ( W ( W q, and, reasoning as before, we can that α( = α(q, which in turn imlies that = ( ( conclude that q = q. We next take a more comlex examle, to exlain (finally! what it means for an ellitic curve to be modular. For instance, we shall consider the curve E : y 2 + y = x 3 x 2 with discriminant = 11 and conductor N = 11. For any ellitic curve V given by a non-singular cubic equation in the lane, we write ˆV for the comlete curve, that is, V { }. We count the number of solutions of this equation modulo for various rimes 11, and add 1 for the oint at, to get # ˆV (F. For any ellitic curve V defined over Q, set # ˆV (F = 1 + a. It is lausible, though not obvious, that one might want to write the number of solutions in this way: for roughly half of the ossible values of x, one exects the left-hand side to have a solution, but in those cases, one can exect there to be 2 solutions, since y 2 + y is a quadratic olynomial. Therefore, one exects roughly solutions to the congruence, lus an additional solution for the oint at infinity, and then we can consider a to measure how far the actual number of solutions deviates from the exected number. We can also motivate the exression 1 a + by thinking in terms of motives. The ellitic curve V affords the motives H 0 ( ˆV, H 1 ( ˆV, and H 2 ( ˆV. The motives in dimensions 0 and 2 are essentially trivial, and contribute 1 and, resectively (via the same analysis that gives the formula #P 1 (F m = 1 + m. The number a corresonds to the non-trivial motive in dimension 1. The statement that an ellitic curve V of conductor N is modular is equivalent to the fact that there is an eigenclass α H 1 (L 2 (N, C such that T (1 (α = a α, N. If you know the usual definition of modular ellitic curve, you can see this as follows: The set H 1 (L 2 (N, C is closely connected to the sace of modular forms of weight 2 and level N. In fact, by a theorem of Eichler and Shimura [18, Ch. 7], any cusidal H-eigenelement α H 1 (L 2 (N, C has the same H-eigenvalues as some newform of weight 2 and level N.

10 10 AVNER ASH AND ROBERT GROSS One can check that the ellitic curve E given above is modular by finding the corresonding α, or equivalently the weight 2 modular form of level 11; see for instance [17, 13]. If we set q = e 2πiτ, and η(τ = e πiτ/12 then the modular form corresonding to E is f(τ = η(τ 2 η(11τ 2 = a n q n n=1 (1 q n, = q 2q 2 q 3 + 2q 4 + q 5 + 2q 6 2q 7 2q 9 2q 10 + q 11 2q q q 14 q 15 4q 16 2q q q 20 + It is amusing to check a few of the a s by hand. For instance, if = 2, we comute the solutions Ê(F 2 = {(0, 0, (0, 1, (1, 0, (1, 1, }. Therefore, #Ê(F 2 = 5 = ( 2, so a 2 should equal 2. It does: the coefficient of q 2 in f(τ is 2. The conjectures stated above imly the well-known modularity conjecture : every ellitic curve defined over Q is modular. Notice that in our first examle, using L 1 (N, we didn t include a oint at infinity on V because V was already comlete, since it consisted simly of 2 geometric oints. In general, including oints at infinity comactifies the geometric sace and simlifies the formulas. For our final examle, we move to a higher dimension. H 3 (L 3 (N, C has been comuted for some small values of N in [1, 2, 10, 9], and certain eigenclasses have been exerimentally related to Galois reresentations. Van Geeman and To have related a few of these classes to varieties in [10]. Here is an examle. Consider ˆV to be the variety given by t 2 = xy(x 2 1(y 2 1(x 2 y 2 + 2xy along with oints at. Then there is an H-eigenclass in H 3 (L 3 (128, C with T (k α = a (k α, k = 0, 1, 2, 3 so that # ˆV (F m, for m 1, corresonds in a recise though rather comlicated way to the air (a (1, a (2 for all rimes 67. (Note that a (0 = a (3 = 1 for all. In articular, define N m(v := #{(x, y, z F 3 m t2 = xy(x 2 1(y 2 1(x 2 y 2 + 2xy} N m(e := #{(v, w F 2 m w2 = v(v 2 + 2v 1} The second quantity corresonds to oints at, and may be considered understood, as E is a modular ellitic curve. Then 6 ( 2 m αi m = N m(v + 2N m(e 2m 2 (1 m + i=1 where the numbers α i are determined by the equation 6 (1 α i X = X 6 c 1 X 5 + c 2 X 4 c 3 X c 2 X 2 4 c 1 X + 6 i=1 = ( X 3 χ(b X 2 + b X χ( 3 ( X 3 χ(b X 2 + b X χ( 3

11 where b = a (1, b = a (2 and χ( = GENERALIZED NON-ABELIAN RECIPROCITY LAWS 11 ( 2. The motive in question here is a 6-dimensional iece of H 2 ( ˆV, which itself is 34-dimensional. There is as yet no roof of this corresondence for all, though there is no reason other than lack of comuter time that the comutations cannot be continued for larger rimes. The status of the generalized recirocity conjecture, restricted as we have treated it, that is, for homology with trivial coefficients, is summarized neatly by the number n. For n = 1, it is essentially equivalent to class field theory for Q or the theory of cyclotomic fields. For n = 2, the results of Breuil, Conrad, Diamond, Taylor, and Wiles cited in the introduction, along with results of Eichler and Shimura [8, 18], give a good iece of the icture. For n 3, the conjecture is mostly unroven. Aendix In this aendix, we introduce Galois reresentations, which are really at the heart of the conjectures described above. We also give one more examle. In this one, the motive conjectured to exist is not yet known. A motive affords a comatible series of l-adic reresentations of G Q, the Galois grou of Q over Q. In an l-adic reresentation ρ, for each finite unramified rime, the characteristic olynomial F of a Frobenius element φ at is well-defined and gives information about how ρ(φ acts in the reresentation. Thus a formula exressing F in some other terms can be thought of as a generalized recirocity law. Moreover, as we have seen above, F is closely related to the #V (F m for the variety V from which the given motive comes. For a last concrete examle, consider the Hecke eigenclass α on L 3 (61 from [1]. If we set ω = (1 + 3/2, then the first few a (1 s were comuted to be a (1 1 2ω 5 + 4ω 2 + 4ω 6ω 2 + 2ω For each, a (2 is the comlex conjugate of a (1. We then conjecture the existence of a continuous 3-dimensional λ-adic reresentation ρ of G Q (where l is a rational rime, and λ a rime in some number field above l unramified outside 61l such that F = X 3 a (1 X 2 + a (2 X 3. ( In [3], we reduced all the coefficients modulo 3 and looked for a ρ : G Q GL(3, F 3. This would be the weakest ossible check of the conjecture for l = 3. Knowing the F s allowed us to infer how ρ(φ must act, and hence (eventually how must slit in the fixed field of ker ρ. For instance, if = 11, then a (1 a (2 1 (mod 3, and 1 (mod 3, so F X 3 + X 2 + X + 1. Thus, ρ(φ must be a matrix in GL(3, F 3 with that characteristic olynomial. Juggling such information, we determined the only ossible ρ, and showed that it matched the given data. We have no idea how to show that ρ is really attached to α (that is, that ( holds for all 3 61, nor how to find ρ. To summarize this examle: we have here recirocity between a certain series of reresentations of G Q and the Hecke information contained in α. It is relatively easy to comute α and then to conjecture these rather dee roerties of G Q. In the case of the roof of Fermat s last theorem, a utative solution of the Fermat equation leads to a certain reresentation of G Q. It is this reresentation that is shown not to exist by roving that it should be controlled through recirocity by a certain Hecke eigenclass in L 2 (2, which is readily checked to be non-existent.

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