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LANGLANDS' CONJECTURES for PHYSICISTS 1. Introduction This Is LANGLANDS’ CONJECTURES FOR PHYSICISTS MARK GORESKY 1. Introduction This is an expanded version of several lectures given to a group of physicists at the I.A.S. on March 8, 2004. It is a work in progress: check back in a few months to see if the empty sections at the end have been completed. This article is written on two levels. Many technical details that were not included in the original lectures, and which may be ignored on a first reading, are contained in the end-notes. The first few paragraphs of each section are designed to be accessible to a wide audience. The present article is, at best, an introduction to the many excellent survey articles ([Ar, F, G1, G2, Gr, Kn, K, R1, T] on automorphic forms and Langlands’ program. Slightly more advanced surveys include ([BR]), the books [Ba, Be] and the review articles [M, R2, R3]. 2. The conjecture for GL(n, Q) Very roughly, the conjecture is that there should exist a correspondence nice irreducible n dimensional nice automorphic representations (2.0.1) −→ representations of Gal(Q/Q) of GL(n, AQ) such that (2.0.2) {eigenvalues of Frobenius}−→{eigenvalues of Hecke operators} The purpose of the next few sections is to explain the meaning of the words in this statement. Then we will briefly examine the many generalizations of this statement to other fields besides Q and to other algebraic groups besides GL(n). 3. Fields 3.1. Points in an algebraic variety. If E ⊂ F are fields, the Galois group Gal(F/E)is the set of field automorphisms φ : E → E which fix every element of F. Let E[x1,x2,...,xn] be the algebra of polynomials in n variables, with coefficients in E. If f1,f2,...,fr ∈ E[x1,x2,...,xn] are polynomials, denote by X(E) the set of solutions n {x ∈ E | f1(x)=f2(x)=···= fr(x)=0} . We may also view the fi as polynomials with coefficients in F, so the set X(F ) of solutions with coordinates in F makes sense. The Galois group Gal(F/E) acts on X(F ), fixing X(E). 1 Then X(F ) is called the set of points of the algebraic variety X in the field of F. We refer to the algebraic variety X as representing the set of solutions over all extension fields of E simultaneously, but we say that X is defined over E. The same applies to the case when the fi are homogeneous polynomials and X(E) is the corresponding set of solutions in projective space Pr−1(E). In particular, if X is an algebraic variety defined over the integers Z then it lives in both worlds: X(C) is a complex algebraic variety, and X (mod p) is an algebraic variety defined over Fp (obtained by reducing the equations for X modulo p). Moreover, for any extension field Fpr one may consider the set of points X (mod p)(Fpr ) with coordinates in Fpr . The amazing relation between these two worlds will be described in §4.3. 3.2. Finite fields. For every prime number p and every positive integer n there is a unique n field Fpn with p elements. If n = 1 then Fp = Z/(p) is the integers modulo p. If p1 and p2 n are prime numbers then F n ⊂ F m if and only if p = p and n divides m. Now let q = p p1 p2 1 2 (with p a prime number). Then px = 0 and xq = x for every x ∈ Fq. We say that Fq has characteristic p. The Galois group Gal(Fqr /Fq)is q isomorphic to Z/(r) and is generated by the Frobenius σq(x)=x . This is an automorphism, for q (x + y)q = xq + qxq−1y + xq−2y2 + ···+ yq. 2 Each of these binomial coefficients is divisible by p so (x + y)q = xq + yq. As r varies, these Frobenius automorphisms are compatible. Taking p = q we obtain a particular element σp ∈ Gal(Fp/Fp). 3.3. Number fields. A number field is a finite extension of Q. Each number field may be obtained by adjoining to Q the roots of a polynomial f(x) with rational coefficients. The algebraic closure Q ⊂ C is the “union” of all number fields: it is the set of all roots of all polynomials with rational coefficients. Just as the rational numbers Q consists of all fractions a/b where a, b ∈ Z, any number field E consists of all fractions a/b where a, b ∈ oE where oE is the ring of integers in E. For example, if E = Q[i] then oE is the set of all a + bi where a, b ∈ Z. 3.4. Local function fields. Let Fq be a finite field and let Fq[[t]] be the ring of formal power series with coefficients in Fq, and with the obvious operations of addition and multiplication. n An element a = Pn≥0 ant is invertible if and only if a0 =06 . In this case the inverse may be m found by writing b = Pm≥0 bmt , expanding the equation ab = 1 and solving the resulting linear equations for the coefficients bm. The mapping Fq[[t]] → Fq, which takes the above element a to its constant term a0, is a surjective ring homomorphism, so Fq is referred to as the residue field. 2 The field Fq((t)) consists of all fractions a/b where a, b ∈ Fq[[t]], and we refer to Fq[[t]] as the ring of integers in the field Fq((t)). It is easy to see that any element A ∈ Fq((t)) may be expressed as a formal Laurent series ∞ n A = X Ant n=N (where N ∈ Z is any integer, possibly negative) with at most finitely many terms involving negative powers of t The valuation of such an element A is the smallest integer n such that An =06 . Similarly one can form the function field C((t)) and its ring of integers C[[t]], however this case is of less interest to number theorists (and of more interest to physicists). 3.5. p-adic fields. Fix a prime number p. A p-adic integer is a formal power series a = 2 a0 + a1p + a2p + ··· where 0 ≤ ai ≤ p − 1. The set of such is denoted Zp. Addition and multiplication are performed using power series manipulations but with a “carry”, meaning that whenever we come across a coefficient a that is greater than p, we write it as a0 + a1p, keep the a0 part, and carry the a1 part on to the next term. As a consequence, the mapping a → a0 defines a surjective ring homomorphism Zp → Z/(p), the residue field. The ring Zp contains most of the rational numbers. It clearly contains the integers, but it also contains the fraction a/b whenever b is not divisible by p. For example, in Z5 the inverse of 3 is 3−1 =2+3· 5+1· 52 +3· 53 +1· 54 + ··· as can be easily seen by multiplying the right hand side by 3. However the number 5 is not invertible in Z5 so in order to make a field we need to include 2 its inverse, as well as that of 5 and so on. This leads to two descriptions of Qp : as set of fractions a/b where a, b ∈ Zp, or as the set of formal Laurent series ∞ n c = X cnp n=N consisting of all formal series with finitely many negative powers of p. The valuation val( c) of c is the “degree” of the leading term in this expansion. In particular, Qp contains Q. The p-adic numbers can also be realized as the (topological) completion1 of the rational numbers 2 Q with respect to a certain norm ||p. With this metric space structure, the field Qp is locally compact. Note that Qp has characteristic 0 and in fact it is possible to embed Qp into the complex numbers C, which, from now on, we will assume to have been done3. There are also many finite extensions of Qp and these are referred to as p-adic fields. Each such may be obtained by “completing” a number field E at a prime ideal p which contains (or “lies over”) p. The p-adic field Ep contains a ring of integers o which then projects to a residue field oE → F which is a finite extension of Fp. If deg(Ep/Qp) = deg(F/Fp) then Ep 3 is said to be an unramified extension of Qp. The field Qp has a unique unramified extension of each degree, but there are other extensions as well. The p-adic fields are referred to as local fields. 3.6. Global Function fields. Let Y be a compact Riemann surface. Then the meromor- phic functions C(Y )onY form a field. Such a Y admits the structure of a (smooth) complex projective algebraic variety which may be realized as a subvariety of P2(C). As such, it has a field of “rational functions”, and this also coincides with C(Y ). If Y = P1(C) then its func- tion field is the field C(x) of “rational functions”, consisting of quotients p(x)/q(x) where p, q are polynomials. If Y 0 → Y is a (possibly ramified) finite covering of compact Riemann surfaces then C(Y 0) is a finite extension of C(Y ). Similarly, suppose Y is a smooth projective algebraic curve defined over a finite field F(q). 1 Let Fq(Y ) be its field of (“rational”) functions.
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