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§2.1 Automorphic Forms for GL(1, AQ)

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§2.1 Automorphic Forms for GL(1, AQ) CHAPTER II AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL(1; AQ) x2.1 Automorphic forms for GL(1; AQ) Let AQ denote the adele ring over Q as in definition 1.3.1. The key point is that GL(1; ) is just ×, the multiplicative subgroup of ideles of . In conformity AQ AQ Q with modern notation we shall use the notation g = fg1; g2; g3;::: g × × to denote an element of the group GL(1; AQ). Here gv 2 Qv for all v and gp 2 Zp for all but finitely many finite primes p. The multiplicative group × is diagonally embedded in × and acts on × by Q AQ AQ left multiplication. Proposition 1.4.6 tells us that a fundamental domain for this action is given by Y ×n × = (0; 1) · ×; Q AQ Zp p where the product is over all finite primes p: An idelic function f : × ! is said to be factorizable if it is determined by AQ C × × local functions fv : Qv ! C (8v ≤ 1), where fp ≡ 1 on Zp for all but finitely many finite primes p, and where Y × (2.1.1) f(g) = fv(gv); 8g = fg1; g2; g3;::: g 2 : AQ v≤∞ Note that this definition is slightly different than the notion of \factorizability of adelic functions" which was given in 1.7.3. Definition 2.1.2 (Unitary Hecke character of ×) A Hecke character of × AQ AQ is defined to be a continuous homomorphism ! : ×n × ! ×: Q AQ C A Hecke character is said to be unitary if all its values have absolute value 1. A unitary Hecke character of × is characterized by the following four properties: AQ (i) !(gg0) = !(g)!(g0); 8g; g0 2 ×; AQ (ii) !(γg) = !(g); 8γ 2 ×; 8g 2 ×; Q AQ (iii) ! is continuous at f1; 1; 1;::: g: (iv) j!j = 1: Typeset by AMS-TEX 1 2 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL(1; AQ) Definition 2.1.3 (Moderate growth) We say that a function φ : × ! is AQ C × of moderate growth if, for each g = fg1; g2; g3;::: g 2 , there exist positive AQ constants C and M such that M φ ftg1; g2; g3;::: g < C (1 + jtj1) for all t 2 R: Definition 2.1.4 (Automorphic form) Fix a unitary Hecke character ! as in (2.1.2). An automorphic form for GL(1; AQ) with character ! is a function φ : GL(1; AQ) ! C which satisfies the following conditions: (1) φ(γg) = φ(g); 8g 2 ×; 8γ 2 ×; AQ Q (2) φ(zg) = !(z)φ(g); 8g 2 ×; 8z 2 ×; AQ AQ (3) φ is of moderate growth as in definition 2.1.3: Let S! denote the set of all automorphic forms for GL(1; AQ) with character ! as in definition 2.1.4. If c1; c2 2 C; are arbitrary complex constants, and φ1; φ2 2 S!, then it is easy to see that c1φ1 + c2φ2 is again automorphic with character !: The space S! is, therefore, a vector space over C: Setting g = f1; 1; 1;::: g it immediately follows from definition 2.1.4 (2) that φ(z) = c!(z); with c = φ(f1; 1; 1;::: g): Thus S! is a one-dimensional space. The reader may ask why we simply do not define an automorphic form as a Hecke character as in (2.1.2)? The reason is that we want to give a uniform definition of automorphic form for GL(n; AQ) that holds for all n = 1; 2; 3;::: In the case of n = 1, definition 2.1.4 (2) becomes superfluous since z; g both lie in the same space. This is not the case for n > 1 as we shall see later. Definition 2.1.4 may seem rather imposing at first sight but it turns out that the automorphic forms for GL(1; AQ) are just classical Dirichlet characters in disguise. For a fixed integer q > 1, a Dirichlet character χ (mod q) is a homomorphism × × (2.1.5) χ :(Z=qZ) ! C : That is, × (2.1.6) χ(ab) = χ(a)χ(b); 8a; b 2 (Z=qZ) : × Because a'(q) = 1 for all a 2 (Z=qZ) ; such a function must take values in the '(q)th roots of unity. In particular, × jχ(a)j = 1; 8a 2 (Z=qZ) : Here '(q) is Euler's ' function. II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL(1; AQ) 3 It is standard practice in analytic number theory (see [Davenport, 2000]) to lift a Dirichlet character χ to Z by defining a new function χ1 : Z ! C which satisfies χ1(a) = 0; 8a 2 Z with (a; q) 6= 1; χ1(a + mq) = χ(a); 8a; m 2 Z with (a; q) = 1; χ1(ab) = χ1(a)χ1(b); 8a; b 2 Z: We shall follow the standard practice of denoting the character χ1 by the symbol χ. Remarkably, it is also possible to lift χ to the idele group ×. Since × is a AQ AQ purely multiplicative group, the value of the lifted character can never be 0. The result is an automorphic form as in definition 2.1.4. We now explicitly describe and prove the existence of this lifting which was found by Tate and appeared in his thesis [Tate, 1950]. Definition 2.1.7 (Idelic lift of a Dirichlet character) Let χ be a Dirichlet character of conductor pf as in (2.1.5), (2.1.6) where pf is a fixed prime power. We define the idelic lift of χ to be the unitary Hecke character χ : ×n × ! × idelic Q AQ C defined as × χ (g) = χ1(g1) · χ2(g2) · χ3(g3) ··· ; g = fg1; g2; g3;::: g 2 ; idelic AQ where 8 1; χ(−1) = 1; <> χ1(g1) = 1; χ(−1) = −1; g1 > 0; > : −1; χ(−1) = −1; g1 < 0; and where m m × χ(v) ; if gv 2 v Zv and v 6= p; χv(gv) = −1 k f χ(j) ; if gv 2 p j + p Zp with j; k 2 Z; (j; p) = 1 and v = p: To see that definition 2.1.7 actually defines a unitary Hecke character satisfying (2.1.2) we make the following observations. First of all, for every prime v ≤ 1; it is clear from the definition that 0 0 χv(gvgv) = χv(gv) · χv(gv) 0 × for all gv; gv 2 Qv : Consequently, χidelic must satisfy (2.1.2) (i). Secondly, if ` 6= p −1 is any finite prime, then χ`(`) = χ(`): Also χp(`) = χ(`) , χv(p) = 1 for any finite prime v and χv(`) = 1 for any finite prime v 6= ` and v 6= p. It follows that χidelic (`) = 1 for all primes `. Also χidelic {−1; −1; −1;:::; g = χidelic f1; 1; 1;:::; g = 1. When combined with (2.1.2) (i), this establishes (2.1.2) (ii). Thirdly, we can see directly that the kernel of χidelic is an open neighborhood of f1; 1; 1;::: g: It is n × also obvious that jχidelic j = 1 since χ has this property on (Z=p Z) : The above 4 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL(1; AQ) observations establish that χidelic is indeed a Hecke character satisfying the four conditions of (2.1.2). Qr fi More generally, every Dirichlet character χ of conductor q = i=1 pi ; where p1; p2; : : : pr are distinct primes and f1; f2; : : : fr ≥ 1 can be factored as r Y χ = χ(i); i=1 (i) fi where χ is a Dirichlet character of conductor pi : It follows that χ may be lifted to a Hecke character χ on × where idelic AQ r Y (2.1.8) χ = χ(i) : idelic idelic i=1 Theorem 2.1.9 Every automorphic form φ on GL(1; AQ), as in definition 2.1.4, can be uniquely expressed in the form it × φ(g) = c · χidelic (g) · jgj ; 8g 2 ; A AQ where c 2 C; t 2 R, are fixed constants, and χidelic is an idelic lift of a fixed Dirichlet character χ as in definition 2.1.7 and (2.1.8). Here, if g = fg1; g2; g3;::: g, then Q jgj = jgvjv is the idelic absolute value. A v≤∞ Proof: It follows from definition 2.1.4 that we may take φ(g) = c · !(g); (8g 2 ×) AQ with c = φf1; 1; 1;::: g; and where ! is a unitary Hecke character satisfying (2.1.2). For each prime v ≤ 1, consider the embedding iv(gv) = f1;:::; 1; gv ; 1 :::; 1g; gv 2 Qv: |{z} vth position Then, if we define !v(gv) := !(iv(gv)); (8gv 2 Qv) × then !v is a character of Qv for every prime v ≤ 1: Furthermore, Y (2.1.10) !(g) = !v(gv) v≤∞ × where !p(gp) ≡ 1; (8gp 2 Zp ) for all but finitely many finite primes p: The Hecke characters ! can then be determined if we can classify the local characters !v for all primes v. × × First of all, every unitary continuous multiplicative character !1 : R ! C is of the form it it (2.1.11) !1(g1) = jg1j1; or !1(g1) = jg1j1 · sign(g1); II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL(1; AQ) 5 +1; if g1 > 0; for some fixed constant t 2 R; where sign(g1) = −1; if g1 < 0: To continue the proof and classify the characters !v for v < 1, we introduce some definitions. Definition 2.1.12 (Unramified local character) Fix a finite prime p. A local character !p, occurring in the decomposition (2.1.10), is said to be unramified if × it !p(u) = 1 for all u 2 Zp : At 1, the local character jg1j1 (for some fixed t 2 R) is said to be unramified.
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