# 1 χ ramified χ unramified

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1 LECTURE 14: LOCAL UNCTIONAL EQUATION LECTURE BY ASI ZAMAN STANORD NUMBER THEORY LEARNING SEMINAR JANUARY 31, 018 NOTES BY DAN DORE AND ASI ZAMAN Let be a local field. We fix ψ P p, ψ 1, such that ψ p d 1 for d " 0 with p the maximal ideal pϖq of. Let q Npϖq {p, and let v be the valuation. We fix dx, dx Haar measures on, respectively with the latter normalized so that O dx 1. We recall the GL 1 theory. Let Φ P Cc 8 p q be a smooth compactly supported function. Then we may define the ourier transform pφpxq Φpyqψpxyq dy. This satisfies the ourier inversion formula Φpxq p Φp xq. or some character χ on GL 1 p q, we define the zeta function ζpφ, χ, sq Φpxqχpxq x s dx and we define the set: We have the following theorem: Theorem 1. Zpχ, sq ζpφ, χ, sq Φ P C 8 c p q (. (A) (γ-factor) There exists a unique gamma factor γpχ, sq P Cpq s q such that and γpχ, sqγpχ 1, 1 sq χp 1q. ζp p Φ, χ 1, 1 sq γpχ, sqζpφ, χ, sq (B) (L-function) The set Zpχ, sq is equal to Lpχ, sqcrq s, q s s. Here, we have: # 1 χ ramified Lpχ, sq `1 χpϖqq s 1 χ unramified (C) (ɛ-factors) The function Lpχ, sq ɛpχ, sq γpχ, sq Lpχ 1, 1 sq satisfies ɛpχ, sqɛpχ 1, 1 sq χp 1q and ɛpχ, sq ɛpχqq pd`fqp 1 sq with ɛpχq 1. If χ is unramified then ɛpχq χpϖq d. If χ is ramified then ɛpχq is a certain Gauss sum depending on χ and ψ. Here f is defined so that the norm of the conductor of χ is q f. 1

2 Remark. The ɛ factor acts as the fudge factor to make the functional equation hold exactly for the L-function. or more precise formulas of the ɛ-factors and the relevant Gauss sums, see [Go, Equations (33) and (40)] or [BH, p.143]. If χ is unramified then the L-function Lps, χq uniquely determines χ. Otherwise, Lps, χq gives no information when χ is ramified; in this case, ɛpχ, sq γpχ, sq so the ɛ-factor encodes all of the data of χ. Remark 3. We suppress the dependence on the choice of ψ everywhere. However, while the ɛ and γ factors do depend on the choice of ψ, the L-function is the same for all appropriate ψ. Now, we will pass to the GL case. Given an irreducible admissible representation π of GL p q, we have the Whittaker and Kirillov models defined by:! Wpπq W : G Ñ C W `p ) x 1 q g ψpxqw pgq, " Kpπq φ: Ñ C π ` a b * φpxq ψpbxqφpaxq. We have a bijection from Wpπq to Kpπq defined by sending W to φ W : x ÞÑ W `p x 0 q. The dual Whittaker functional W P Wpπq is defined to be the function g ÞÑ W pgwq for w ` 1 0, the generator of the Weyl group. Recall Cc 8 p q Ď Kpπq always. Now, define a zeta integral similar to before: ZpW, χ, sq φ pxqχpxq 1 W x s 1 dx The function φ W can be thought of as a test function varying over W P Wpπq. Remark 4. This is defined in vague analogy to the definition given above in the GL 1 case, but the analogy isn t perfect. or example, in the GL 1 case, ζpφ, χ, sq is defined for Φ P C 8 c p q, but here we consider φ W defined on rather than on. A closer parallel to the GL 1 theory can be drawn using functions Φ on M p q; see [BH, Section 4] for details. Before we discuss the GL local functional equation, we prove the existence of a Whittaker functional such that it and its dual have Kirillov elements both supported away from zero, while also having a non-trivial zeta integral. Lemma 5. There exists W P Wpπq such that φ W P C 8 c p q, φ W P C 8 c p q and ZpW, χ, sq 0. Proof. rom [Go, Lemma 7, page 1.13], there exists non-zero W P Wpπq with φ W P Cc 8 p q, φ W P Cc 8 p q satisfying φ W pxuq φ W pxqχpuq for all u P O. By direction computation, we verify that the zeta integral does not vanish: ZpW, χ, sq φ pxqχpxq 1 W x s 1 dx 8ÿ n 8 8ÿ n 8 O φ W pϖ n uqχpϖ n uq 1 ϖ n s 1 du φ W pϖ n qχpϖ n q 1 q ps 1qn.

3 Note the sum is actually finite since φ W is compactly supported away from zero. Thus, ZpW, χ, sq is a non-zero element of Crq s, q s s if and only if φ W ı 0. The latter is true as W is non-zero. This finishes Step. Now, analogous to Theorem 1(A) in the GL 1 case, we have the following theorem: Theorem 6. (i) ZpW, χ, sq converges for Repsq " 0. (ii) ZpW, χ, sq admits an analytic continuation to a meromorphic function with at most poles. (iii) There exists some γ π pχ, sq P Cpq s q such that: for all W P Wpπq. Also, Zp W, ω π χ 1, 1 sq γ π pχ, sqzpw, χ, sq (1) γ π pχ, sqγ π pχ 1 ω π, 1 sq ω π p 1q. () Proof. (i) When π is supercuspidal, this part is easy. We have that φ W P Kpπq Cc 8 p q, so the integral converges due to the compact support away from zero. We may split the integral into a finite sum according to the valuation of x P. Suppose π is not supercuspidal. rom an earlier description of the Kirillov model [Go, Section 10], it follows that φ W P Kpπq is a sum of terms like x 1{ λpxqfpxq and x 1{ vpxqλpxqfpxq, where λ: Ñ C is some character and f P Cc 8 p q, with v the valuation. We claim that this implies that the integral converges. Let us consider one such integral. Since f is locally constant near 0, it follows that fpxq fp0q for x ď q N with N ě 1 sufficiently large. Moreover, as f is compactly supported in, it follows that fpxq 0 for x ě q M with M ě 1 sufficiently large. Thus, x 1{ λpxqfpxq x s 1 dx fp0q x s 1{ λpxq dx ` p q x ďq N q N ă x ďq M The second integral over q N ă x ď q M can be written as a finite sum over m vpxq with N ď m ď M and therefore it converges for all s P C. or the first integral, we also divide it according to the valuation n vpxq and observe that x s 1{ λpxq dx ÿ x q n q n{ ns λpxqdx x ďq N or x q n, we may write x ϖ n y with y P O. Since λ is a character of, it follows that λpyq ď 1 and thus λpxq λpϖq n for x q n. Moreover, x q n dx dx 1 by our normalization of the Haar measure. Hence, the above expression is O bounded in absolute value by ÿ q n{ n Repsq q n{ n Repsq λpϖq n. něn něn x q n λpxq dx ď ÿ As λpϖq is some fixed power of q, the above infinite sum converges once Repsq " 0. 3 něn

4 (ii) Proof postponed 1. We will not utilize this result until after Theorem 8. (iii) This follows from three steps. Step 1: Show (1) holds for any W P Wpπq with φ W P C 8 c p q. (We will prove this step last.) Step : Show () holds. Choose W from Lemma 5. rom Steps 1 and, we may apply Equation 1 twice to see that: ω π p 1qZpW, χ, sq Zp W, χ, sq γπ pχ 1 ω π, 1 sqzp W, χ 1 ω π, 1 sq γ π pχ 1 ω π, 1 sqγ π pχ, sqzpw, χ, sq Since ZpW, χ, sq ı 0, we may divide both sides by ZpW, χ, sq to deduce () holds. Step 3: Show (1) holds for any W P Wpπq. or every W P Wpπq, there exists W 1, W P Wpπq such that φ W φ W1 ` φ W and φ W1, φ W P C 8 c p q. This follows from the arguments leading to [Go, Section 10, Equation (144)]. Thus, we may apply the functional equation (1) to each of ZpW 1, χ, sq and Zp W, χ, sq and use linearity to deduce (1) for ZpW, χ, sq. The remainder of the proof is to establish Step 1. irst, we make a reduction. Claim 7. Any f P C 8 c p q is a linear combination of functions of the form: with λ: Ñ C some character and 1 O λpxq1 Opxq the indicator function on O. Proof of Claim 7: Recall Z O. Since f is compactly supported and locally constant, there exists positive integers N, M ě 1 (dependingly only on f) such that for every x P, fpxq fpϖ n uq for some unique integer n P r N, Ns and unique u chosen from a fixed set of coset representatives Ω of O {p1 ` ϖm O q. Therefore, fpxq ÿ ÿ NďnďN upω fpϖ n uq1 pxϖ n u`ϖ q. M O By orthogonality of characters on the finite quotient group O {p1 ` ϖm O q, the indicator function 1 u`ϖ pyq can be written as a finite linear combination of λpyq1 M O Opyq where λ is a character of with conductor at most q M. This proves the claim. 1 It is not apparent to me that a complete proof is provided in [Go, Section 1]. rom the functional equation, ZpW, χ, sq is meromorphic in Repsq ě A and Repsq ď 1 A for some large positive A but, without additional work, it is not obvious why it extends to the strip 1 A ď Repsq ď A. Instead, this extra input will follow implicitly from Theorems 8 and 1(B). 4

5 Continuing the proof of Step 1, recall we assume φ W P C 8 c p q. By Claim 7 and the linearity of ZpW, χ, sq in W, it suffices to show the functional equation for φ W of the form: φ W pxq λpxq1 Opxq with λ: Ñ C some arbitrary character. During the course of our computations, it is crucial that the calculated γ factor depends only on π, χ, and s. In particular, γ should be independent of the arbitrary character λ. Now, as λ and χ are characters on, the maps λ O and χ O are (necessarily unitary) characters on the compact group O. Hence, by orthogonality of characters (for the compact group O because O If λ O χ O ZpW, χ, sq O ), λpxqχpxq 1 dx dx 1 via our choice of normalization. # 1 λ O 0 else χ O then one may again verify by a similar computation that Zp W, ω π χ 1, 1 sq 0. Thus, in this case, any choice of γ π pχ, sq would satisfy a functional equation. Otherwise, we have reduced to the case when φ W pxq χpxq1 Opxq. In particular, φ W now depends only on χ. Therefore, for this φ W, we define γ π pχ, sq : Zp W, ω χ 1 π, 1 sq φ pxqω 1 W π χpxq x 1 s dx ZpW, χ, sq with the latter equality following from (3) and the definition of the zeta integral. Evidently, γ π satisfies the functional equation for this choice of φ W and, since φ W depends only on χ, we see that γ π depends only on χ and s. Therefore, we ve shown that for all W P Wpπq with φ W P C 8 c p q that (1) holds. This completes the proof of Step 1 and hence Theorem 6. (3) Note that the operation W ÞÑ W is not actually anything to do with a ourier transform: the duality in the functional equation appearing in Theorem 6 comes from the action of the Weyl group (which of course is trivial in the GL 1 case). In the next theorem, the ourier transform plays a role. Theorem 8. Define: $ 1 π cuspidal & L`χ 1 µ 1, s 1 L`χ 1 µ, s 1 π π µ1,µ L π pχ, sq, µ 1 {µ, 1 L`χ 1 µ 1, s 1 π π µ1,µ, µ 1 {µ % L`χ 1 µ, s 1 π π µ1,µ, µ 1 {µ 1 With this definition, we have: ZpW, χ, sq W P Wpπq ( Lπ pχ, sq Crq s, q s s 5

6 Thus, if π is induced from µ 1 b µ on T B{N» G m G m, then the L-function is the product of the GL 1 L-functions for µ 1, µ. If π is cuspidal, then just like in the ramified case for GL 1, the L-factor carries no information. Proof. If π is cuspidal, there is not much to show. We have: ZpW, χ, sq φ pxqχ 1 W pxq x s 1 dx Since φ W P Kpπq C 8 c p q, the integral breaks into a sum of finitely many terms based on the valuation of x, so ZpW, χ, sq P Crq s, q s s. If π π µ1,µ with µ 1 {µ 1,, 1 then we have: φ W pxq x 1{ `µ 1 pxqφ 1 pxq ` µ pxqφ pxq with Φ j P Cc 8 p q and µ 1, µ : Ñ C characters. Then we have: ZpW, χ, sq ζ Φ 1, χ 1 µ 1, s 1 ` ζ Φ, χ 1 µ, s 1 We will write z s 1 here and in the future. By Theorem 1(B), this is contained in: Lpχ 1 µ 1, zqcrq s, q s s ` Lpχ 1 µ, zqcrq s, q s s Lpχ 1 µ 1, zqlpχ 1 µ, zqcrq s, q s s The claimed equality holds since µ 1 µ implies the two GL 1 L-functions have different poles (as one can see by looking at the defining formula). or the cases when µ 1 {µ 1,, 1, similar arguments hold but with minor variations due to the precise characterization of the Kirillov models Kpπq. See [Go, pp ] for details. Now, we give a computation of the γ factors in the case of principal series. Paralleling the GL 1 theory for ramified characters, the case of cuspidal representations is more subtle and involves an analogue of the Gauss sum. See [BH, Section 5] for details. Theorem 9. If π π µ1,µ, then: γ π pχ, sq γ χ 1 µ 1, s 1 γ χ 1 µ, s 1 Proof. Choose W from Lemma 5 and let φ φ W. We have the following claims: Claim 10. There exists some Φ P C 8 c p q which extends to Φ P B µ1,µ such that Φpw 1 ` 1 y q Φpyq and the ourier transform p Φ P C 8 c p q satisfies ZpW, χ, sq ζpφ, χ 1 µ, zq. Here B µ1,µ is the space of locally constant functions ϕ : G Ñ C satisfying ϕ` p a 0 b q g µ 1 paqµ pbq a{b 1{ ϕpgq. 6

7 Proof of Claim 10: Set Φpxq p µ 1 pxq x 1{ φ W pxq P Cc 8 p q. By ourier inversion, Φp yq Φpxqψpxyqdx P Cc 8 p q. By the ourier transform [Go, Equation (148)], it follows that Φ P B µ1,µ and ` Φpw 1 1 y q Φpyq. This proves the claim. Claim 11. Choose Φ as in the proof of Claim 10. Define Φ w by g ÞÑ Φpgwq so its restriction Φ w P Cc 8 p q satisfies x ÞÑ Φ`w 1 p 1 0 x 1 q w. Then, with z s 1, Z W, χ 1 ω π, 1 s ζ pφw, χµ 1 1, 1 z. Proof of Claim 11: By a direct substitution of the definitions of Z, ζ and Φ, Z W, χ 1 ω π, 1 s φ pxqχω 1 W π pxq x 1 s dx ζ pφpxqµ pxq x 1{ χpxqµ 1 1 pxqµ pxq 1{ x 1 z x 1{ dx pφpxqχµ 1 1 pxq x 1 z dx. pφw, χµ 1 1, 1 z This proves the claim. Claim 1. Choose Φ as in the proof of Claim 10. Then ζ`φ w, χ 1 µ 1, z pµ 1 χ 1 qp 1qζ`Φ, χµ 1, 1 z Proof of Claim 1: By Claim 11, Φ w pyq Φ ` w 1 1 y w Φ 1 1{y 1{y 0 0 y w 1 1{y As Φ P B µ1,µ by Claim 10, the above expression is equal to µ 1 pyq y 1 Φ w ω p 1qµ 1 pyq y 1 π Φp 1{yq. 1 1{y Substituting this formula into the integral ζpφ w, χ 1 µ 1, zq, it follows by the change of variables 1{y ÞÑ y that ζ`φ w, χ 1 µ 1, z Φ pyqχ 1 w µ 1 pyq y z dy ω π p 1q Φp 1{yqχ 1 µ pyq y z 1 dy χ 1 µ 1 p 1q Φpyqχµ 1 pyq y 1 z dy χ 1 µ 1 p 1qζpΦ, χµ 1, 1 zq. 7.

8 This proves the claim. Now, we have, by repeatedly applying the claims as well as the GL 1 theorems: pµ χ 1 1 qp 1qγpµ χ 1, zqzpw, χ, sq pµ χ 1 1 qp 1qγpµ χ 1, zqζ pφ, χ 1 µ, z pµ χ 1 1 qp 1qζ`Φ, χµ 1, 1 z ζ`φ w, χ 1 µ 1, z γpxµ 1 1, 1 zqζp x Φ w, χµ 1 1, 1 zq γpχµ 1 1, 1 zqz W, χ 1 ω π, 1 s By Theorem 6, the righthand side equals γpχµ 1 1, 1 zqγ π pχ, sqzpw, χ, sq. Our choice of W from Lemma 5 satisfies ZpW, χ, sq ı 0 so we may divide this term from both sides to deduce that γ π pχ, sq µ χ p 1q γpµ χ 1, zq γpµ 1 1 χ, 1 zq. Applying Theorem 1(A) to the denominator proves the theorem. Theorem 13. Define ɛ π pχ, sq γ π pχ, sq L πpχ,sq L πpχ 1 ω π,1 sq and ɛ π pχ, sq aq bs for some a P C and b P Z. ɛ π pχ, sqɛ π pχ 1 ω π, 1 sq ω π p 1q. Then we have the functional equation: inally, in the principal series case, we can compute the ɛ factors. or the cuspidal case, the ɛ factor equals the γ factor (as the L-functions defining ɛ π are trivial) so we again refer the reader to [BH, Section 5] for details. Theorem 14. If π π µ1,µ with µ 1 {µ, 1, then we have: ɛ π pχ, sq ɛpχ 1 µ 1, s 1 q ɛpχ 1 µ, s 1 q where the ɛ factors on the right are the ɛ factors from the GL 1 theory. Proof. This follows from Theorems 9 and 13 as well as the relationship between ɛ factors and γ factors in the GL 1 case. or the cases when µ 1 {µ, 1, see [Go, pages ]. Now, we will prove Theorem 13: Proof. The equation ɛ π pχ, sqɛ pχ 1 π ω π, 1 sq ω π p 1q follows directly from the definition and the functional equation for γ given in Theorem 6. Now, again by Theorem 6, we have: ZpW, χ, sq L π pχ, sq ɛ πpχ, sq 8 Z W, χ 1 ωπ, 1 s L π pχ 1 ω π, 1 sq

9 By Theorem 8, the right-hand side is in Crq s, q s s, so it is entire in s. Now, choose W such that ZpW, χ, sq L π pχ, sq. This implies that ɛ π pχ, sq P Crq s, q s s as well. Similarly, we see that ɛ π pχ 1 ω π, 1 sq P Crq s, q s s. Since their product is ω π p 1q, we have: ɛ π pχ, sq P Crq s, q s s Thus, ɛ π pχ, sq aq bs for some a P C and b P Z. References [BH] Colin Bushnell, Guy Henniart, The Local Langlands Conjecture or GLpq, Springer-Verlag, Berlin, 006. [Go] R. Godement, Notes on Jacquet-Langlands Theory, The Institute for Advanced Study, 1970, available at conrad/conversesem/refs/godement-ias.pdf. 9