Lecture Notes: Tate s thesis

Lecture Notes: Tate s thesis September 9, 2 Motivation To prove the analytic continuation of the Riemann zeta function (85), we start with the Gamma function: Γ(s) Substitute: Γ(s/2) And now put πn 2 x for x: π s/2 n s Γ(s/2) e x x s dx x e x x s/2 dx x e πn2x x s/2 dx x Now sum over n (valid for real part s greater than ): Λ(x) π s/2 Γ(s/2)ζ(s) where ω(x) Now set n e n2 πx θ(x) + 2ω(x) n Z ω(x)x s/2 dx x e n2 πx This is one of Jacobi s theta functions. Its transformation law is a consequence of the Poisson summation formula: if f : R C is a Schwartz

function (infinitely differentiable and x a f (b) (x) for all b, a R), we have the Fourier transform ˆf(y) f(x)e 2πixy dx, R and then f(n) ˆf(n). n Z n Z The idea now is to observe that f(x) e πx2 is its own Fourier transform: ˆf(y) e πx2 2πixy dx R e π(x+iy)2 +y 2 dx R f(y) Therefore or so that θ(x) n Z f(n x) ( ) n f θ(/x) x x x n Z θ(/x) xθ(x) ω(/x) 2 + 2 θ(/x) 2 + x 2 θ(x) 2 + x 2 + xω(x) 2

Now apply to the Λ-function: Λ(s) ω(x)x s/2 dx x ω(x)x s/2 dx x + ω(x)x s/2 dx x ω(/x)x s/2 dx x + ω(x)x s/2 dx x [ 2 x s/2 + 2 x s/2 /2 + ω(x)x s 2 + ω(x)x s 2 s( s) + ( x s/2 + x ( s)/2) ω(x) dx x ] dx A priori this is valid only for Rs >, but the integral converges for all s. Consequently Λ(s) is meromorphic for all s with simple poles at and. What happened? We constructed a function which was its own Fourier transform, summed it over a discrete group Z and took its Mellin transform; Poisson summation was applied to get the F. 2 The local story 2. Generalities on locally compact groups Let G be a locally compact abelian group. The dual group Ĝ is another such thing. Its elements are characters c: G S. A sequence of such cs converges if it convergers uniformly on compact subsets of G. G has a Haar measure dx, and this is unique up to scaling. For the moment, if y Ĝ, write x, y for y(x). Given an integrable function f, we can form the Fourier transform ˆf(y) f(x) x, y dx x G with respect to dx; this is a function on Ĝ. inversion formula: ˆf(x) γf(x ) There is generally a Fourier for some γ which does not depend on x. One can choose dx so that γ ; this is called a self-dual measure. The Fourier transform puts into bijection (suitable) functions on G and on Ĝ. If G Z then Ĝ S. If G R or C then Ĝ G. 3

2.2 Harmonic analysis on local fields In fact the additive group of a local field k is always dual to itself. We choose special isomorphisms k ˆk as follows, for k Q p or k R. If k R, let x ψ x be the isomorphism G Ĝ with ψ x (y) e 2πixy If k Q p, let x ψ x be the isomorphism G Ĝ with ψ x (y) e 2πi{xy} where {z} Z[/p]/Z is the fractional part of z Q p ; that is, z {z} Z p. In each case write ψ ψ. (Neat property: If y Q, then the product of ψ(y) over the ψ coming from each Q v is.) If G is the additive group of a finite extension k/q v, put ψ k (y) ψ Qv (Tr y). Let µ dx be the self-dual Haar measure on k. On Q p, we have Z p dx. For k/q p we have O k dx (ND) /2, where D k is the different ideal (the orthogonal complement to O k under x, y ψ k (xy)). We also choose a Haar measure d x on k. This is dx/ x if k is archimedean, and is normalized to satisfy d x (ND) /2 if k is non-archimedean. O k 2.3 Quasi-characters These are continuous homomorphisms c: k C. They are called characters if c(k ) S. Crucially, tate views the set of quasi-characters on k as a complex manifold (albeit generally one with infinitely many components). Say c is unramified if it vanishes on { α }. If K is nonarchimedean then the unramified characters are of the form c(α) α s, where s is only well-defined up to 2πi/ log q. Such a guy can be factored as c(π) χ(α) α σ, where χ(α) α s σ is a character. If K R or C the unramified characters are c(α) α s. In all cases σ Rs is well-defined. We call it the real part of c. Two quasicharacters are equivalent if they differ by an unramified character. We view each equivalence class of qcs as a complex manifold via the above. 4

2.4 The local zeta function In Tate s theory, one attaches a zeta function to a pair (f, c), where c is a quasi-character and f is a test function. Tate considers a class of functions having the property that. f and ˆf are continuous and L. 2. f(α) α σ are integrable on k for σ >. Similarly for ˆf. Given such an f and a quasi-character c, Tate defines ζ(f, c) f(α)c(α)d α k This is an analytic function of c for Rc >. xample: if k Q p, f is the char function of Z p, and c is unramified of positive real part, so that c(ϖ) <, we get ζ(f, c) c(α)d α n n Z p c(p) n p n Z p p n Z p d α c(p) n ( c(π)) If we write c(α) χ(α) α σ this becomes ζ(f, χ σ ) χ(p) p σ (Note this was the result of integrating c against the self-dual function.) On the other hand if c is ramified of conductor p f then ζ(f, c) c(α)d α Z p c(α)d α n p n Z p p n Z p c(p) n c(α)d α n Z p 5

If k R, let f(x) e πx2. Let c(x) x s be unramified. Then ζ(f, c) x s dx x R e πx2 π s/2 Γ(s/2). 3 Local functional equation For a quasi-character c, put ĉ(α) α c(α). Thus if c(α) α s, then ĉ(α) α s. In particular Rĉ Rc. If f z and < Rc <, then ζ(f, c) and ζ( ˆf, ĉ) both converge. Tate s local functional equation is: ζ(f, c) extends to a meromorphic function on all c. There exists a meromorphic ρ(c) with ζ(f, c) ρ(c)ζ( ˆf, ĉ) The proof combines two computations: One is that for < Rc <, ζ(f, c)ζ(ĝ, ĉ) ζ( ˆf, ĉ)ζ(g, c). This is a formal calculation. We have ζ(f, c)ζ(ĝ, ĉ) f(α)ĝ(β)c(αβ ) β d αd β Now substitute (α, β) (α, αβ): ζ(f, c)ζ(ĝ, ĉ) f(α)ĝ(αβ)c(β ) αβ d αd β ( ) f(α)ĝ(αβ) α d α c(β ) β d β Remember that d α dα/ α. The inner integral is equal to f(α)g(γ)ψ(αβγ)dαdβ, and this is symmetric in f and g. Thus ρ(c) ζ(f, c)/ζ( ˆf, ĉ) is independent of f so long as the denominator is nonzero. The second computation the evaluation of this quotient for a suitable f (relative to c) which makes the denominator nonzero. Tate shows that the 6

quotient is always a meromorphic function of all c (which he makes explicit in terms of familiar functions). Therefore ζ(f, c) can be defined even for Rc <, as the product ρ(c)ζ( ˆf, ĉ) (because Rĉ Rĉ > > ). If c(α) α s, let L(s) ζ(f, c), where f is the self-dual function. If k Q p, then L(s) α s d α Z p p s (converges for Rs >, but has obvious analytic continuation). Thus ρ(c) ζ(f, c) ζ(f, ĉ) L(s) L( s) ps p s. If k R and c(α) α s, then f(α) e πα2 so that and L(s) Γ R (s) π s/2 Γ(s/2) ρ(c) Γ R (s)γ R ( s) 2 s π s cos(πs/2)γ(s). (In the p-adic case, if c is ramified, then ρ(c) is a Gauss sum.) 4 Adeles, Ideles 4. Restricted direct products Let G v be a collection of locally compact abelian groups. For almost all v, let H v G v be an open compact subgroup. Let S be the (finite) set of indices for which H v is not defined. For S a finite set of indices containing S, let G S v S G v v S H v This is a topological group under the product topology. (Recall that the topology of a product of spaces v X v is generated by boxes of the form v U v with U v X v open, subject to the restriction that U v X v for almost all v.) The G S form a directed system of topological groups: When S S, 7

there is an inclusion G S G S as an open subgroup. The restricted direct product is the direct limit G Gv lim S G S. Thus an element of G v is an element of one of the G S ; i.e. it is a collection (x v ) of elements x v G i with x v H v for all but finitely many v. A subset U G v is open if and only if U G S is open for all S. G is locally compact, because a compact neighborhood of in any G S serves as a compact neighborhood of in G. A sequence converges to an element in G S if and only if almost all elements of the sequence lie in G S, and that part of the sequence converges in G S. A (quasi-)character χ of G is one and the same as a collection of (quasi- )characters χ v of G v, subject to the restraint that χ v (H v ) for almost all v. Thus Ĝ is isomorphic as a topological group to the restricted direct product of the Ĝv with respect to the Hv Ĝv. Choose Haar measures dx v on G v for all v such that H v dx v for almost all v. We can then define a Haar measure dx on G which is essentially the product of the dx v. (This is a matter of defining a compatible system of Haar measures dx S on the G S. The G S are a compact group times a finite product of the G v for v S, so this is unproblematic.) In the end, if f is an integrable function on G then f(x)dx lim f(x)dx. S G S 4.2 Adeles and ideles G Let K be a number field or function field. The adeles A K are the restricted direct product of the K v with respect to the O v K v (for v nonarchimedean), and K A K is an embedding of K as a discrete closed subring of A K. The ideles A K are the restricted direct product of the K v with respect to the O K v K v. K embeds in A K as a discrete closed subgroup. It so happens that the abstract group A K is the same as the multiplicative group of the ring A K, but it topology is not the same as that induced on it as a subset of A K. Lang writes J for the idele group. 8

There is an absolute value map : J/K R > given by (α v ) v α v. Let J be the kernel. The quotient J /K is compact. 4.3 Harmonic analysis on the adele group We have a collection of additive characters ψ v on K v for all v, and these have the property that ψ v (O v ) for almost all v, and ψ v (a) v for any a K. Under the self-duality K v ˆKv, we have that Ov Dv, where D v is the different. Thus the dual of A K is the restricted direct product of the K v with respect to the Dv, which is just A K again! If f is a rapidly decreasing function on A K, then as usual its Fourier transform is ˆf(y) f(x)ψ(xy)dx A K where dx is the self-dual Haar measure on A K with respect to ψ. 4.4 Poisson summation, Riemann-Roch formula This is a phenomenon that occurs for a pair (V, L), where V is a locally compact commutative ring together with a character ψ which induces a selfduality V ˆV, and L V is a discrete subring which is co-compact and satisfies L L. Such is the story with (V, L) (R n, Z n ) or (V, L) (A K, K). The final ingredient is a fundamental domain D for V/L. A key observation is that our ψ identifies the dual of V/L with L. Theorem 4. (Poisson summation). Let f be a continuous integrable function on V such that. x L f(x + v) is uniformly convergent for all v V. 2. x L ˆf(x) converges. 9

Then x L ˆf(x) x L f(x) Here s the proof: let φ(v) x L f(x + v), so that φ(v) is a periodic function on V. Such functions are generally not integrable on V (they tend not to be rapidly decreasing!), but φ will be integrable on D, which is compact. Write ˆφ for its Fourier transform (this is to be considered as a function on L): ˆφ(y) φ(v)ψ(vy)dv For y L, this is ˆφ(y) D f(x + v)ψ(vy)dy D x L f(v)ψ(vy)dy x L D+x f(v)ψ(vy)dy V ˆf(y) Here we have used the fact that y L, so every xy L, so the change of variable we did on the second line doesn t affect the ψ(vy) factor. Our first condition on f guarantees that the Fourier inversion formula applies to φ. It reads: φ() ˆφ() φ(y) ˆ ˆf(y) y L y L which is exactly what we set out to prove. The Riemann-Roch theorem is a consequence of the Poisson formula in the context where V is a ring. Theorem 4.2 (Riemann-Roch theorem). Let f be a continuous integrable function on A K, satisfying the conditions:. For all a A K, x A K, x K f(a(x + v)) converges uniformly for v A K.

2. x K ˆf(av) is convergent for all a A K. Then for all a A K : a a K ( v ˆf a) f(av) a K To prove the theorem, one applies Poisson summation to g(x) f(ax), noting that ĝ(y) f(ax)ψ(xy)dx A K f(x)ψ(xy/a)dx a A K a ˆf ( y ) a 5 The global functional equation 5. Hecke characters Recall that J A K and J J is the kernel of the absolute value map J R >. A Hecke character χ is a quasi-character of J/K. Since J /K is compact, the restriction of χ to J /K is unitary. Thus χ(a) only depends on a modulo J, which is to say that χ(a) only depends on a. There must be some σ R with χ(a) a σ ; this σ is called the real part of χ. This might be a good time to classify Hecke characters in the case of K Q. The short story is that the finite-order Hecke characters are essentially the primitive Dirichlet characters, and that any Hecke character is of the form χ s, where s C. First, notice we have an isomorphism J/Q Ẑ R >. Given any idele a (a v ), one can find a rational number b Q with a v v b v for all finite places v (this is because Q has class number one), and furthermore b can be chosen to have the same sign as a. Then b a Ẑ R > J shall be the image of a. The map is well defined because

the intersection of Q with Ẑ R > is {}. The inverse is the inclusion Ẑ R > J followed by the quotient J J/Q. Now suppose ψ is a primitive Dirichlet character on (Z/NZ). We can then define a Hecke character χ of finite order on J/Q via J/Q Ẑ R > Ẑ (Z/NZ) C, where the last map is χ. One should work through the details of how χ works locally. If χ v is the restriction of χ to Q v, then (for instance) χ is the unique character of R with χ ( ) ψ( mod N). Here s another example: if p is a finite prime, let N be the largest divisor of N which is relatively prime to p. Let p be an integer with p p (mod N ) and p (mod p). Then χ p (p) ψ(p ). 5.2 Global zeta function Tate defines a class z of functions f on A K which have the following nice properties:. f and ˆf are continuous and integrable. 2. The average x K f(a(x + v)) converges uniformly as a function of x D and a in any compact subset of J. Similarly for ˆf 3. a f(a) a σ is integrable on I for all σ >. For such f, and for a quasi-character c of J/K, Tate defines the zeta function ζ(f, c) f(a)c(a)da J This is convergent as long as the real part of c is >. Just as in the local theory, one divide quasi-characters c of J/K into equivalence classes, each class being parameterized by a complex variable s. It then makes sense to ask whether ζ(f, c) can be analytically continued to all s. Theorem 5.. The zeta function ζ(f, c) has an analytic continuation to quasi-characters. The only poles are at c(a) and c(a) a ; these are 2

simple and the poles are κf() and κ ˆf(), respectively. We have the functional equation ζ(f, c) ζ( ˆf, ĉ). Here κ is the volume of a fundamental domain for J/k : κ 2r (2π) r 2 hr dk w Proof. We have the absolute value map J R >. It is helpful to provide a splitting for this map, R > J, for instance as t (t,,,... ) (where the first slot is one of the archimedean places), so that J R > J. Choose a measure d b for J which satisfies d a (dt/t)d b. (This is the measure inherent in the definition of κ above. We won t be going into the evaluation of κ; this is quite analogous to the derivation of the class number formula for the Dedekind zeta function.) Then ζ(f, c) [ ] dt f(tb)c(tb)d b J t ζ t (f, c) dt t, say. Now use the fact that J /K is compact. Let be a fundamental domain for this quotient. We may write ζ t (f, c) f(tb)c(tb)d b x K x K x f(xtb)c(tb)d b [ ] f(xtb) c(tb)d b f() c(tb)d b x K [ ( x ) ] ˆf c(tb)d b f() c(tb)d b tb tb x K [ ( ) ] xb ˆf ĉ t x K ζ /t ( ˆf, ĉ) f() 3 ( ) b d b f() t c(tb)d b + ˆf() ĉ c(tb)d b ( ) b d b t

As for the remaining integrals: they are zero unless c is trivial on J, which is to say that c(a) a s for some s. If that is the case then c(tb)d b κt s. Plugging this all in to our original expression for ζ(f, c) gives: ζ(f, c) ζ t (f, c) dt t + ζ t (f, c) dt t + ζ t (f, c) dt t + ζ t (f, c) dt t ζ /t ( ˆf, ĉ) dt t + κ ζ t ( ˆf, ĉ) dt t + κ { } ˆf() s f() s { } ˆf() s f() s where the expression in braces means that it is only included when c(a) a s. Recall that the defining integral for ζ(f, c) converged for Rc >. So of course the first integral converges on that region as well. But the integrand in the first integral gets smaller as Rc decreases, so that in fact the first integral converges for all c, period. The same argument holds for the second integral: this integral converges a priori for Rĉ >, but all the better for Rĉ smaller than this. Therefore the whole expression converges for all c, save for the poles at s and s when c(j ). Finally, the expression is unchanged when the pair (f, c) is replaced by ( ˆf, ĉ). 5.3 The finale Let χ be a character of J/K. Tate now applies his global functional equation to a special function f, relative to χ, which is essentially the product of his special f v relative to the components χ v of χ. An important point is that if χ is unramified at the finite place v, which is the case for almost all v, then f v is simply the characteristic function of O v, in which case the local zeta function is (up to a factor ND /2 v ζ(f v, χ v s ) which only appears finitely many times) ( χ(π ) v) Nπv s The Hecke L-function L(s, χ) is the product of the above factors over all unramified places v. 4

The zeta function of Tate s f differs from L(s, χ) at finitely many factors: ζ(f, χ s ) v S ζ(f v, χ v s v ) v S ND /2 v L(s, χ) Here S is the set of ramified places of χ, together with the infinite places. Tate s main theorem on the global zeta function will now imply an analytic continuation and functional equation for L(s, χ); the huge advantage is that the constant appearing in the functional equation can be factored as a product of local factors, derived from the quantities ρ(c) appearing in the local theory. 5