ZETA FUNCTIONS ATTACHED TO PREHOMOGENEOUS VECTOR SPACES. Disc(g x) = (det g) 2 Disc x. Stab(x) 1 Disc(x) s. Aut R 1 Disc R s

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1 ZETA FUNCTIONS ATTACHED TO PEHOMOGENEOUS VECTO SPACES TAKASHI TANIGUCHI. Introduction Let G GL 2 and let V be the space of binary cubic forms. G acts on V by the action (g x)(u, v) x((u, v)g), det g where x V and g G. The discriminants are related by Definition. (Shintani). for s >. Disc(g x) (det g) 2 Disc x. ξ ± (s) x G Z \V Z ±Disc(x)> is a cubic ring ± Disc() Stab(x) Disc(x) s Aut Disc s Theorem.2. () (Analytic Continuation; AC) ξ ± (s) has an analytic continuation to C. In particular, (s ) 2 (s 5 6 )(s 7 6 )ξ± (s) is entire. (2) (Principal Part; PP) Indeed, ξ ± (s) is holomorphic except for simple poles at s, 5 6, with explicit residue formulas. (3) (Functional Equation; FE)There is a functional equation between ξ ± ( s) and ˆξ ± (s), where ˆξ is represented to the dual representation. Proof. () and (3) follow from general theory of prehomogeneous vector spaces. For (2), we need some careful analysis. The goal of this lecture is to give an overview of the proof of this theorem. An application (cf. Thorne s talk): let ξ ± (s) a ± n. Then an example of something n s that can be proved: a ±n r ± X + r ± x O(X ε), 6 <n<x 6 and in 975, Shintani was able to separate contributions of irreducible and reducible forms from ξ ±.

2 2. A proof of AC and FE for ξ Let f S(), where S() denotes smooth functions on of rapid decay. Its Fourier transform is defined by ˆf(y) f(x)e 2πixy dx. Then Poisson summation formula gives We claim that for t, we have x Z f(x) y Z x Z f(tx) t y Z ˆf(y) ˆf(t y). The proof is simple: Let f t (x) f(tx). Then ˆf t (y) t ˆf(t y), to which we apply the Poisson summation formula. Definition 2. (Local zeta function). Let f S() and s C. We define Φ(f, s) x s f(x)dx, which is holomorphic if (s) >. Proposition 2.2. () Φ(f, s) has a meromorphic continuation to C, and Φ(f, s) is entire; Γ(s) (2) Φ( ˆf, s) c(s)φ(f, s), where c(s) : (2π) s Γ(s)(e iπs 2 + e iπs 2 ). We have ζ(s)φ(f, s) x s f(x) dx n s n x x n n s f(x) dx x x nx x s f(nx) dx x n x x x s f(nx) dx x, n Z\ where the last equality follows from writing +. Let Z(f, s) t s f(tx)d t and Z + (f, s) where d t denotes dt t, for s >. 2 x Z\ t s x Z\ f(tx)d t,

3 Lemma 2.3. Z + (f, s) is entire. Proof. Since t, the convergence is better when s is smaller. Now, consider Z(f, s) Z + (f, s) That is, we have t s x Z\ f(tx)d t t s (t y Z\ ˆf(t y) + t ˆf() f())d t t s y Z\ ˆf(t x)d t + ˆf() Z + ( ˆf, s) + s + s. Z(f, s) Z + (f, s) + Z + ( ˆf, s) + ˆf() s + ˆf() s by using the substitution (f, s) ( ˆf, s). The functional equation comes from the identities and Z(f, s) ζ( s)φ(f, s) Z( ˆf, s) ζ(s)φ( ˆf, s) ζ(s)c(s)φ(f, s) t s d t f() Z( ˆf, s) t s d t and equating them, where the final equality comes from Proposition 2.2. Now we consider the residues: es s Z(f, s) ˆf() f(x)dx, where Z(f, s) ζ(s)φ(f, s) and Φ(f, ) x f(x)dx f(x)dx, from which we get that es s ζ(s). Proof of Proposition 2.2. () Γ(s) Φ (f, s) Γ(s) int. by parts Γ(s + ) x s f(x)dx sγ(s) ( [x s f(x)] Γ(s + ) Φ (f, s + ) the last of which is holomorphic for s > n. 3 (x s ) f(x)dx ) x s f (x)dx ( )n Γ(s + n) Φ(f (n), s + n),

4 (2) which implies that Φ(f t, s) x s f(tx)dx x t x t s Φ(f, s), Φ( ˆf t, s) t Φ( ˆ f t, s) t s Φ( ˆf, s) By the uniqueness of relative invariant distribution on homogeneous spaces, the maps must coincide up to a constant c c(s). f Φ( ˆf, s), f Φ(f, s) There is a simpler proof of FE using a clever choice of f: Let f Cc ( ), and f : d f dx f. Then ˆf(y) ˆf (y) y ˆf (y), which implies that f() ˆf(). Then Poisson summation yields from which we get x Z\ f(x) y Z\ ˆf(y) Z(f, s) Z + (f, s) + Z + ( ˆf, s), so it is entire. So we get AC from the expression Z(f, s) ζ(s)φ(f, s) ζ(s)(s )Φ(f, s ). A question arises: can we generalize Proposition 2.2? Definition 2.4 (M. Sato). Let (G, V ) be a finite dimensional representation of an algebraic group over k. It is a prehomogeneous vector space if there exists x V k such that G k x V k is Zariski open. Further P k[v] is a relative invariant polynomial if for all g G, x V with χ : G G m. Sato showed that for P [V ], Φ () (f, s) P (gx) χ(g)p (x), V () P (x) s f(x)dx has AC and FE. Sato-Shintani showed that there exists a zeta function associated to (G, V ). Example 2.5. Examples of prehomogeneous vector spaces: () G GL acting on V A P (x) x, so ζ(s) is the associated zeta function. (2) (G, V ) representing the binary cubics P (x) Disc(x), from which we get ξ ± (s) is the associated zeta function. 4

5 3. Shintani zeta function Let G GL 2, and V the vector space representing binary cubics. prehomogeneous vector space, with P (x) Disc(x). Let V {x V : P (x) }, Then (G, V ) is a i.e. the set of x in V with no multiple roots in P. (Let S {x V : P (x) }). While V C has a single G C -orbit, V admits a G -orbit decomposition V V + V, where V + {x V : P (x) > }. The associated local zeta function is given by Φ ± (f, s) P (x) s f(x)dx, which converges when s >, at least. For x, y V, let V ± x, y x y 4 3 x 2y x 3y 2 x 4 y. Then this pairing satisfies gx, g y x, y, where g det g ˆf(y) f(x)e 2πi x,y dx. V Proposition 3.. () (f, s) is entire. Γ(s) 2 Γ(s 6 )Γ(s+ 6 )Φ± (2) There exists some 2 2 matrix M(s) satisfying [ Φ + ( ˆf, s) Φ ( ˆf, s) Proof. There exists Q( ) such that x ] M(s) Q( x )e x,y P (y)e x,y, where b(s) s 2 (s )(s + ). 6 6 Proof of AC and FE for ξ ± (s) follows: Z(f, s) det g 2s G /G Z x G Z /V Z,P (x) x V P (x) [ Φ + (f, s) Φ (f, s) g. Let ]. Q( x )P (x)s b(s)p(x) s, Stab(x) P (x) s f(gx)dg G P (gx) s f(gx)dg, where P (x) depends only on the G -orbit of x. 5

6 emark 3.2. φ(gx)dg m ± dy φ(y) G 2π G x P (y), where m ± is the degree of covering G G x : g gx. We have P (gx) s f(gx)dg m ± P (y) s dy f(y) G 2π G x P (y) Now, for f C c from which f S ˆf S. m ± 2π Φ± (f, s), and Z(f, s) [ ξ + (s) ξ (s) ] [ ] π Φ+ (f, s). π Φ (f, s) (V ), with f Q( )f x, we have ˆf(y) P (y) ˆf (y), The hard part is to analyze the integral Z(f, s) Z + (f, s) Z + ( ˆf, s) det g 2s ( det g 2 ˆf(g y) x V Z Sf(gx))dg G /G Z, det g y VZ S x ( ˆf() 2s 2 f() 2s ) Vol(G /G(Z)) + ( )dg y x The cubic case was computed by Shintani (972), and the quartic case by Yukie (992). emark 3.3. We can impose congruence conditions f(x) e 2πi x,y ˆf(y) N 4 x x +NV Z y N V Z 6