The determinant and the discriminant
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2 ( R) carries a PGL 2 ( R)-invariant measure (the quotient on Haar measures of PGL
2 ( R) and PSO 2 ( R)) unique up to scalar; we denote it by µ M 0,(1)
2 . Also
M 0,(1)
2 ( R) has two connected components, namely the two PGL + 2 ( R)-orbits of ± ✓ 0 1 1 0 ◆ ; these components are interchanged by conjugation of any matrix of determinant 1.
38 2. THE DETERMINANT AND THE DISCRIMINANT Corollary 2.1. Given any m 2 M
0,(1) 2 ( R), the set {⇢ g,g (m), g 2 M
(n) 2 ( Z)} becomes equidistributed on the connected component of M 0,(1) 2
R) containing m w.r.t. the measure µ M 0,(1)
2 as n
! +1. In other terms, for ' 1 , ' 2 continuous functions, compactly supported on this connected component and such that µ M 0,(1) 2 (' 2 ) 6= 0, one has P g
(n) 2 ( Z) ' 1 (⇢ g (m)) P g 2M (n) 2 ( Z) ' 2 (⇢ g (m)) ! µ M 0,(1) 2 (' 1 ) µ M 0,(1)
2 (' 2 ) , n
! 1. More precisely there exist > 0 depending on the choice of the Haar measure µ M 0,(1) 2 such that for any ' 2 C c (M 0,(1) 2 ( R)) X g n 2M (n) 2 ( Z) '(⇢ g n ,g n (m)) = (µ M 0,(1)
2 (') + o(1)) |SL 2
Z)\M (n)
2 ( Z)|, n ! +1. Proof. Let H denote the stabilizer of m in G = PSL 2 ( R); this is a compact subgroup of G conjugate to PSO 2 (
of M 0, ±1 2 ( R) containing m is homeomorphic to G/H via the map gH 2 G/H 7! g.m and the (restriction of) the measure µ M 0,(1) 2 on this component is the quo- tient measure, µ G/H
. Since any compactly supported function on G/H may be identified with a compactly supported function on G which is right H- invariant, the result now follows. ⇤ 2.1.1. Equidistribution on two-sheeted hyperboloid. We can now visualize the equidistribution of the {⇢ g (m), g 2 M
(n) 2 ( Z)} by identifying M 0,(1)
2 ( R) with the a ne variety V 1 ( R) = {(a, b, c) 2 R 3 , ac
b 2 = 1 }, via the map (a, b, c) 7! ✓ b a c b ◆ . 3. The discriminant We consider now the ternary quadratic form: disc(a, b, c) = b 2 4ac, to be called the discriminant as it corresponds to the discriminant of the binary quadratic form f a,b,c
(X, Y ) = aX 2 + bXY + cY 2 4. REPRESENTATIONS BY THE DISCRIMINANT 39 Figure 1. n = 6632, (a, b, c) = (1, 0, 1) or in fancier terms, one has an isometry of the quadratic spaces ( Q 3 , disc)
' (Sym 2 ( Q), disc), the space of 2 ⇥ 2 binary quadratic forms enquiped with the discriminant. Another interesting isometry is the following (3.1) (Sym
2 ( Q), disc) ' (M 0 2 ( Q), det)
aX 2 + bXY + cY 2 7! ✓ b 2c 2a b ◆ . Thus SO disc
' SO M 0,(1) 2 ' PGL
2 and the action of PGL 2 on the space of bi- nary quadratic form intertwining with conjugation on M 0 2 is given explicitly for g =
✓ u v
w z ◆ by (3.2) g.f (X, Y ) = det(g) 1 f (uX + wY, vX + zY ) = det(g) 1 f ((X, Y )g), or if we represent the quadratic form aX 2 + bXY + cY 2 by the symmetric matrix ✓
b/2 b/2
c ◆ , the intertwining actions are g ✓ b 2c 2a b ◆ g 1 ! 1 det(g) g ✓ a b/2 b/2
c ◆ t g. We are therefore essentially reduced to the study of the traceless integral matrices (with even entries on the anti-diagonal) of given discriminant. 4. Representations by the discriminant For the discriminant quadratic form, the existence of representations is easy:
40 2. THE DETERMINANT AND THE DISCRIMINANT Proposition 4.1. An integer n is represented by disc (ie. n is a dis- criminant) if and only if n ⌘ 0, 1(mod 4). Moreover, the number of such representations is infinite. Proof. Necessity is evident since 0, 1 (mod 4) are exactly the squares in Z/4Z. Conversely, if d ⌘ 0, 1 (mod 4), then d = b 2 + 4a for b = 0 or 1 and disc(a, b, 1) = d. Moreover for any integer k, (a k k 2 , b + 2k, 1) is another solution. ⇤ From now on, we valid change notations and replace the letter “n” by “d” (for “discriminant”). We denote by R disc
(d) the representation of d by the discriminant quadratic form and by R ⇤ disc
(d) the set of primitive repre- sentations (ie. such that (a, b, c) = 1). It follows from the explicit action of PGL 2
binary forms Sym 2 ( Z) is stable by PGL 2 ( Z), thus PGL 2 ( Z) act on R disc
(d) and on R
⇤ disc
(d). While these sets are infinite, the set of PGL 2 ( Z)-orbits is finite: this is the content of Gauss reduction theory: Theorem 4.1 (Gauss). The set of primitive orbits PGL 2 ( Z)\R ⇤ disc (d) is finite.
Proof. Specifically Gauss proved (using the fact that SL 2 ( Z) is gener- ated by the matrices ✓ 1 1
0 1 ◆ and ✓ 0 1 1 0 ◆ ) that any such orbit has a representative (a, b, c) such that ( |d 1/2 2 |c|| < b < d 1/2 if d > 0
0 |b| |a| |c| if d < 0 . ⇤ By the discussion in §5 of the previous chapter (PSL 2 (
in PGL 2 ( Z)), it follows from Dirichlet class number formula and Siegel’s theorem that Theorem 4.2. Given d a discriminant, one has |PGL
2 ( Z)\R ⇤ disc
(d) | = |d|
1/2+o(1) |PGL
2 ( Z)\R disc (d)
| = |d| 1/2+o(1)
4.1. Discriminant and quadratic fields. The representations of in- tegers by the dicriminant are closely related to quadratic fields: we discuss this relation in details in the present section. Let d be a discriminant which is not a perfect square; let (a, b, c) 2 R
disc (d) be a primitive representation, and let (4.1) m = m
a,b,c = ✓ b 2c 2a b ◆ by the trace zero matrix associated to it via the map (3.1), since m 2 = dId 4. REPRESENTATIONS BY THE DISCRIMINANT 41 this defines an embedding of the quadratic field (d is not a square) K = Q( p d) into M 2 ( Q) ◆ m : K 7! M 2 ( Q) u + v p n 7! uId + v.m Let
O m := M 2 ( Z) \ ◆ m (K)
be the order associated with m, one has ◆ 1 m (O m ) = O d = Z[ d +
p d 2 ] is the order of discriminant d. In other terms ◆ m is an optimal embedding of O d into M 2 ( Z). ◆ m (u + v d + p d 2 ) =
✓ u + v
d+b 2 av cv u + v
d b 2 ◆ Since d ⌘ b(2), it is clear that ◆ m (O
) ⇢ M
2 ( Z); conversely if ◆ m (u + v
d+ p d 2 ) belongs to M 2 ( Z) one has v 2 1 (a, c) Z, u + v
d b 2 2 Z, v 2 1 b Z and by primitivity v 2 Z, from which follows that u 2 Z (since d b
2 2 Z).
Proposition 4.2. The above defines a bijection between Primitive representations up to sign: ±(a, b, c) 2 R ⇤ disc (d)/ {±1}
and Optimal embeddings ◆ : O d ,
2 ( Z). The group PGL 2 ( Z) acts on both sides (by conjugation on the set of optimal embeddings) and the above bijection induces a bijection between the corresponding orbits: PGL 2 ( Z)\R ⇤ disc (d) and the GL 2 ( Z)-conjugacy classes of optimal embeddings. Finally, we have the following Proposition 4.3. There is a bijection between The GL 2
Z)-conjugacy classes of optimal embeddings of O d and the ideal class group Pic(O d ) = {[I] = K ⇥ .I, I ⇢ K a proper O d -ideal }. Proof. Given a proper O d -ideal I
⇢ K, one choose a Z-basis I = Z.↵ + Z. which give an identification ✓ : I
Z 2 u↵ + v 7! (u, v) 42 2. THE DETERMINANT AND THE DISCRIMINANT This identification induces the embedding ◆ : K ,
! M 2 ( Q) defined by ◆( )(u, v) = ✓( .(u↵ + v )), (or in other terms, such that ✓( .x) = ◆( )✓(x)). Since O d
⇢ I, one has ◆(O d ) Z 2 ⇢ Z 2 , that is ◆(O d )
2 ( Z) and the fact that I is a proper O d -ideal is equivalent to the fact that ◆ is an optimal embedding of O d . If we replace the Z-basis (↵, ) by another basis, then (↵ 0
0 ) = (u↵ + v , w↵ + z ) with ✓
w z ◆ 2 GL 2 ( Z) and one see that ◆ is replaced by a GL 2 ( Z)- conjugate. Finally if I is replaced by an ideal in the same class I 0 = .I 2 K ⇥ , then one check esily that the corresponding GL 2 ( Z)-conjugacy classes coincide: [◆ I 0 ] = [◆ I ]. The inverse of the map [I]
7! [◆ I ] is as follows: given ◆ : K 7! M
2 ( Q) an optimal embedding of O d , let e
1 = (1, 0) 2 Z 2 be the first vector of the canonical basis 1 of Z 2 , the map ✓ : K
Q 2 7! ◆( ).e 1 is an isomorphism of Q-vector spaces; let I = ✓ 1 ( Z 2 ), this is a lattice in K which is invariant under multiplication by O d : I is an O d -ideal and it being proper is equivalent to ◆ being optimal. ⇤ 5. Equidistribution of representations To investigate the distribution of representations, we proceed as before and introduce the a ne varieties of level ±1 V
±1 ( R) = {(a, b, c) 2 R 3 , b
2 4ac =
±1}. Given d a non zero discriminant, we may consider the projection of R disc (d)
on the variety of level ±1 = sign(d): |d| 1/2
R disc
(d) ⇢ V
disc, ±1 ( R). Observe that V disc, 1 (
1 ( R) in §2.1.1. V disc,1
( R) is a one sheeted (ie. connected) hyperboloid and V disc, 1 (
two sheeted hyperboloid (the two components being determined by the sign of a)
By Witt’s theorem, both are acted on transitively by the orthogonal group SO
disc ( R) ' PGL 2 ( R) and therefore one has the identification V disc,
±1 ( R) ' SO disc ( R)/ SO disc ( R) x ±1 1 we could have choosen any primitive vector in Z 2 6. TRANSITION TO LOCALLY HOMOGENEOUS SPACES 43 for some choice of point x ±1 2 V
disc, ±1 ( R) with stabilizer SO disc
( R) x ±1 . Because of this V disc, ±1 ( R) admit a natural SO disc
( R)-invariant measure (a quotient of Haar measures -cf. Chap. ??-) well defined up to positive scalars, µ disc, ±
for ⌦ ⇢ V
disc, ±1 ( R) an open subset, let C(⌦) = {r.x, x 2 ⌦, r 2 [0, 1]} be the solid angle supported by ⌦, then µ disc, ± (⌦) = µ
R 3 ( C(⌦)) were µ
R 3 is the Lebesgue measure. One has then the following equidistribution statement: Theorem 5.1. As d ! 1, |d| 1/2
R disc
(d) becomes equidistributed on V disc, ±1 ( R) (±1 = sign(d)) w.r.t. µ disc, ±1 in the following sense: for ' 1 , '
2 2 C c (V disc, ±1 ( R)) such that µ disc . ±1 (' 2 ) 6= 0, then P x 2R disc (d)
' 1 ( |d| 1/2
x) P x 2R disc
(d) ' 2 ( |d|
1/2 x))
! µ disc, ±1 (' 1 ) µ disc, ±1 (' 2 ) , d
! 1. More precisely, there is a positive constant > 0 depending only on the choice of the measure µ disc, ±1
2 C c (V disc, ±1 ( R)), (5.1)
X x 2R disc (d)
'( |d|
1/2 x) = (µ
disc, ±1 (') + o(1)) |d| 1/2+o(1)
. 6. Transition to locally homogeneous spaces The starting point of the proof is the group theoretic interpretation of the problem. Let Q denote the quadratic form disc. By Witt’s theorem, the varieties V Q, ±1 ( R) are acted on transitively by the orthogonal group SO Q ( R) ' PGL 2 ( R) =: G; so the choice of some point x 0 = (a
0 , b
0 , c
0 ) 2 V Q, ±1 ( R), induces an homeo- morphism
(6.1) V Q, ±1 ( R) = G.x 0 ' G/H
where H := Stab x 0 (G) denote the stabilizer of x 0 . To be specific, we will take x 0 = (0, 1, 0) in the +1 case and x 0 = (1/2, 0, 1/2) in the 1 case; under the identification (3.1) correspond to the choice of the matrices m 0 = ✓ 1 0 0 1 ◆ , and m 0 = ✓ 0 1 1 0 ◆ so that H is is either - the split torus A = diag 2 ( R) ⇥ / R ⇥ .Id (ie. the image of the diagonal matrices in PGL 2 ( R)), - the non-split torus K := PSO 2 (
2 ( R)/{±Id}. 44 2. THE DETERMINANT AND THE DISCRIMINANT The choice of Haar measures µ G , µ H on (the unimodular group) G and H then determine a left G-invariant quotient measure µ G/H on G/H ' V
Q, ±1 ( R); that measure correspond to (a positive multiple of) µ Q, ±1 . 6.1. A duality principle. It follows from the previous discussion that each representation (a, b, c) 2 R
Q (d), or its projection |d| 1/2
(a, b, c) 2 V Q, ±1 ( R) is identified with some class g a,b,c
H/H 2 G/H or what is the same to an orbit g a,b,c
H ⇢ G for some g a,b,c 2 G such that g a,b,c
x 0 = |d| 1/2
(a, b, c). Let
= PGL 2 ( Z); as we have seen R Q (d) decomposes into a finite disjoint union of -orbits; we denote by [R Q (n)] = \R Q (d) the set of such orbits and by [a, b, c] = \ (a, b, c) 2 [R Q (d)];
one has R Q (d) = G [a,b,c] 2[R Q (d)] .(a, b, c) and (6.1) identifies |d| 1/2
.R Q (d) with G [a,b,c]
2[R Q (d)] g a,b,c
H/H ⇢ G/H;
thus the problem of the distribution of |d|
1/2 .R Q (d) inside V Q, ±1 ( R) is a
problem about the distribution of a collection of -orbits inside the quotient space G/H. We note the tautological equivalence (6.2)
gH/H ! gH ! \ gH, between (left) -orbits on G/H and (right) H-orbits on \G. From this equivalence and the previous identification, one could expect that studying the distribution of |d|
1/2 .R Q (d) inside V Q, ±1 ( R) is tantamount to studying the distribution of some collection of right-H orbits, indexed by [R Q (d)] inside the homogeneous space \G namely Y d
[ [a,b,c]
2[R Q (d)] x [a,b,c]
H ⇢ \G
with x [a,b,c]
= \ g
a,b,c .
6. TRANSITION TO LOCALLY HOMOGENEOUS SPACES 45 6.2. The shape of orbits. Let us describe more precisely the structure Download 398.7 Kb. Do'stlaringiz bilan baham: 
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