From continued fractions and quadratic functions to modular forms

0.19 MB·English

by  Paloma Bengoechea

From continued fractions and quadratic functions
to modular forms
Paloma Bengoechea
3
January 30, 2013
1
0
2

n
Abstract
a
J

9 In this paper we study certain real functions defined in a very simple way
2
byZagier [Z4]assumsofinfinitepowersofquadraticpolynomialswithinteger

coefficients. These functions give the even parts of the period polynomials of
]
T
themodularformswhicharethecoefficients inFourierexpansionofthekernel
N
function for Shimura-Shintani correspondence. We prove two conjectures of
.
h Zagier showing that the sums converge exponentially. We also prove unex-
at pected results on the representation of these functions as sums over simple
m or reduced quadratic forms and the positive or negative continued fraction of
[ the variable. These arise from more general results on polynomials of even

degree. Especially we give the even part of the Eichler integral on x R of
1
∈
v any cuspform for PSL(2,Z) in terms of theeven partof its periodpolynomial
4
and the continued fraction of x.
2
0
7
.
1 1 Introduction
0
3
1
In [Z4], D. Zagier studied certain functions defined as sums of powers of quadratic
:
v
polynomials with integer coefficients and discovered that these functions have many
i
X surprising properties. For example, for x R, take the sum over all quadratic
∈
r functions Q(X) = aX2+bX+c with integer coefficients and fixed discriminant that
a
are negative at infinity and positive at x. Then one finds (for convenience we write
“Q( ) < 0” to mean that a = a is negative):
Q
∞
Theorem 1.1 Let D be a positive non-square discriminant. Then the sum
A (x) := Q(x) (1.1)
D
discr(Q)=D
X
Q(∞)<0<Q(x)
converges for all x R and has a constant value α independent of x.
D
∈
1
For example, one finds A (x) = 2, and more generally α = 5L( 1,χ ),
5 D D
− −
where L(s,χ ) is the Dirichlet L-series of the character χ (n) = (D/n) (Kronecker
D D
symbol). We denote
= aX2 +bX +c a,b,c Z, b2 4ac > 0, b2 4ac not a square ,
Q | ∈ − −
(cid:8) (cid:9)
and for any positive non-square discriminant D,
= aX2 +bX +c b2 4ac = D ,
D
Q ∈ Q | −
x = Q Q( ) < 0 < Q(x) .
D (cid:8) D (cid:9)
Q h i { ∈ Q | ∞ }
The sum in (1.1) is the special case k = 2 of the function
A (x) := Q(x)k−1 (k 2). (1.2)
k,D
≥
Q∈XQDhxi
Theorem 1.1 is still true for k = 4 with α now replaced by L( 3,χ ), but this fails
D D
−
forevenk 6becauseoftheexistence ofcuspformsofweight 2k onthefullmodular
≥
group. More explicitly, for even k 6, the function A (x) is a linear combination
k,D
≥
of a constant function and the functions n1−4ka (n)cos(2πinx), where f runs
n≥1 f
over the normalized Hecke eigenforms in S (PSL(2,Z)) and a (n) denotes the n-
2k f
P
th Fourier coefficient of f. The function A (x) arose from studying the modular
k,D
form of weight 2k which is the D-th coefficient in Fourier expansion of the kernel
function for the Shimura and Shintani lifts between half-integral and integral weight
cusp forms ([Shim],[Shin]). The function A (x) is (modulo a constant multiple of
k,D
X2k−2 1) the even part of its Eichler integral on R and so gives the even part of
−
its period polynomial ([KoZ1], [KoZ2] and [Z4]).
The convergence of A (x) is not immediate at all. As Zagier observed, one has
k,D
Q(x) = O(1/a )foralltheQoccuringin(1.2)andoneseeseasilythatthereareonly
Q
O(1) quadratic functions Q for each a-value. Therefore the series (1.2) converges at
most like a1−k if k 4. In the case k = 2, however, this argument fails and
a>0 ≥
Zagier could only deduce the convergence of (1.1) from the fact that the function is
P
finite and constant (= 5L( 1,χ )) when x is rational. In fact, if for some value of
D
− −
x the sum diverged, then, since all summands are positive, there would be finitely
many quadratic functions Q whose sum at x already exceeded 5L( 1,χ ), and
D
− −
the sum of their values at a sufficiently nearby rational number would also exceed
5L( 1,χ ).
D
− −
Following this proof, the sum might converge extremely slowly. However, exper-
iments carried out for the value x = 1/π suggested that the series (1.1) (and hence
also the series (1.2)) converge extremely rapidly. More precisely, for x = 1/π and
D = 5, Zagier found experimentally that the elements of x belong to the union
D
Q h i
of two lists, each having values that tend to 0 exponentially quickly. Each quadratic
function Q in each list is obtained from the preceding one by applying a fairly simple
element of PGL(2,Z) giving the positive continued fraction of x. The first functions
2
in the lists correspond to the opposite of the simple forms coming from the reduction
theory ofbinary quadraticformswith fixed discriminant 5: [1, 1, 1]and[1,1, 1].
− − −
The following tables give the first five functions Q, and the corresponding values of
Q(1/π) for each list:
Q Q(1/π)
[ 1,1,1] 1.216989
−
[ 11,7, 1] 0.113636
− −
[ 541,345, 55] 0.00215
− −
[ 117731,74951, 11929] 0.000008
− −
[ 133351,84893, 13511] 0.000008
− −
Sum: 1.332791
Q Q(1/π)
[ 1, 1,1] 0.580369
− −
[ 5,5, 1] 0.084943
− −
[ 409,259, 41] 0.001896
− −
[ 5959340757998441,3793834156817819, 603807459328429] 6.856501 E-17
− −
[ 7755390254828071,493723477865040, 785785320227431] 1.568047 E-18
− −
Sum: 0.667208
Zagier conjectured that all functions in one list and some in the second one occur
in A (x), but he found no criterion to decide which ones. He suggested there is a
5
similar situation in the general case.
We will prove Zagier’s conjectures, giving a criterion for the second list, and
establish a similar result for the general case, obtaining the exponential convergence
for A (x) and a direct proof for the convergence in the case k = 2. We also
k,D
prove other descriptions of A (x) analogous to the one conjectured by Zagier: first
k,D
we replace the simple forms with the reduced forms and the usual algorithm of
positive continued fraction with a different one (with sign + as well). Later we keep
the simple forms but we consider the (usual) negative continued fraction. We give
correspondences between all these situations. This is done in the third section.
In the fourth section we prove an unexpected result: the values at x of the
functions in the lists that do not appear in the sum A (x) cancel each other
k,D
out. Thus the sum over all functions in the lists is equal to A (x). A similar
k,D
phenomenon holds for the description with the negative continued fraction. This is
a consequence of the more general Corollary 4.2 which gives the even part of the
Eichler integral on x R of a cusp form for PSL(2,Z) in terms of the even part of
∈
its period polynomial and the continued fraction of x.
3
2 Reduction theories
In this section we give the main connections between reduction theory of binary
quadratic forms with positive non square discriminant for PSL(2,Z) and the con-
tinued fractions of a real number. We also recall some simple properties of positive
and negative continued fractions and give the non usual algorithm that we use in
the next sections.
We denote
0 1 1 0 0 1 1 1
ε = , σ = − , S = − , T = .
1 0 0 1 1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
The matrices ε,σ,T generate the group Γ = PGL(2,Z) and S,T generate the group
Γ = PSL(2,Z). Clearly
1
ε2 = σ2 = S2 = εσS = 1
so 1,ε,σ,S form a Klein 4-group. We write Γˆ to mean Γ or Γ . The group Γˆ acts
1
{ }
on the set of binary quadratic forms by
r s
Q γ(X,Y) = Q(rX +sY,tX +uY) (γ = Γˆ). (2.1)
| t u ∈
(cid:18) (cid:19)
For a positive non-square fixed discriminant D, a reduction theory of binary
quadratic forms with discriminant D for the action of the group Γ consists in
1
giving a finite system of such forms (called a system of reduced forms) and an
R
algorithm such that:
(i) each form with discriminant D is Γ -equivalent to some element of by
1
R
applying the algorithm a finite number of times;
(ii) the image by the algorithm of an element of still belongs to . In other
R R
words, the elements of form cycles such that two elements of are Γ -equivalent
1
R R
if and only if they belong to the same cycle. In particular, the number of cycles is
the number of Γ -equivalence classes for D.
1
Usually(andinallcasesweconsider)thereductionalgorithmforbinaryquadratic
forms Q = [a,b,c] (:= aX2 + bXY + cY2) is obtained by applying a reduction
algorithm for real numbers (usually some sort of continued fraction algorithm) to
one of the roots of Q(X,1)
b √D b+√D
w = − − , w′ = − ,
Q 2a Q 2a
where √D denotes the positive square root.
We use two different sets of forms [a,b,c]:
R
a > 0, c > 0, b > a+c
4
which we call reduced (see [Z3] for details), and
a > 0 > c
which we call simple.
The bijection [a,b,c] [ a, b, c] exchanges the simple forms with positive
7→ − − −
and negative values of a+b+c (the value 0 cannot occur for non-square D). The
bijection
reduced simple with a+b+c > 0
→
[a,b,c] [a,b 2a,c b+a]
7→ − −
proves thatthereareexactly halfasmanyreduced formsassimple forms. Wedenote
by Red and Sim the sets of quadratic polynomials that correspond to the reduced
and simple forms respectively
Red = Q(X) Y2Q(X/Y)is reduced ,
∈ Q|
Sim =(cid:8) Q(X) Y2Q(X/Y)is simple (cid:9).
∈ Q|
We can translate the inequa(cid:8)lities for the coefficients of simple(cid:9)or reduced forms into
inequalities for the corresponding quadratic irrationalities:
Q(X,Y) reduced w < 1 < w′ < 0,
⇐⇒ Q − Q
Q(X,Y) simple w < 0 < w′ .
⇐⇒ Q Q
The positive and negative continued fraction of a real number x, denoted by
1
x = n + (n Z, n 1 i 1),
0 1 i ∈ i ≥ ∀ ≥
n +
1 1
n +
2 ...
1
x = m (m Z, m 2 i 1),
0 − 1 i ∈ i ≥ ∀ ≥
m
1 − 1
m
2 − ..
.
are also produced by reduction algorithms:
1
x = x, n = x , x = = εT−ni(x ) (i 0), (2.2)
0 i i i+1 i
⌊ ⌋ x n ≥
i i
−
1
x = x, m = x +1, x = = ST−mi(x ) (i 0), (2.3)
0 i i i+1 i
⌈ ⌉ m x ≥
i i
−
where x and x arerespectively thefloorandceilparts. Thepositiveandnegative
⌊ ⌋ ⌈ ⌉
continuedfractionsofw areperiodicforaquadraticformQ. Moreover, thenegative
Q
5
continued fraction of w is purely periodic if and only if Q is reduced. Thus the
Q
cycle of reduced forms that are equivalent to a given form Q corresponds to the
cycle of real quadratic irrationalities x given by the algorithm (2.3) with x = w .
i 0 Q
In a similar way, a quadratic form Q is simple if and only if w is purely periodic
Q
for the algorithm ([ChZ])
x +1 = T(x ) if x 0,
i i i
≤
 x
x0 = x, xi+1 =  1 ix = T−1ST−1(xi) if 0 < xi < 1,
 i
 −

x 1 = T−1(x ) if x 1.
 i − i i ≥



This gives an expansion in negative continued fraction that is slower than the usual
one.
The cycle of simple forms which are equivalent to a given form Q corresponds
to the cycle of real quadratic irrationalities x given by the algorithm above with
i
x = w .
0 Q
Clearly eachx in(2.2)istheimageofxbya matrixγ = γ Γ, givenexplicitly
i i i,x
∈
by
0 1 0 1
γ = γ := Id, γ = γ := (i 1) (2.4)
0 0,x i i,x 1 n ··· 1 n ≥
i−1 0
(cid:18) − (cid:19) (cid:18) − (cid:19)
and recursively by
γ = Id, γ = εT−niγ (i 0). (2.5)
0 i+1 i
≥
We denote
Γ(x) := γ ,γ ,γ ,... Γ.
1 2 3
{ } ⊂
There is an explicit description of Γ(x) in terms of the convergents of x. The i-th
p
i
convergent of x is denoted by = [n ,...,n ]. The integers p and q satisfy the
0 i i i
q
i
recurrence
p = 0 p = 1, p = n p +p (i 0),
−2 −1 i i i−1 i−2
≥
q = 1, q = 0, q = n q +q (i 0),
−2 −1 i i i−1 i−2
≥
the equation
p q p q = ( 1)i (2.6)
i+1 i i i+1
− −
and the inequalities
(1) q q 0 for all i 0 and q > q > 0 for all i 2,
i i−1 i i−1
≥ ≥ ≥ ≥
(2) p p for all i 2 and p > p for all i 3.
i i−1 i i−1
| | ≥ | | ≥ | | | | ≥
6
The numbers δ (i 1) defined by
i
≥ −
δ = ( 1)i(p q x) (2.7)
i i−1 i−1
− −
satisfy the recurrence
δ
i−1
δ = x, δ = 1, δ = n δ +δ with n =
−1 0 i+1 i i i−1 i
− δ
(cid:22) i (cid:23)
and the inequalities 1 = δ > δ > ... 0. If x is rational, then x = p /q for some
0 1 i i i
≥
i and the recurrence stops with δ = 0; if x is irrational, the δ are all positive and
i+1 i
converge to 0 with exponential rapidity. With these notations, one has
p p x δ
γ−1 = i−1 i−2 , γ = i−1 . (2.8)
i q q i 1 δ
i−1 i−2 i
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Now we consider the slower version of the algorithm of reduction (2.2)
x +1 = T(x ) if x 0,
i i i
≤
 1
x = x, x =  1 = T−1ε(x ) if 0 < x 1, (2.9)
0 i+1  x − i i ≤
 i

x 1 = T−1(x ) if x > 1,
 i − i i


such that with this algorithm the expansion of x in continued fraction is
1
x = 1 1+
± ±···± 1
1+ +1+
|n0| ··· 1
1+ +1+
| {z } n1 ··· ...
| {z } n2
where n0,n1,n2,..., are given in (2.2) and each |equ{azls th}e sign of n0 .
± | |
Each x in algorithm (2.9) is the image of x by a matrix γ′ = γ′ Γ given
i i i,x ∈
recursively by
Tγ′ if x 0,
i i ≤

γ′ = Id, γ′ = T−1εγ′ if 0 < x 1, (2.10)
0 i+1  i i ≤



T−1γ′ if x > 1.
i i


We denote 


Γ(x)′ := γ′,γ′,γ′,... Γ.
{ 1 2 3 } ⊂
We note that
Γ(x)′ = T−kγ , 1 k n .
i ≤ ≤ i i≥1
The following two propositions(cid:8), whose proofs are g(cid:9)iven in [Be], describe the sets
Γ(x) and Γ(x)′ for x R Q as subspaces of elements of Γ defined by certain simple
∈ −
linear inequalities:
7
Proposition 2.1 For all x R irrational, the set Γ(x) equals W (W W ),
1 2
∈ − ∪
where
W = γ Γ 1 γ( ) 0, γ(x) > 1 ,
{ ∈ | − ≤ ∞ ≤ }
0 1
W = γ W γ( ) = 0, det(γ) = 1 = − ,
1 { ∈ | ∞ } 1 1 n
0
(cid:26)(cid:18) − − (cid:19)(cid:27)
1 1+n
− 0 if n 2,
W = γ W γ( ) = 1, det(γ) = 1 = 1 n 1 ≥
2 0
{ ∈ | ∞ − − }  (cid:26)(cid:18) − (cid:19)(cid:27)
if n = 1.
 1
∅
Remark 2.2 Proposition 2.1 is also true for x Q if we allow the value for
∈ ∞
γ(x), when γ W.
∈
Proposition 2.3 For all x R irrational, the set Γ(x)′ equals W′ W′, where
∈ − 1
W′ = γ Γ γ( ) 1, γ(x) > 0
{ ∈ | ∞ ≤ − }
and
1 n
W′ = γ W′ γ( ) = 1, det(γ) = 1 = − 0 .
1 { ∈ | ∞ − } 1 n +1
0
(cid:26)(cid:18)− (cid:19)(cid:27)
Each x in (2.3) is the image of x by a matrix γ˜ = γ˜ Γ defined by
i i i,x 1
∈
q˜ p˜
γ˜ = Id, γ˜ = ST−miγ˜ = − i−1 i−1 (i 0), (2.11)
0 i+1 i q˜ p˜ ≥
i i
(cid:18) − (cid:19)
p˜
i
where = [m ,...,m ] is the i-th convergent of x. The set of matrices from (2.11)
0 i
q˜
i
will be denoted by
Γ (x) := γ˜ ,γ˜ ,γ˜ ,... Γ .
1 1 2 3 1
{ } ⊂
The integers p˜ and q˜ satisfy the recurrence
i i
p˜ = 0, p˜ = 1, p˜ = m p˜ p˜ (i 0),
−2 −1 i i i−1 i−2
− ≥
q˜ = 1, q˜ = 0, q˜ = m q˜ q˜ (i 0),
−2 −1 i i i−1 i−2
− − ≥
the equation
p˜q˜ p˜ q˜ = 1, (2.12)
i i+1 i+1 i
−
and the inequalities
(1) q˜ q˜ 0 for all i 0 and q˜ > q˜ > 0 for all i 1,
i i−1 i i−1
≥ ≥ ≥ ≥
(2) p˜ p˜ for all i 1 and p˜ > p˜ for all i 2.
i i−1 i i−1
| | ≥ | | ≥ | | | | ≥
8
˜
In a similar way to the positive continued fraction, the numbers δ (i 1)
i
≥ −
defined by
˜
δ = p˜ q˜ x (2.13)
i i−1 i−1
−
satisfy the recurrence
˜
δ
˜ ˜ ˜ ˜ ˜ i−1
δ = x, δ = 1, δ = m δ δ with m =
−1 0 i+1 i i − i−1 i & δ˜i '
˜ ˜
and the inequalities 1 = δ > δ > ... 0. If x is rational, then x = p˜/q˜ for some
0 1 i i i
˜ ≥ ˜
i and the recurrence stops with δ = 0; if x is irrational, the δ are all positive and
i+1 i
converge to 0 with exponential rapidity.
3 Combining reduction theories: proofs of the
conjectures
For d 2N, the group Γˆ acts on homogeneous polynomials of degree d by (2.1) or,
∈
equivalently, on the space of polynomials of degree d in one variable by
≤
rx+s r s
(P γ)(x) := (tx+u)dP (γ = Γˆ).
|−d tx+u t u ∈
(cid:18) (cid:19)
(cid:16) (cid:17)
We write
= P Z[X] P has exactly 2 real roots, both irrational .
d ≤d
F { ∈ | }
Given P , we denote by w and w′ its two real roots such that
∈ Fd P P
sign(P( )) w < sign(P( )) w′ .
∞ · P ∞ · P
If is a Γˆ-equivalence class in , we define
d
A F
Red = P P( ) > 0, P( 1) < 0 < P(0) = P w < 1 < w′ < 0 ,
A { ∈ A | ∞ − } { ∈ A | P − P }
Sim = P P(0) < 0 < P( ) = P w < 0 < w′ ,
A { ∈ A | ∞ } { ∈ A | P P}
x = P P( ) < 0 < P(x) .
Ah i { ∈ A | ∞ }
In the special case d = 2, we have = and
2
F Q
Red = Red, Sim = Sim.
A A∩ A A∩
In this case, Red and Sim are both finite. If is a Γˆ-equivalence class in , we
A A A Q
define
A (x) = Q(x)k−1 (x R, k 2).
k,A
∈ ≥
Q∈Ahxi
X
9
Then the sum A defined in (1.2) is given by
k,D
A (x) = A (x). (3.1)
k,D k,A
A∈XQD/Γˆ
The two theorems below are stated in the general case with even d 2.
d
F ≥
Theorem 3.1 For d 2N, a Γ-equivalence class in and x R, the following
d
∈ A F ∈
bijection holds
(P,γ) Sim Γ(x) : ∼=
∈ A × x
P(γ( )) < 0 < P( γ(x) ) −→ Ah i
(cid:26) ∞ ⌊ ⌋ (cid:27)
(P,γ) P γ.
7→ |
Proof. We only have to check that P( γ(x) ) > 0 implies P(γ(x)) > 0 to prove
⌊ ⌋
that the map is well defined. This follows from P( γ(x) ) > 0 and γ(x) > 0,
⌊ ⌋ ⌊ ⌋
which imply γ(x) > w′ , so γ(x) > w′ , and hence P(γ(x)) > 0.
⌊ ⌋ P P
We now prove that the map is a bijection. Let P satisfy P( ) < 0 < P(x).
p ∈ A ∞
For j 0, the convergents j of x belong to (w′ ,w ), because x (w′ ,w ).
≫ q P P ∈ P P
j
p p
Moreover, if two consecutive convergents j and j+1 belong to (w′ ,w ), so do all
q q P P
j j+1
the later convergents.
p
If x (w′ ,w ), we define i to be the unique positive integer such that i−1
⌊ ⌋ 6∈ P P q 6∈
i−1
p p
(w′ ,w ) but i, i+1 (w′ ,w ). If x (w′ ,w ), we set i = 0. Since P( ) < 0,
P P q q ∈ P P ⌊ ⌋ ∈ P P ∞
i i+1
in both cases we have
p p p
i−1 i i+1
P < 0 < P , P > 0. (3.2)
q q q
i−1 i i+1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Put
q p
γ = i−1 − i−1 ,
q p
i i
(cid:18)− (cid:19)
and R = P γ−1. We have
|
p p
i−1 i
R(0) = P < 0, R( ) = P > 0.
q ∞ q
i−1 i
(cid:16) (cid:17) (cid:16) (cid:17)
p
i+1
The inequality P > 0 is equivalent to the condition R( γ(x) ) > 0 because
q ⌊ ⌋
i+1
p (cid:16) (cid:17)
i+1
P = R(n ) and n = γ(x) . The condition P( ) < 0 is equivalent to
i+1 i+1
q ⌊ ⌋ ∞
i+1
th(cid:16)e cond(cid:17)ition R(γ( )) < 0. Thus (R,γ) belongs to the left hand set of the map in
∞
Theorem 3.1.
The uniqueness of the preimage (R,γ) follows from the equivalence between the
p
i+1
condition R( γ(x) ) > 0 and the inequality P > 0, together with the unique-
⌊ ⌋ q
i+1
ness of i satisfying (3.2). (cid:16) (cid:17)
10