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ЧЕБЫШЕВСКИЙ СБОРНИК Том 12 Выпуск 2 (2011)
Труды VIII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной 190-летию Пафнутия Львовича Чебышева и 120-летию Ивана Матвеевича Виноградова
ON TWISTED L-FUNCTIONS OF ELLIPTIC CURVES
VIRGIN 1.1A GARBALIAUSKIENE (Siauliai University, Faculty of Mathematics
and Informatics)
ANTANAS LAURINCIKAS (Vilnius University, Faculty of Mathematics and
Informatics)*
1. Introduction
Let E be an elliptic curve over the field of rational numbers given by the Weierstrass equation
2 3 7
y = x + ax + b,
with integers a and b, and non-zero diseriminant A = — 16(4a3 + 27b2, For each prime p, denote by Ep the reduction of the curve E modulo p, i. e,, a curve over the finite field Fp, Now define real numbers A (p) by
|E (Fp )| = p +1 — A (p),
where |E(Fp)| is the number of points of Ep.
The L-function Le(s), s = a + it, of the elliptic curve E is defined by the Euler product
Le(s) = n(l — ^)-‘ n(l — APp) + ^ r . d)
p|A v 1 y p\A 1 1 y
p
|A(p)| < 2VP. (2)
Therefore, the infinite product in (l)eonverges uniformly on compact subsets of the half-plane {s G C : a > |}, and define there an analytic function without no zeros. Moreover, since the Taniyama-Shimura conjecture is true [2], the function Le(s) has analytic continuation to an entire function, and satisfies the functional equation
* Partially supported by grant No MP-94 from the Research Council of Lithuania
^2pj r(s)LE(s) = w|^ ^ r(2 — s)Le (2 — s),
where r(s) denotes the Euler gamma-funetion, N is the conductor of the curve E, and w = ±1,
It is know that the function Le(s) has a limit distribution in the probabilistic sense. For D = {s G C : a > 1}, denote bv H(D) the space on analytic function on D with the topology of the uniform convergence on compacta. Let B(S) stand for the class of Borel sets of the space S, Then we have that the probability measure
—meas {t G [0, T] : Le(s + ir) G A} , A G B(H(D)),
converges weakly to the explicitly given probability measure on (H(D), B(H(D))) as T —— to, The latter statement follows from a general limit theorems for the Matsumoto zeta-function [12] which is defined by general Euler product over primes. Now let x be a Dirichlet character modulo q, The twist Le(s, x) of the function Le(s) with charaeter x is defined, for a > |, by
Le(s,x) = n(1 — ^)-1 n (1 — A^ + 0 '
p|A V P ' pfA V P P
and by analytic continuation elsewhere.
For any fixed q, as the case of Le(s),the probability measure
—meas {r G [0, T] : Le(s + ir, x) G A} , A G B(H(D)),
also converges weakly to the explicitly given probability measure on (H(D), B(H(D))) as T — to.
q
not fixed. Suppose that q is a prime ^^mber, denote by xo the principal character modulo q, and,for Q > 2, define
mq = E E 1.
q<Q X = x(modq)
X = X0
Moreover, for brevity, we put
„q(...) = mq1 E E 1-
q<Q x=x(modq)
Xi=X 0
where in place of dots a condition satisfied by a pair (q,x(modq)) is to be written. We will consider the asymptotic behavior for
^Q(/(Le(s,x)) G A)
with some function f as Q — to, More precisely, we study probabilistic limit theorems for LE(s, x) with respect to the increasing parameter q.
The first result in this direction, for Diriehlet L-function L(s, x) was obtained by Chowla and Erdos, They proved [3] a limit theorem for Diriehlet L-function L(1, x)
x
converges weakly to -some distribution function as Q — to.
When L(s,x) = 0, 1 < a < 1, let argL(s,x) denote a value of the argument of L(s, x) defined by continuous displacement from the point s = 2 along an arc on which L(s, x) does not vanish. Thus, argL(s,x) is only defined to within the addition of an integer multiple of 2ni. In [5], a limit theorem, for argL(s,x) has been obtained. Before the statement of this theorem, we remind the definition and convergence of distribution functions mod1.
A function G(x) is said to be a distribution function mod1 if and only if it
satisfies the conditions:
1o
2o
30 G(x) = 1 if x > 1, and G(x) = 0 if x < 0.
A distribution function Gn(x) mod1, n G N, converges weakly mod1 as n — to if there exists a distribution function G(x) mod1 such that, at all points x1,x2, 0 < x1 < x2 < 1, which are continuity points of G(x), we have
G(x)
Now we state the theorem from [5]. Theorem 2. For each s G {s G C : a > 2},
converges weakly to a continuous distribution function mod1 as Q — to. Moreover, the Fourier transform of the limit distribution function is of the form
Then in [14], the following theorem was proved.
Theorem 1. Suppose that a > Then the distribution function
vq(Hx)r1|L(s,x)| <x)
lim (G„(x2) — G„(xi)) — G(x2) — G(xi).
E, Stankus generalized [14] Theorems 1 and 2 for probability measures on (C, 0(C)), Let be a probability measure on (C, 0(C)), Then the function
w(r,k)= J |z|iTeikargzdP, t G R, k G Z,
C\{0}
is called the characteristic transform of the measure P. It is known [11] that the measure P is uniquely determined by w(t,k). Let cT,k(m) denote the multiplicative function defined, for primes ^d m G N, by
(pm) = ^« +!)...« + k - !)
m!
with £ = iT+^, Define the function wP (t, k) by
, M ^ CT,k(m)cT,-k(m)
wp (T,k)^^ —m^—,
m=1
Then we have the statement [14],
Theorem 3, Suppose that a > Then
Vq((L(s, x)) G A), A G 0(C),
converges weakly to the probability measure P on (C, 0(C)) as Q — to defined by the characteristic transform wP (t, k).
Theorems 1-3 are examples of limit theorems with respect to a parameter q. There are known other results with respect to some parameter. We will remind two of them. The first result is related to the universality of Diriehlet L-functions with respect to a character, and was obtained independently in [1], [8] and [6],
Theorem 4. Let K c{s G C : 1 <a< 1}6ea compact subset with connected complement, and f (s) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every e > 0,
1 I x is a Dirichlet character modp . limini-------fl<x : i \Tt \ a m } > 0-
p — 1 I and sup |L(s,x) — f (s)| <e
I sEK
The second result [9] is connected with growth of the function L(1,x),where x
x>1
Dx = ^{d < x : d is a fundamental discriminant}
and
$x(t) = ft{d < x : d is a fundamental discriminant and L(1, xd) > eY0t},
Dx
where Y0 is the Euler constant, and xd is a eharaeter mod d.
Theorem 5 [9], Uniformly in t < log log x,
$x(t) = exp{ — e—-} ^ 1 + O
where C has an explicit integral representation, C = 0.8187....
More general results that Theorem 5 are given in [10].
2. A limit theorem for the modulus of Le(s,x)
For a > 2, we can rewrite (1) in the form
Le (s,x) = n(1 — ^ )-1 n(' — ^ )-1 n(1 — “ )-1 ■
p|A V P ' pfA \ r V PS
(3)
where, for primes p f A, a(p) and ft(p) are complex conjugate numbers such that a(p) + ft(p) = A(p) and a(p)ft(p) = p.
We put
, , iT ^
n = n(T) = y, t G R,
and define, for primes p and k G N,
d-(pk) =n(n + 1)---(n +k —1}, d(1) = 1.
Moreover, for p f A and k G N, let
k
a- (pk) = X d- (p1)«1 (p)d- (pk-1)ftk-1(p)
1=0
and
k
b- (pk) = X]d- (p1)a1(p)d- (pk-1 )ftk- (p^
T /nk-M^k
1=0
where z denotes the conjugate of a complex number z, while, for p | A and k G N,
a- (pk) = b- (pk) = dT (pk )Ak (p).
mN
a-(m) = a-(p1), b-(m) = b-(p1),
where, as usual, p1 || m means that p1 | m tat p1+1 f m.
Let PR be the probability measure on (R, 0(R)) defined by the characteristic transforms
aT(m)bT(m) 3 _
W0(T) = W1(T) = £ T( ' • a > 2. T G R.
m=1
where
wk(t) = J |x|iTsgnkxdPR, k = 0,1.
R\{0}
Then we have the following statement [7],
Theorem 6, Suppose that a > !• Tften
Pq,r =fjM|£e(s,x)I G A), A €0^),
converges weakly to PR as Q — to.
Sketch of ifte proof. Let wq(t) be the characteristic transform of the measure
Pq,r ^w0q(t) = w1q(t) =wq(t) because |Le(s, x)I > 0^. We have that
wQ(T) = X X |Le(s,x)|iT, T G R. (4)
Q 9<Q X = x(modq)
X = X0
Since
|Le (s,x)|iT = (Le (s,x)) т (Le (s, x)) ^ in view of (3) we find that, for a > | + £, £ > 0,
|Le(s,x)r = П (l - П (l - “) X
p|A V У ' p|A V V /
X П^х _ a(p)x(p) ^ ^ _ e(p)x(p) ^ -n X
X ^ _ a(p)x(p) ^ ^ _ e(p)x(p) ^ -n =
= ar (m) 6r (n) ,
ms 2-^/ ns , m=1 n=1
where aT (m) and bT (m) are multiplicative functions given, for primes p \ A and k Є N, by
k
aT(pk) = X dT(Рг)a1(p)x(p1 )dT(pk-1 )ek-1 (p)x(pk-1)
1=0
and
k
„k\ _ A ^k-1\^k-1/ / k-1
b- (pk ) = d- (p1 )a1(p)x(p1)d- (pk 1)ft (p)x(pk 1),
1=0
p | A k G N
a- (pk) = d- (pk )Ak (p)x(pk),
and
b- (pk) = d- (pk )Ak (p)x(pk).
It is easily seen that
aT (m) = aT (m)A(m), bT (m) = bT (m)A(m).
Therefore, we deduce from (4), (5) and (2) that, for a > 2+£, |t| < ^d N = log Q,
wq(t)= XX ^ ^^ X X x(m)x(n)+o(1)
m<N n<N Q q<Q X = x(modq)
X = X0
as Q — to Considering separately the cases m = n and m = n, and using the equality
/ x_/ x f q — 1 if m = n(modq),
X x<m)x<n) H 0 if m = n(modq).
X=x(modq)
hence we obtain that, for |t| < ^d a > 2,
aT (m)bT (m)
m2 m=1
as Q — to. This and a continuity theorem for characteristic transformations [13] prove the theorem,
3. A limit theorem for the argument of Le(s,x)
The estimate (2) implies the bounds
|a(p)| < Vp, and |ft(p)| < ^p.
Therefore, Le(s, x) = 0 for a > 2, and we can define argL(s, x) similarly to the case of Le(s,x)-
Denote by y the unit circle on the complex plane, and let PY be a probability measure on (y, 0(y)) defined by the Fourier transform
n\def f km ^ «k(m)bk(m) 3
g(k) = I xkdPY = ^ m2a , k G Z a > 2.
m=1
Here ak (m^d bk (m) are multiplicative functions defined in the same manner as aT (m^d bT (m). More precisely, let
k
6 = 6(k) = -, k G Z,
and, for primes p and l G N,
dk(p1) = 6(6 + 1)..l(6 +l —1), dk(1) = 1.
Then, for p f A and l G N,
1
«k (p1) = X dk (p7 )aj (p)dk (p1-7 )ft1-7 (p), j=0
1
bk (p1) = X d-k (p7 )a7 (p)d-k (p1-7 )ft1 7 (p),
7=0
p | A l G N
ak (p1) = dk (p1 )A1(p), bk (p1) = d-k (p1 )A1(p).
Theorem 7. Suppose that a > !• Tften
pQ,y(A) ==^Q (exp{iargLE(s,x)} G A) , A G 0(Y),
converges weakly to PY as Q — to.
3 2
Corollary. Suppose that a > §. Tften
^Q (2nargLE(s,x) < x(mod1)
mod1 mod1
transform g(k).
Proof of Theorem 7 runs in a similar way as that of Theorem 6. We have that the Fourier transform gQ (k) of Pq,y is
1 v ^ iargLE (s,x)
Q q<Q X = x(modq)
X = X0
From the equality
e“"iE(*’x) = (Le(s,x))2 (i£(s,x))-2
= j A — A(p)x(pA e j ^ — A(p)x(pA0 x
p|A ' p ' p|A ' p /
x ^(1 — a(p)x(p)) -0 (1 — ft(p)x(p)) -0 x
x J-J (1 — a(p)yp)y (1 _ ft(p)yp)y (6)
pfA ' p ' ' p /
we easily find that, for a > | and k G Z,
^ ak(m)bk(m) , m wq (k) = X ——+o(1)
m=1
as Q — to. From this, the theorem follows,
4. A limit theorem for Le(s,x) on the complex plane
In this section, we connect Theorems 6 and 7, For t G R and k G Z, let
£ = «t, ±k) = ,
and, for primes p and l G N,
d (p1 ) = £(£ + 1)...(£ + l — 1) d M) = 1 d—,±k(p ) l! , d—,±k(1) 1.
Let PC be a probability measure on (C, 0(C)) defined by the characteristic transform
w(T,k)= f |z|iTeikargzdPc = X a—,k(m), T G R, k G Z, a > 3,
C\{0} m=:
where aT,k(m^d bT,k(m) are multiplicative functions given, for p f A and l G N, bv
k (p1) = X d—,k (p1)a7 (p)d—,k (p1 7 )ft1 7 (P),
7=0
b—,k(p1) = Xd—,-k(p1 )a7(p)d—,-k(p1 7)ft 7(p),
7=0
p | A l G N
a—,k (p1) = d—,k (p1 )A1(p), b—,k (p1) = d—,-k (p1)A1 (p).
Theorem 9. Suppose that a > 2- Then
Pq,c(A) =Vq (Le(s, x) G A), A G 0(C),
converges weakly to PC as Q — to.
Sketch of the proof Let wq(t, k) be the characteristic transform of Pq,c, i. e.
wq(t,-)= J |z|iTeikargzdPQ,c, t G R, k G Z.
C\{0}
Then we have that
wq(t,-) = X X |Le(s,x)|iTe-k^gLE(s,x).
Q 9<Q X = x(modq)
X=X0
Using, for a > !, the formula
|Le (s,x)|i—eikargLE (s,x) =
k
A(p)x(pU -,T+5 (, A(p)x(pA -,T-2
= n i1 — i1 x
k • k x j ^ — q(p)x(p^",T+2 ^ — P(p)x(p^",T+5 x
-iT - 2
x j ^ — a(p)x(p^ iT 2 ^ — p(p)x(p)y ^
we find similarly as in Section 2 that, for |z| < ^d k G Z,
, M ^ a—,k(m)b—,k(m) ,
wq(t, k) = z, —m^—+o(1) m=1
as Q — to. This and a continuity theorem for characteristic transforms of probability C, 0(C)
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2
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Received 12.10.2011
