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Revista de la Unión Matemática Argentina

versão impressa ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

 

Exponents of Modular Reductions of Families of Elliptic Curves

Igor E. Shparlinski

Abstract. For some natural families of elliptic curves we show that "on average" the exponent of the point group of their reductions modulo a prime p grows as  1+o(1) p .

2000 Mathematics Subject Classification. 11B57, 11G07, 14H52

Key words and phrases. Elliptic curves; Group exponent; Farey fractions.

1. Introduction

For integers a and b such that 4a3 + 27b2 ⁄= 0 , we denote by E a,b the elliptic curve defined by the affine Weierstraß equation:

 2 3 Ea,b : Y = X + aX + b.

For a basic background on elliptic curves, we refer to [11].

For a prime p > 3 , we denote by Fp the finite field of p elements, which we identify with the set of integers {0, ±1, ...,±(p - 1)∕2} .

When  3 2 p ∤ 4a + 27b , the set Ea,b(Fp) , consisting of the Fp -rational points of Ea,b together with a point at infinity O , forms an abelian group under an appropriate composition rule called addition, and the number of elements in the group Ea,b(Fp) satisfies the Hasse bound:

 √ -- |#Ea,b(Fp ) - p - 1| ≤ 2 p
(1)

(see, for example, [11, Chapter V, Theorem 1.1]).

It is well-known that Ea,b(Fp ) is of rank at most two, that is, Ea,b(Fp) is isomorphic to

 ~ Ea,b(Fp) = ℤ ∕m ℤ × ℤ ∕nℤ
(2)

for unique integers m and n with m | n and #Ea,b (Fp) = mn . The number n is called the exponent of Ea,b(Fp) which we denote by ℓa,b(p) . In other words, ℓa,b(p) is the smallest positive ℓ such that ℓP = O for all points P ∈ E (F ) a,b p .

We also put ℓa,b(p) = 0 if  3 2 p | 4a + 27b .

Thus we see that (1) and (2) imply the following trivial bound

 1∕2 1∕2 ℓa,b(p) ≥ (Ea,b(Fp )) ≥ p - 1.
(3)

The exponent of elliptic curves has been studied in a number of works, see [478910], with a variety of results, each of them indicating that in a "typical case" the exponent tends to be substantially larger than the bound (3) (and its analogue for curves over arbitrary finite fields) guarantees.

W. Duke [4], among other results, has proved that, assuming the Generalised Riemann Hypothesis, for every fixed integer a and b with 4a3 + 27b2 ⁄= 0 , and arbitrary small ε > 0 , the bound

ℓa,b(p) ≥ p1-ε
(4)

holds for all but o(T∕ log T ) of primes p ≤ T .

It is also shown in [10] that (4 ) holds for all but  2 o(p ) pairs (a,b) ∈ Fp × Fp .

Here we use a combination of the results and ideas of [110] to prove unconditionally that (4 ) is satisfied for almost all pairs (a,b) with |a| ≤ A , |b| ≤ B for A and B relatively small compared to p .

Theorem 1. For any fixed ε > 0 and all integers A , B satisfying the inequalities

 1∕2 1+ε 1∕4+ ε AB ≥ p and B ≥ p

or

 1∕2 1+ε 1∕4+ε A B ≥ p and A ≥ p

the bound

ℓa,b(p) ≥ p1-ε

holds for all but o(AB ) pairs (a, b) with |a| ≤ A , |b| ≤ B

In particular, Theorem 1 is nontrivial if

max {A, B } ≥ p7∕8+ε and min {A,B } ≥ p1∕4+ε

or

AB ≥ p4∕3+ε.

We also show that averaging over p gives some additional saving.

Theorem 2. For any fixed ε > 0 and all integers A , B and T satisfying the inequalities

 ∘ ----------- Tε ≤ A, B ≤ T1-ε and AB ≥ T 1+ε min {A, B }.

the bound (4 ) holds for all but o(ABT ∕log T) triples (a,b,p) with |a| ≤ A , |b| ≤ B , p ≤ T .

We note that the condition A,B ≤ T 1-ε from [1], where it is used to simplify the error term, is not neccessary. One can easily extend Theorem 2 for A and B beyond this range, however since (as in [1]) small values of A and B are of main interest we have not done this.

We remark that in [5] some of the results of [4] have been extended to hyperelliptic curves. It would also be interesting to obtain analogues of our result for natural families of hyperelliptic curves.

We also consider the set of Farey fractions

F (W ) = {u∕v : gcd(u, v) = 1, 1 ≤ u,v ≤ W }.

In particular

 ( 6 ) #F (W ) = -2-+ o(1) W 2. π

For t = u ∕v with gcd (v,p) = 1 and two polynomial A (X ),B(X ) ∈ ℤ[X ] , the reduction EA (t),B (t)(Fp ) is correctly defined. Various questions concerning the behaviour of the curves EA (t),B(t)(Fp ) on average over p ≤ T and t ∈ F (W ) have been studied in [2]. Here we continue to study this family of curves. Certainly the most interesting case is when W is small compared to T .

Theorem 3. Assume that the discriminant

ΔA,B (t) = - 16(4A (t)3 + 27B (t)2)

is nonzero and the j -invariant

 3 jA,B(t) = - ---6912A-(t)----- 4A(t)3 + 27B (t)2

is nonconstant. Then for any fixed ε > 0 and all integers W and T with

 1∕2+ε W ≥ T

the bound

ℓA (t),B(t)(p) ≥ p1-ε

holds for all but o(W T∕ log T) pairs (t,p ) with t ∈ F (W ) , p ≤ T .

2. Preparations

The following result follows immediately from the more precise statement of [10, Theorem 3.1].

Lemma 4. For any ε > 0 , the number of triples (a,b) ∈ Fp × Fp with

ℓa,b(p) < p1-ε

is at most o(p2) .

Let d = gcd(p - 1,6) p and put

 { ||H∑ ||} σ (H ) = max 1, || χ (n )|| , p χdp=χ0 | | χ⁄=χ0 n=1

where the maximum is taken over all non-principal multiplicative characters χ modulo p such that χdp is the principal character χ 0 .

Similarly, we define ep = gcd (p - 1,4 ) and put

 { ||∑H ||} ρ (H ) = max 1,|| χ(n)|| , p χep=χ0 | | χ⁄=χ0 n=1

where the maximum is taken over all non-principal multiplicative characters χ modulo p such that χep is the principal character χ0 . For an arbitrary subset S ⊆ Fp × Fp , we denote by Np (S,A, B ) the number of pairs such that (a,b) ∈ S with |a| ≤ A and |b| ≤ B . We also denote

E (A, B; p) = min {A σp (B ) + B1 ∕2p, B ρp(A ) + A1∕2p} .

The following estimate is given in [1].

Lemma 5. For all primes p > 3 , integers 1 ≤ A, B ≤ (p - 1)∕2 , and subsets S ⊆ Fp × Fp such that whenever (r,s) ∈ S the isomorphism Ea,b(Fp ) ~= Er,s(Fp ) implies (a,b) ∈ S , the following bound holds:

|| || |Np(S, A,B ) - 4AB--#S |≪ E (A, B; p). | p2 |

Moreover, it is shown in [1] that E (A, B; p) is small "on average" over p .

Lemma 6. The following bound holds:

∑ E(A,B; p) ≪ ABT 1∕2+o(1) + AB7 ∕8T + B1∕2T 2 p≤T

For a prime p and an integer t with 1 ≤ t < p we denote by RT,p(t) the number of fractions u∕v ∈ F (T ) with gcd(v,p ) = 1 and u ∕v ≡ t (mod p) .

It is shown in [3] that R (t) T,p is close to its expected value #F (T)∕p on average over t = 1,...,p - 1 . More precisely, we have:

Lemma 7. We have,

 | | p∑-1| 6 W 2| ( 2 -1 1∕2+o (1)) ||RW,p (t) - π2-⋅ -p-|| = O W p + W p . t=0

3. Proof of Theorem 1

Let Sp (ε) be the set of pairs (a,b) ∈ Fp × Fp for which ℓa,b(p) ≤ p1-ε . Then it is enough to show that

Np (Sp (ε),A,B ) = o(AB ).

Since by Lemma 4 we have  2 #Sp (ε) = o(p ) , invoking Lemma 5 we see that it is enough to check that E(A, B; p) = o(AB ) .

Assume that B ≥ p1∕4+ε then by the Burgess bound, see [6, Theorems 12.5 and 12.6], we have σp (B ) = o(B ) . Also, if AB1 ∕2 ≥ p1+ ε then have B1∕2p = o(AB ) .

Similarly, if  1∕4+ε A ≥ p then ρp(B ) = o(B) , and if  1∕2 1+ε A B ≥ p then have  1∕2 A p = o(AB ) .

 

4. Proof of Theorem 2

As before, let Sp(ε) be the set of pairs (a,b) ∈ Fp × Fp for which ℓ (p) ≤ p1-ε a,b . Then it is enough to show that

∑ Np (Sp (ε ),A, B ) = o(ABT ∕log T). p≤T
(5)

Let us assume that A ≥ B since the case A < B is similar.

Using the trivial bound Np (Sp(ε),A,B ) ≤ AB for primes p ≤ 2A + 1 , we deduce

∑ ∑ Np (Sp(ε),A, B ) = Np(Sp(ε),A, B ) + O (A2B ). p≤T 2A+1 <p≤T
(6)

Noticing that for p > 2A + 1 the set Sp(ε) satisfies the conditions of Lemma 5 , we obtain

 ∑ Np(Sp(ε),A, B ) 2A+1<p≤x ( ) ∑ ∑ =4AB #Sp-(ε)+ O E(A, B;p) . 2A+1 <p≤T p2 2A+1 <p≤T
(7)

By Lemma 4 we have

 ∑ #Sp-(ε) = o(T ∕logT ). 2A+1<p≤T p2
(8)

Substituting (7) and (8) in (6), we obtain

∑ Np (Sp (ε),A, B ) p≤T ( ∑ ) = o (ABT ∕ logT ) + O E (A,B; p) + A2B . 2A+1 <p≤T

We now easily verify that under the conditions of the theorem, Lemma 6 implies the desired bound (5).

5. Proof of Theorem 3

As before, we use Sp(ε) to denote the set of pairs (a, b) ∈ Fp × Fp for which ℓa,b(p) ≤ p1-ε .

Let T (ε) A,B,p be the set of t ∈ F p such that

( 4 6) A (t)λ ,B (t)λ ∈ Sp(ε).

for some λ ∈ F*p .

Obviously, for any t ∈ Fp and λ ∈ F* p we have

ℓA (t),B(t)(p) = ℓA(t)λ4,B(t)λ6(p)

(since the corresponding curves are isomorphic, see [11, Section III.1]).

We also note that the system of equations

 4 6 A (t)λ = a, B (t)λ = b

leads to the equation

b2A (t)3 = a3B (t)2

which has O (1 ) solutions (by the condition on the j -invariant jA,B(t) ).

Therefore

 #S (ε) #TA,B,p (ε) ≪ ----p--. p

Using Lemma 7, we obtain

 ∑ W--2#TA,B,p(ε) 2 - 1 1∕2+o(1) 2 RW,p(t) ≪ p + W p + W p = o (W ) t∈TA,B,p

which concludes the proof.

References

[1]    W. D. Banks and I. E. Shparlinski, 'Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height', Israel J. Math., (to appear).        [ Links ]

[2]    A. Cojocaru and C. Hall, 'Uniform results for Serre's theorem for elliptic curves', Internat. Math. Res. Notices, 2005 (2005), 3065-3080.        [ Links ]

[3]    A. Cojocaru and I. E. Shparlinski, 'Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average', Proc. Amer. Math. Soc., 136 (2008), 1977-1986.        [ Links ]

[4]    W. Duke, 'Almost all reductions modulo p of an elliptic curve have a large exponent', Comptes Rendus Mathematique, 337 (2003), 689-692.        [ Links ]

[5]    K. Ford and I. E. Shparlinski, 'On finite fields with Jacobians of small exponent', Preprint, 2006 (available from http://arxiv.org/abs/math.NT/0607474).        [ Links ]

[6]    H. Iwaniec and E. Kowalski, On curves over finite fields with Jacobians of small exponent. Intern. J. Number Theory, 4, 2008, 819-826.        [ Links ]

[7]    F. Luca, J. McKee and I. E. Shparlinski, 'Small exponent point groups on elliptic curves', J. Théorie des Nombres Bordeaux, 18 (2006), 471-476.        [ Links ]

[8]    F. Luca and I. E. Shparlinski, 'On the exponent of the group of points on elliptic curves in extension fields', Intern. Math. Research Notices, 2005 (2005), 1391-1409.        [ Links ]

[9]    R. Schoof, 'The exponents of the group of points on the reduction of an elliptic curve', Arithmetic Algebraic Geometry, Progr. Math., vol. 89, Birkhäuser, Boston, MA, 1991, 325-335.        [ Links ]

[10]    I. E. Shparlinski, 'Orders of points on elliptic curves', Affine Algebraic Geometry, Contemp. Math., vol. 369, Amer. Math. Soc., Providence, RI, 2005, 245-252.        [ Links ]

[11]    J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, Berlin, 1995.         [ Links ]

Igor E. Shparlinski
Department of Computing, Macquarie University, North Ryde,
Sydney, NSW 2109, Australia
igor@ics.mq.edu.au

Recibido: 7 de octubre de 2007
Aceptado: 21 de mayo de 2008