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Revista de la Unión Matemática Argentina
versión impresa 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 grows as
.
2000 Mathematics Subject Classification. 11B57, 11G07, 14H52
Key words and phrases. Elliptic curves; Group exponent; Farey fractions.
For integers and
such that
, we denote by
the elliptic curve defined by the affine Weierstraß equation:
For a basic background on elliptic curves, we refer to [11].
For a prime , we denote by
the finite field of
elements, which we identify with the set of integers
.
When , the set
, consisting of the
-rational points of
together with a point at infinity
, forms an abelian group under an appropriate composition rule called addition, and the number of elements in the group
satisfies the Hasse bound:
![]() | (1) |
(see, for example, [11, Chapter V, Theorem 1.1]).
It is well-known that is of rank at most two, that is,
is isomorphic to
![]() | (2) |
for unique integers and
with
and
. The number
is called the exponent of
which we denote by
. In other words,
is the smallest positive
such that
for all points
.
We also put if
.
Thus we see that (1) and (2) imply the following trivial bound
![]() | (3) |
The exponent of elliptic curves has been studied in a number of works, see [4, 7, 8, 9, 10], 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 and
with
, and arbitrary small
, the bound
![]() | (4) |
holds for all but of primes
.
It is also shown in [10] that (4 ) holds for all but pairs
.
Here we use a combination of the results and ideas of [1, 10] to prove unconditionally that (4 ) is satisfied for almost all pairs with
,
for
and
relatively small compared to
.
Theorem 1. For any fixed and all integers
,
satisfying the inequalities
or
the bound
holds for all but pairs
with
,
In particular, Theorem 1 is nontrivial if
or
We also show that averaging over gives some additional saving.
Theorem 2. For any fixed and all integers
,
and
satisfying the inequalities
the bound (4 ) holds for all but triples
with
,
,
.
We note that the condition from [1], where it is used to simplify the error term, is not neccessary. One can easily extend Theorem 2 for
and
beyond this range, however since (as in [1]) small values of
and
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
In particular
For with
and two polynomial
, the reduction
is correctly defined. Various questions concerning the behaviour of the curves
on average over
and
have been studied in [2]. Here we continue to study this family of curves. Certainly the most interesting case is when
is small compared to
.
Theorem 3. Assume that the discriminant
is nonzero and the -invariant
is nonconstant. Then for any fixed and all integers
and
with
the bound
holds for all but pairs
with
,
.
The following result follows immediately from the more precise statement of [10, Theorem 3.1].
Lemma 4. For any , the number of triples
with
is at most .
Let and put
where the maximum is taken over all non-principal multiplicative characters modulo
such that
is the principal character
.
Similarly, we define and put
where the maximum is taken over all non-principal multiplicative characters modulo
such that
is the principal character
. For an arbitrary subset
, we denote by
the number of pairs such that
with
and
. We also denote
The following estimate is given in [1].
Lemma 5. For all primes , integers
, and subsets
such that whenever
the isomorphism
implies
, the following bound holds:
Moreover, it is shown in [1] that is small "on average" over
.
Lemma 6. The following bound holds:
For a prime and an integer
with
we denote by
the number of fractions
with
and
.
It is shown in [3] that is close to its expected value
on average over
. More precisely, we have:
Let be the set of pairs
for which
. Then it is enough to show that
Since by Lemma 4 we have , invoking Lemma 5 we see that it is enough to check that
.
Assume that then by the Burgess bound, see [6, Theorems 12.5 and 12.6], we have
. Also, if
then have
.
Similarly, if then
, and if
then have
.
As before, let be the set of pairs
for which
. Then it is enough to show that
![]() | (5) |
Let us assume that since the case
is similar.
Using the trivial bound for primes
, we deduce
![]() | (6) |
Noticing that for the set
satisfies the conditions of Lemma 5 , we obtain
![]() | (7) |
By Lemma 4 we have
![]() | (8) |
Substituting (7) and (8) in (6), we obtain
![]() |
We now easily verify that under the conditions of the theorem, Lemma 6 implies the desired bound (5).
As before, we use to denote the set of pairs
for which
.
Let be the set of
such that
for some .
Obviously, for any and
we have
(since the corresponding curves are isomorphic, see [11, Section III.1]).
We also note that the system of equations
leads to the equation
which has solutions (by the condition on the
-invariant
).
Therefore
Using Lemma 7, we obtain
which concludes the proof.
[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