SELMER GROUPS IN TWIST FAMILIES OF ELLIPTIC CURVES
ILKER INAM
Abstract. The aim of this article is to give some numerical data related to
the order of the Selmer groups in twist families of elliptic curves. To do this
we assume the Birch and Swinnerton-Dyer conjecture is true and we use a
celebrated theorem of Waldspurger to get a fast algorithm to compute LE(1).
Having an extensive amount of data we compare the distribution of the order
(log log( 1+εX))
of the Selmer groups by functions of type α with ε small. We
log(X)
discuss how the ”best choice” of α is depending on the conductor of the chosen
elliptic curves and the congruence classes of twist factors.
1. Introduction
The purpose of this article is to give some numerical data related to the order of
the Selmer groups in twist families of elliptic curves. This article is of experimental
type. It would be most interesting to give theoretical explanations for the statistical
observations we make.
Till today there is no algorithm that computes the order of the Selmer group of
a random elliptic curve defined over Q and even assuming the Conjecture of Birch
and Swinnerton-Dyer it is a hard computational problem to determine this order.
The situation will become easier if we restrict ourselves to twist families of some
specific elliptic curves. Working over Q we can use the theory of modular forms to
get an analytic function for the L-series of the discussed curves.
Assuming the Birch and Swinnerton-Dyer conjecture is true, we are able to
exploit a result of Waldspurger, which has a crucial role in this article. It yields an
efficient way to compute the order of the Selmer groups in twist families of elliptic
curves, if one can find an eigenform of weight 3/2 attached to the elliptic curve
via the Shimura-Shintani lift. Examples for this together with an explanation of
how to apply Waldspurger’s result are discussed in [1] (see Section 4). We use
these examples and compute the orders of the Selmer groups of twists of these
curves up to D ≤ 107. To do this one begins with a curve E and compares the
order of the Selmer groups of two twisted elliptic curves ED and ED with twist0 1
factors D0 and D1 in the same quadratic congruence class modulo 4.NE where
NE is the conductor of E. If one chooses the ED with D0 small then its Selmer0
group can be computed rather easily. So one can compute the order of the Selmer
groups for the elliptic curve ED by a fast computation described in Subsection1
3.3. After these computations and with many data, it is a natural question to study
the distribution of members in twist families for which the Selmer groups have the
same order, say k times the order of the torsion of E and to find simple functions
2000 Mathematics Subject Classification. 14H52, 11G40, 14G10.
Key words and phrases. Elliptic curves, Birch, Swinnerton-Dyer Conjecture, Zeta-functions
and related questions.
1
arXiv:1203.2791v1  [math.NT]  13 Mar 2012
2 ILKER INAM
that approximate this distribution. Through the article, we only interested in k in
order to compare the Selmer groups of different elliptic curves. In this article we
give numerical evidence that only constants have to be changed for different twist
families. We are interested in twisted elliptic curves which have rank zero, but
one has to be careful about the cases where twisted elliptic curves have (analytic)
positive rank. We define k = 0 to mean that the corresponding twisted elliptic
curve has positive (analytic) rank. In this case the torsion subgroup doesn’t play
any role by definition.
1.1. Overview. In Section 2, we present some necessary definitions. The notation
used in the article is introduced. Section 3 consists of four subsections. In the first
subsection, a statement of Waldspurger’s Theorem which plays a pivotal role in the
article is given. In Subsection 3.2., we describe how to compute d(n, n0). Proof
is given which can be deduced from some well-known facts. In Subsection 3.3. we
describe the algorithm to compute the order of the Selmer groups in twist families
of elliptic curves. Furthermore, the approximation function is introduced in this
subsection. We take the quotients of the distribution functions and formulate a
conjecture. Finally in Section 4, we give examples of our numerical results and in
particular tables listing constants α occurring in the approximating functions.
Lastly we plot a graph showing the behavior of the distribution function and the
approximating function.
1.2. Acknowledgements. This article was partly written during my visit at the
Institut für Experimentelle Mathematik in Universität Duisburg-Essen. I wish to
express my gratitude for the support and warm hospitality by this institution which
made the visit a very pleasant one and especially Prof.Dr.Gerhard Frey who sug-
gested this nice problem and made valuable comments and important improvements
on this article. Also I would like to thank Prof.Dr.Gabor Wiese who made com-
ments on an early version. This article has grown out of my PhD thesis. This
article is supported by the The Scientific and Technological Research Council of
Turkey (TUBITAK) Research Project, Project No: 107T311. I wish to thank the
referees for their helpful suggestions.
2. Background Material
Let E/Q be an elliptic curve and assume that D is a square-free integer. With
ED we denote the quadratic twist of E with D. For E given in ”short” Weierstrass
form
y2 = x3 − g2x− g3.
ED is given by
y2 = x3 − g D22 x− g3D3. √
ED is the elliptic curve defined over Q isomorphic to E over Q( D) but not over
Q.
We recall that E is modular and call the attached eigenform fE with q-expansion
∞
∑
fE = q + a q
n
n .
n=2
This is a newform in S2(NE , χ1) where S2(NE , χ1) is the space of cusp forms of
weight 2, level NE and χ1 is the trivial character.
SELMER GROUPS 3
The attached eigenform of ED is the twist of fE by the quadratic character χD
∞
∑
: fE := fE ⊗ χD = χD(n)a nnq ∈ S2(NE ) (and NE divides NE .D2). So theD D D
n=1
Hasse-Weil L−function of ED is
∑∞
LE (s) = χD(n)ann
−s.
D
n=1
In this paper, we shall give numerical data related to the order of the Selmer
groups of twist families {ED}. In particular we are interested in the number of
twists for which there are infinitely many points in ED(Q). Recall the theorem of
Mordell which states that
E(Q) ∼= E(Q) rtor × Z ,
where the torsion subgroup E(Q)tor is finite and the rank r of E(Q) is a non-
negative integer.
For any given elliptic curve, it is possible to describe quite precisely the torsion
subgroup [11]. The rank is much more difficult to compute, and in general there is
no known procedure which is guaranteed to yield an answer. But if the rank of E
is positive then a celebrated theorem of Kolyvagin states that LE(1) = 0. So we
are sure that if LE(1) 6= 0 then E(Q) is finite. The converse result is not known
today but it should be true. One part of the celebrated Birch and Swinnerton-Dyer
Conjecture (BSD) is that the order of vanishing of LE(s) at s = 1 (”the analytic
rank”) is equal to the rank of E(Q). BSD states much more. It interprets the value
of the first non-vanishing derivative of LE at s = 1 in terms of arithmetical objects
attached to E. We shall be interested in this prediction only in the case that the
analytic rank of E is 0.
We begin defining the Selmer and the Tate-Shafarevich group of elliptic curves
by using the Kummer sequence of elliptic curves: Let E be an elliptic curve over Q.
Let Q be an algebraic closure of Q and GQ := AutQ(Q) the absolute Galois group
of Q. Consider the abelian group E(Q) of all points on E defined over Q. One can
consider the Galois cohomology groups Hm(GQ, E(Q)) for m ∈ N.
For all n ∈ N, we have the exact sequence of GQ−modules
0 −→ E(Q)[n] −→ nE(Q) −→ E(Q) −→ 0.
As it is well known [9], there is an associated long exact sequence of Galois
cohomology groups. We need a consequence of the beginning of this sequence [11]
0 → αE(Q)/nE(Q) → H1(GQ, E(Q)[n]) → H1(GQ, E(Q))[n] → 0.
This sequence is called the Kummer Sequence associated to E. For each prime
p we choose an extension of the corresponding p−adic valuation. Let Gp be the
corresponding decomposition group in GQ which is in a canonical way isomorphic to
GQ . Let γp,n be the restriction map from H
1(GQ, E(Q)[n]) to H
1(Gp, E(Qp))[n]p
and P the set of primes. The Tate-Shafarevich group of E is denoted by ShaQ(E)
and defined by
⋃
ShaQ(E) := ShaQ(E)[n],
n∈N
where
⋂
Sha(E)[n] := ker(γp,n).
p∈P
4 ILKER INAM
The Selmer group of E is denoted by SQ(E) and defined by
⋃
SQ(E) := SQ(E)[n],
n∈N
where
S (E)[n] := α−1Q (ShaQ,S(E)[n]).
So we have the exact sequence
0 −→ E(Q)/nE(Q) −→ SQ(E)[n] −→ ShaQ(E)[n] −→ 0.
We are now ready to state the part of BSD which is of importance for us.
Conjecture 1. [2] LE(1) 6= 0 iff E(Q) is finite group, and then the Selmer group
of E is finite and the following equality holds:
 
 
∫
∏
| | #SQ(E) LE(1) = ω E c   p ,
#(E(Q))3
E0( ) p|N.∞R
where E0(R) is the connected component of E(R), ωE is the Néron differential of
E, c∞ = [E(R) : E
0(R)], and for primes p, cp = [E(Qp) : E
0(Qp)]. The numbers
cp are called local Tamagawa numbers.
We remark that all terms different from the order of the Selmer group are com-
putable more or less easily. But in some special cases it is possible to compute the
order of the Selmer groups (sometimes one has to assume its finiteness), and then
one can verify BSD. So there is numerical evidence for its truth.
Convention: Without further notice we always shall assume in this paper
that BSD holds and use the analytic theory of modular forms to compute both the
order of SQ(E) and LE(1) conditionally.
A good test for the exactness of algorithms is a result of Cassels for the order of
SQ(E):
Theorem 1. [3] Let E/Q be an elliptic curve. There exists an alternating, bilinear
pairing
Γ : ShaQ(E)× ShaQ(E) −→ Q/Z
whose kernel is precisely the group of divisible elements of Sha.
In particular if SQ(E) is finite, then k = #SQ(E)/#(E(Q)) is a perfect square.
3. Waldspurger Theorem and Its Consequences
3.1. Statement of Waldspurger’s Theorem. Assume that the rank of E is
equal to zero. As said above one can compute the order of the Selmer group
and hence of the Tate-Shafarevich group of E by using BSD. Note that the local
Tamagawa numbers cp as well as ωE can be computed easily (the latter value is
transcendental and hence has to be computed up to a desired precision). The most
time consuming item is the computation of LE(1). For this, there is a routine in
the computer algebra system MAGMA [7].
It turns out that computing LE(1) with the necessary precision (again this is a
transcendental) for an elliptic curve E with large conductor takes a long time.
For instance, computing LE(1) for the elliptic curve
E : y2 = x3 − 87662765543106x+ 572205501116432432042932656
SELMER GROUPS 5
which has conductor 11520793560025904, one needs at least 1000 hours in a lap-
top computer1 with MAGMA which doesn’t guarantee to answer. Another hard
numerical problem is to decide by computation whether LE(1) = 0.
The situation is much better in families of twists of a given elliptic curve. The
elliptic curve E from above is a member of such family, and we shall see in Section
3.3 how this can be used to accelerate the computation dramatically. The reason
is Waldspurger’s Theorem which is crucial for our work:
Theorem 2. [12] Let E be an elliptic curve over Q with attached new form fE.
Assume that FE ∈ S ′3/2(N ,χ1) is an eigenform and S(FE) = fE where S is the
Shimura-Shintani lifting.
Let an be the n−th Fourier coefficient of FE. Then for square-free natural num-
∏ 2
bers n and n with n ≡ n mod Q∗ and n.n prime to N ′0 0 p 0 we have
p|N ′
√
a2
√
n nLE (1) = a
2
n n0LE (1),0 −n −n0
Hence we get: If an =6 0 then LE (1) is determined by an, a0 −n n and L0 E−n (1).0
∏ 2
In particular, LE (1) = 0 for all n ≡ n ∗0 mod Qp iff LE−n (1) = 0, and−n 0
p|N ′
else LE (1) =6 0 iff an 6= 0.−n
Corollary 1. [1] Assume that the Birch and Swinnerton-Dyer conjecture holds for
E−n and E−n , n, n0 as in the theorem and that an ·LE−n (1) =6 0. Then E−n(Q)0 0 0
is finite iff an 6= 0 and
a2
#SQ(E−n) = d(n, n0) ·#SQ(E n−n ) ,0 a2n0
where d(n, n0) is easily computed as explained in Subsection 3. 2 and essentially a
power of 2 depending on the divisor structure of n, n0.
3.2. Computing d(n, n0). We continue to assume that E−n and E−n are twists0
of E satisfying the conditions of Corollary 4.
We want to compute the numbers d(n, n0). By definition d(n, n0) depends on
the Tamagawa numbers and the torsion subgroups of the two elliptic curves E−n
and E−n .0
To be explicit one has to use some easy facts about twists of elliptic curves.
d(n, n0) does not depend on the real period since t√he tw√isting factors n0 and n are
odd and congruent modulo 4 and ωE−n/ωE−n = n0/ n and hence cancel in the0
formula in Corollary 4.
Independence of torsion elements:
It is well known and obvious that for all pairs of elliptic curve E and twists E−n
we have E(Q)[2] = E−n(Q)[2].
Moreover for given E there are only finitely many (in fact only very few) twists
of E which have torsion points of order > 2 over Q. Avoiding these twists is easy
and so one can assume without falsifying the statistic, that all members of the
twist families have only Q−rational torsion points of order dividing 2. In fact, in
the chosen examples below this holds for all non-trivial twists of the treated curves
E. So we can assume that the order of E−n (Q) is equal to the order of E0 −n(Q)
and hence d(n, n0) is independent of torsion elements.
1With the properties: Intel Core 2 Duo Mobile, 2GB DDR2, 2.00GHz
6 ILKER INAM
The next observation is that the groups of connected components of twists of an
elliptic curve E over the reals are equal, and so d(n, n0) is computed by looking at
the non-Archimedean Tamagawa numbers.
Let us denote the Tamagawa numbers for E−n at a prime p by cn,p and the
Tamagawa numbers for E−n at a prime p by c0 n ,p.0
First observe that for p prime to n0 · n · N ′ both twists have good reduction
modulo p and so the Tamagawa numbers are equal to 1.
By assumption −n0 and −n lie in the same class of squares in all completions
with respect to divisors of N ′E and so the Néron models are equal at all primes
dividing N ′E .
Now let p be a divisor of, say, n prime to N ′E . Since E has good reduction
modulo p and p is odd we can use the table 15.1 in [11], p. 359 to see that E has
Kodaira symbol I0 and so cn,p = 4. The same result holds of course for prime
divisors of n0.
Hence we get
Lemma 1. Let E be an elliptic curve and E−n and E−n be twists of E with0
#E−n(Q) = #E−n (Q) < ∞ and n.n0 prime to N ′E and a square in all completions0
with respect to divisors of N ′E . Then
∏
c 4#div(n0)n ,p
d(n, n0) =
0
∏ = ,
c 4#div(n)n,p
where #div(−) denotes the number of prime divisors of −.
For using Waldspurger Theorem for members of the twist family {E−n} one has
to find an eigenform FE as above. Then one has to implement a fast algorithm for
computing the Fourier coefficients of FE in a large range.
3.3. Computing Fourier Coefficients and the Selmer Group. Recall the sit-
uation. We have an elliptic curve E with eigenform fE and the Shimura-Shintani lift
FE given in a concrete way. In particular we shall consider the following examples
from [5]:
FE E
(Θ(X2 + 11Y 2)−Θ(3X2 + 2XY + 4Y 2)).Θid,11 11a1
(Θ(X2 + 14Y 2)−Θ(2X2 + 7Y 2)).Θid,14 14a1
(Θ(3X2 − 2XY + 23Y 2)−Θ(7X2 + 6XY + 11Y 2)).Θid,17 17a1
(Θ(X2 + 20Y 2)−Θ(4X2 + 5Y 2)).Θid,20 20a1
(Θ(X2 + 17Y 2)−Θ(2X2 + 2XY + 9Y 2)).Θid,17 34a1
∞
∑ 2
where Θ(.) is the theta series of a binary quadratic form and Θψ,t := ψ(n)q
tn
n=−∞
is a Fourier series for the Dirichlet character ψ.
The elliptic curve E is given as in Cremona’s Table [4].
∞
∑
Let FE ∈ S3/2(N ′, χ1) as above with Fourier expansion a qnn .
n=1
Strategy
1) Calculate the q−expansion of FE up to an upper bound M , construct the list
L := {(n, an)|n ∈ {1, · · · ,M} squarefree}.
2) Choice of Congruence Classes: To apply Waldspurger’s theorem we compare
twists with twist factors −n, −n0 with n and n0 odd and prime to N ′ which are
SELMER GROUPS 7
∏ 2 ∏
congruent modulo Q∗p . This is satisfied if n ≡ n0 mod 8 · p and hence we
p|N ′ 26=p|NE
shall investigate twist families with twist factors in such congruence classes. First
we determine the twist families (with respect to the above congruences) which
consist of odd elliptic curves and so have positive analytic rank by looking at the
parity of the twist characters. We delete these congruence classes.
We simplify the situation in the cases NE = 11 and NE = 17. To apply Wald-
spurger’s theorem we have to look at congruence classes modulo 88 and respectively
136. We check that for all pairs of these congruence classes which become equal
modulo 44 respectively 68 there are n n′0 0 with the same number of prime divi-
sors, the Fourier coefficients an = an′ and the same order of the Selmer groups0 0
and hence we can investigate in these cases twist families with families with twist
factors running over congruences modulo 44 respectively 68.
We list the resulting congruence classes in Table A.
Elliptic Curve Modulo n0
11a1 44 1, 3, 5, 15, 23, 31, 37
14a1 56 1, 15, 23, 29, 37, 39, 53
17a1 68 3, 7, 11, 23, 31, 39
20a1 40 1, 21, 29
34a1 136 1, 13, 19, 21, 33, 35, 43, 53, 59, 67, 69, 77,
83, 89, 93, 101, 115, 117, 123
Table A.
3) For the integer M and fixed n0 calculate
xn (M) := #{n : n ≤ M, n is square-free, n ≡ n0 (modN ′)},0
sn ,0,E(M) := #{n : n ≤ M , n is square-free, n ≡ n0 (mod N ′), an = 0},0
and plot the function sn ,0,E(M)/xn (M).0 0
4) For n0, find α ∈ R and ǫ ∈ [−0.02, 0.02] such that
(log log(x 1+ǫn (M)))
σ(xn (M)) := α
0 .
0 log(xn (M))0
approximates sn ,0,E(M)/xn (M) ”well”.0 0
5) If an = 0 then replace n0 by the minimal n in the congruence class such0
that an =6 0. Calculate LE (1), #E−n (Q)tors and #SQ(E−n ) by using the0 −n0 0 0
BSD-conjecture and fE .
6) For n0 and n ≤ M , compute d(n, n0) as described in Subsection 3.2.
7) For n0 and n ≤ M , compute
#S (E ) · a2Q −n n · d(n, n0 0)sE := .−n a2n0
which is, according to the BSD-conjecture, the order of SQ(E−n).
8) Compute t := #E(Q). Recall that twisting E doesn’t change the order of the
torsion subgroup of E.
9) For M,k, t and n0 compute
sE
sn ,k,E(M) := #{n : n ≤ M , n is square-free, n ≡ n0 (mod N ′), −n = k}.0 t
10) For n0, plot the function sn0,k,E(M)/xn (M).0
8 ILKER INAM
11) For n0, find α ∈ R and ǫ ∈ [−0.02, 0.02] such that
(log log(x (M)))1+ǫn
σ(x 0n (M)) = α0 log(xn (M))0
approximates sn ,k,E(M)/xn (M) ”well”.0 0
Remark 1. 1) All data can be found in http://homepage.uludag.edu.tr/˜inam/
2) Having computed #SQ(E−n), d(n, n0) and LE (1), one can use the BSD-−n0
conjecture again to compute LE (1) as−n
LE (1) ·#SQ(E−n −n)
LE (1) =
0
−n #SQ(E−n ) ·
.
d(n, n0)0
This is much faster than to compute LE (1) directly. We have included these−n
values in our lists.
We come back to the example in Section 3.1. Recall that we wanted to compute
the value of LE(1) of the curve
E : y2 = x3 − 87662765543106x+ 572205501116432432042932656.
It is the twist of the elliptic curve 11a1 with the twist factor n = 8090677, and by
the method described above, we get very fast
LE(1) = 2.10072023061090418110927626775
approximately in 360 seconds.
We now fix an elliptic curve E as well as n0 and k.
We sketch how to determine an approximation function for qn0,k,E . We choose
α and ε in the following way: In this work, using the data obtained up to the
bound M = 107, we construct a family {Ii} of subintervals of I := [0,M ] defined
by Ii = [0, 50000i] for i = 1, 2, · · · , 200 such that
I1 ⊆ I2 ⊆ · · · ⊆ I200 = I.
s (M ) log(log(x (M ))
First we calculate the value n0,k,E ix (M ) and afterwards
n0 i . Com-
n0 i
xn (M )0 i
paring these two values, the constant αi can be obtained for each Ii. Using the
weighted average for the constants αi we determine α (depending on M). By means
of these constants α, we compare the values
sn ,k,E(Mi) log(log(xn (M0 0 i))and α .
xn (Mi) xn (M )0 0 i
After this step we choose for fine-tuning ε ∈ [−0.02, 0.02] such that the approx-
imation is getting better.
4. Numerical Results
We use the considerations of Subsection 3.3 for extensive computations and ob-
serve that in all our examples the functions
sn0,k,Eqn0,k,E := (M)xn0
are fairly well approximated by
log(log(x 1+εn (M))
α 0
log(xn (M))0
where α > 0 and ε ∈ [−0.02, 0.02].
SELMER GROUPS 9
This observation confirms predictions stated by Birch and lead to
Conjecture 2. For all elliptic curves E, E′ over Q, all n , n′0 0 satisfying the con-
q
ditions of Theorem 3 and all k, k′ the asymptotic behavior of n0,k,Eq is well ap-n′ ,k′,E′
0
proximated by a constant times a factor log(log(x(M))δ where x(M) is the number
of square-free numbers ≤ M and δ is a real number with small absolute value.
Of course one should be much more precise and predict how the factor depends
on the parameters. In our context we shall restrict ourselves to a discussion of
the reals α we get out of our data by the approximation process in the algorithm
described above.
4.1. Observations. Conjecture 7 predicts that the type of the approximation func-
tion is independent of k. But the constants α vary and so a finer analysis seems
necessary in order to find reasons or patterns for the size of α.
But let us begin with a word of caution. In our examples we computed α for
k ≤ 961. For large k we do not have enough material for any statistical statement.
(The record, k = 68121 occurs just one time).
We now discuss examples of the weighted average values which are given in
Section 4.3.
4.1.1. Some Examples. Considering the values of α in our examples we see that it
depends significantly on the congruence classes modulo 4 respectively 8.
Case 1. For the elliptic curve 11a1, we have the class K := {1, 5, 37} which have
members congruent to 1 modulo 4 and the class L := {3, 15, 23, 31} with members
congruent to 3 modulo 4. As example, we take k = 1 and see that the values of
α for n0 in class K are respectively 0.296, 0.299, 0.300, where as for n0 in class L
they are respectively 0.458, 0.459, 0.469, and 0.464 (See Section 4.3).
Case 2. For the elliptic curve 14a1, we have one class modulo 8 in which all twists
are odd curves, namely n congruent to 3 modulo 8. The other classes modulo
8 separate our congruence classes modulo 56 into K := {1}, L := {29, 37, 53},
M := {15, 23, 39}. As example we take k = 9 and get the values of α for n0 in class
K are 0.485, for n0 in class L are respectively 0.195, 0.185 and 0.192 whereas for
n0 in class M they are respectively 0.559, 0.568 and 0.536.
Case 3. For the elliptic curve 17a1. In this case all congruence classes modulo 68
which are congruent to 1 modulo 4 are odd, and for all classes congruent 3 modulo
4 the values α are around 0.33 and hence of the same size for k = 0 as an example.
Case 4. For the elliptic curve 20a1, all congruence classes modulo 40 which are
congruent to 3 modulo 4 are odd, and for all classes congruent 1 modulo 4 the
values α are around 0.1 and hence of the same size for k = 225 as an example.
Case 5. For the elliptic curve 34a1, we looked at congruence classes modulo
8. If n is congruent 7 modulo 8 we get odd curves. The other classes mod-
ulo 8 consist of K := {1, 33, 89}, L := {19, 35, 43, 59, 67, 83, 115, 123}, M :=
{21, 53, 69, 77, 93, 101, 117}. Again take k = 1, then the values of α are around
0.38 and for every n0 ∈ L, the values of α are around 0.47 and for every n0 ∈ M ,
the values of α are around 0.41.
4.2. Some Actual Values. We give some examples of actual values. Here, all
the notation given before is valid and σ(xn (M)) is defined by σ(xn (M)) :=0 0
(log log(xn (M)))
1+ǫ
α 0log(x . The values of α are given in Subsection 4.3.n (M))0
10 ILKER INAM
For the elliptic curve 11a1, n0 = 3, k = 4 and ε = 0.005 we have
M sn ,k(xn (M))/xn (M) σ(xn (M))0 0 0 0
50000 0.106452 0.099558
1500000 0.074267 0.066593
3000000 0.066195 0.062381
4000000 0.062997 0.060786
5000000 0.060743 0.059604
10000000 0.053981 0.056209
For the elliptic curve 14a1, n0 = 1, k = 16 and ε = 0.005 we have
M sn0,k(xn (M))/xn (M) σ(xn (M))0 0 0
100000 0.082313 0.09115
1400000 0.066638 0.066978
2000000 0.06462 0.064665
5000000 0.060241 0.059394
8000000 0.056955 0.057011
10000000 0.055412 0.055946
For the elliptic curve 17a1, n0 = 7, k = 324 and ε = 0.005 we have
M sn ,k(xn (M))/x0 0 n (M) σ(xn (M))0 0
100000 0 0.016898
5000000 0.009965 0.010903
6000000 0.010771 0.010726
7000000 0.011213 0.01058
8000000 0.011651 0.010458
10000000 0.012272 0.010259
For the elliptic curve 20a1, n0 = 1, k = 100 and ε = 0.005 we have
M sn ,k(xn (M))/xn (M) σ(xn (M))0 0 0 0
500000 0.026748 0.038045
3000000 0.029427 0.031896
5000000 0.029764 0.030491
6000000 0.029958 0.030019
7000000 0.030039 0.029632
10000000 0.030132 0.028772
For the elliptic curve 34a1, n0 = 1, k = 36 and ε = 0.005 we have
M sn ,k(xn (M))/xn (M) σ(x0 0 0 n (M))0
3000000 0.066667 0.071691
5000000 0.069827 0.069099
6000000 0.068564 0.067903
7000000 0.0682 0.066922
8000000 0.067812 0.066096
10000000 0.067339 0.064759
4.3. Weighted average values of α’s.
SELMER GROUPS 11
E n0 0 1 4 9 16
11a1 1 0.140221 0.295669 0.204751 0.309679 0.184184
11a1 3 0.214424 0.458141 0.296646 0.445157 0.244439
11a1 5 0.139959 0.299438 0.199569 0.308824 0.186938
11a1 15 0.211029 0.458734 0.299005 0.442075 0.244968
11a1 23 0.208441 0.468673 0.304083 0.440648 0.23676
11a1 31 0.208064 0.46357 0.205327 0.441027 0.23975
11a1 37 0.14234 0.300449 0.205431 0.314227 0.18237
E n0 25 36 49 64 81
11a1 1 0.312985 0.179629 0.239533 0.144443 0.235818
11a1 3 0.423453 0.208141 0.27803 0.141689 0.258489
11a1 5 0.320314 0.180445 0.238248 0.140963 0.234221
11a1 15 0.415875 0.207029 0.278424 0.146673 0.249818
11a1 23 0.411152 0.20807 0.283056 0.149102 0.250205
11a1 31 0.412866 0.205186 0.277763 0.150807 0.254808
11a1 37 0.314151 0.171081 0.235285 0.143295 0.230991
E n0 100 121 144 169 196
11a1 1 0.149895 0.188637 0.115865 0.171031 0.091912
11a1 3 0.144478 0.144478 0.099708 0.158332 0.068375
11a1 5 0.151277 0.183852 0.117823 0.165406 0.089266
11a1 15 0.1391 0.1391 0.096762 0.156685 0.069026
11a1 23 0.140066 0.190515 0.09152 0.159746 0.070316
11a1 31 0.141375 0.185663 0.098614 0.162711 0.071718
11a1 37 0.152059 0.184835 0.112431 0.172438 0.08812
E n0 225 256 289 324 361
11a1 1 0.203897 0.076341 0.135676 0.07271 0.117004
11a1 3 0.178166 0.054526 0.113502 0.047742 0.097803
11a1 5 0.201578 0.076127 0.132841 0.070794 0.116838
11a1 15 0.185691 0.055885 0.117211 0.0487 0.094476
11a1 23 0.179255 0.058757 0.112958 0.048998 0.0973
11a1 31 0.18378 0.054368 0.116274 0.049585 0.0968
11a1 37 0.208334 0.079288 0.132626 0.06951 0.117505
E n0 0 1 4 9 16
14a1 1 0.283019 0.386791 0.349053 0.485425 0.289702
14a1 15 0.319039 0.483879 0.409463 0.559197 0.319645
14a1 23 0.336754 0.461198 0.402525 0.567646 0.312323
14a1 29 0.442312 0.172938 0.560877 0.194746 0.485192
14a1 37 0.42059 0.175757 0.589686 0.185407 0.492192
14a1 39 0.312339 0.493768 0.407928 0.536247 0.328624
14a1 53 0.447676 0.171374 0.571985 0.191641 0.472537
12 ILKER INAM
E n0 25 36 49 64 81
14a1 1 0.28595 0.326485 0.190388 0.148085 0.292196
14a1 15 0.314341 0.322687 0.235968 0.181454 0.290788
14a1 23 0.314975 0.331965 0.236908 0.173742 0.299374
14a1 29 0.099951 0.567019 0.071551 0.310347 0.086056
14a1 37 0.108121 0.544933 0.076132 0.32644 0.081628
14a1 39 0.327089 0.316292 0.251003 0.186261 0.27054
14a1 53 0.104698 0.561543 0.073051 0.321251 0.086696
E n0 100 121 144 169 196
14a1 1 0.142119 0.149737 0.159668 0.119627 0.087451
14a1 15 0.139186 0.150902 0.143504 0.116318 0.076208
14a1 23 0.134807 0.144059 0.150461 0.112789 0.072932
14a1 29 0.265114 0.039297 0.303978 0.03107 0.174705
14a1 37 0.276584 0.042321 0.285788 0.029985 0.18166
14a1 39 0.140236 0.14836 0.135962 0.11914 0.079435
14a1 53 0.264155 0.036761 0.306407 0.029676 0.174854
E n0 225 256 289 324 361
14a1 1 0.147626 0.073324 0.081276 0.090333 0.069123
14a1 15 0.13048 0.060237 0.071448 0.066382 0.058287
14a1 23 0.138795 0.059647 0.070181 0.074594 0.054617
14a1 29 0.033083 0.144352 0.017318 0.190917 0.011085
14a1 37 0.031929 0.141464 0.01651 0.180291 0.011341
14a1 39 0.129477 0.062409 0.074189 0.06682 0.059717
14a1 53 0.03286 0.137962 0.01691 0.194275 0.012076
E n0 0 1 4 9 16
17a1 3 0.337432 0.477285 0.501361 0.41195 0.402614
17a1 7 0.333173 0.480345 0.512958 0.411449 0.397752
17a1 11 0.331548 0.470597 0.506991 0.41595 0.398308
17a1 23 0.324727 0.469987 0.510093 0.413703 0.409981
17a1 31 0.332091 0.482396 0.496191 0.410295 0.405293
17a1 39 0.335686 0.481485 0.50061 0.403566 0.403538
E n0 25 36 49 64 81
17a1 3 0.291697 0.298922 0.227993 0.217491 0.190683
17a1 7 0.294852 0.305199 0.224537 0.209766 0.191861
17a1 11 0.29197 0.307093 0.224453 0.212266 0.194131
17a1 23 0.302838 0.296815 0.223042 0.212759 0.197424
17a1 31 0.299154 0.303459 0.219757 0.213895 0.19307
17a1 39 0.29654 0.302868 0.21993 0.218869 0.19364
E n0 100 121 144 169 196
17a1 3 0.155873 0.138493 0.129301 0.109835 0.086544
17a1 7 0.153253 0.134025 0.127087 0.109443 0.08902
17a1 11 0.152146 0.142299 0.1244 0.115795 0.090622
17a1 23 0.149471 0.136131 0.125797 0.115204 0.088695
17a1 31 0.153817 0.141537 0.128656 0.10723 0.086032
17a1 39 0.155061 0.139321 0.12622 0.110953 0.084326
SELMER GROUPS 13
E n0 225 256 289 324 361
17a1 3 0.10133 0.066479 0.070144 0.054392 0.063161
17a1 7 0.102284 0.066453 0.07182 0.052183 0.061862
17a1 11 0.101815 0.06621 0.073766 0.053459 0.060803
17a1 23 0.104502 0.066214 0.069106 0.055098 0.060021
17a1 31 0.106572 0.068254 0.071002 0.057448 0.058929
17a1 39 0.108346 0.065278 0.073044 0.055616 0.058537
E n0 0 1 4 9 16
20a1 1 0.268253 0.3465 0.315475 0.427111 0.27129
20a1 21 0.266462 0.337876 0.32056 0.431508 0.272359
20a1 29 0.267792 0.343135 0.317666 0.425236 0.271235
E n0 25 36 49 64 81
20a1 1 0.254326 0.307296 0.210761 0.179752 0.245513
20a1 21 0.253463 0.304903 0.209301 0.178783 0.246748
20a1 29 0.258567 0.308143 0.20873 0.178674 0.252364
E n0 100 121 144 169 196
20a1 1 0.144222 0.141449 0.171656 0.115768 0.095252
20a1 21 0.146098 0.144796 0.165373 0.115249 0.09443
20a1 29 0.142937 0.14178 0.1634 0.115021 0.09569
E n0 225 256 289 324 361
20a1 1 0.141739 0.076533 0.082182 0.091834 0.066588
20a1 21 0.147373 0.078015 0.082924 0.091549 0.067251
20a1 29 0.14228 0.081021 0.081558 0.091311 0.067867
E n0 0 1 4 9 16
34a1 1 0.300968 0.385865 0.387258 0.462396 0.28402
34a1 13 0.290206 0.415303 0.352209 0.474225 0.272241
34a1 19 0.353435 0.475157 0.436218 0.505167 0.317592
34a1 21 0.29167 0.415613 0.359182 0.472539 0.274045
34a1 33 0.30458 0.388798 0.381037 0.440285 0.291886
34a1 35 0.357437 0.47347 0.42035 0.504132 0.326558
34a1 43 0.355486 0.466179 0.44077 0.503479 0.323861
34a1 53 0.281215 0.413834 0.357105 0.470171 0.283536
34a1 59 0.345971 0.471297 0.436144 0.50406 0.327326
34a1 67 0.351665 0.467335 0.427308 0.512714 0.326024
34a1 69 0.290429 0.408492 0.366839 0.478386 0.275193
34a1 77 0.293768 0.41554 0.350608 0.478178 0.272873
34a1 83 0.352644 0.475251 0.440215 0.500611 0.328119
34a1 89 0.305732 0.396955 0.372179 0.443078 0.296228
34a1 93 0.283804 0.42395 0.358696 0.479956 0.279951
34a1 101 0.29705 0.412981 0.359811 0.476887 0.286045
34a1 115 0.34747 0.476572 0.438912 0.505538 0.321909
34a1 117 0.291683 0.420945 0.355004 0.476725 0.278145
34a1 123 0.354215 0.475638 0.437478 0.495364 0.32921
14 ILKER INAM
E n0 25 36 49 64 81
34a1 1 0.247423 0.309932 0.194351 0.177262 0.225383
34a1 13 0.271392 0.294956 0.199301 0.166857 0.230411
34a1 19 0.271006 0.324198 0.191454 0.171407 0.214279
34a1 21 0.265973 0.301452 0.19956 0.159942 0.230879
34a1 33 0.267835 0.311831 0.193117 0.172183 0.229097
34a1 35 0.275831 0.327544 0.189914 0.17779 0.220039
34a1 43 0.272699 0.317971 0.202658 0.174474 0.211269
34a1 53 0.277814 0.310885 0.201981 0.161572 0.229304
34a1 59 0.267928 0.327115 0.193429 0.175461 0.215779
34a1 67 0.273065 0.324959 0.192272 0.172805 0.212269
34a1 69 0.267456 0.29777 0.207129 0.162831 0.232521
34a1 77 0.272458 0.297814 0.201069 0.167444 0.227964
34a1 83 0.275118 0.314132 0.199511 0.174251 0.210163
34a1 89 0.255453 0.318219 0.207944 0.165966 0.226874
34a1 93 0.259963 0.297419 0.204218 0.165222 0.234308
34a1 101 0.254251 0.303388 0.197465 0.15854 0.23436
34a1 115 0.270414 0.323107 0.196365 0.177784 0.19878
34a1 117 0.260653 0.2887 0.202377 0.157916 0.244395
34a1 123 0.270226 0.335145 0.200594 0.170866 0.208101
E n0 100 121 144 169 196
34a1 1 0.128304 0.122231 0.147564 0.102664 0.082786
34a1 13 0.129592 0.130917 0.143784 0.100675 0.07895
34a1 19 0.130772 0.109944 0.140669 0.086182 0.077919
34a1 21 0.132753 0.126583 0.143338 0.106306 0.080138
34a1 33 0.131932 0.123085 0.140728 0.099784 0.082223
34a1 35 0.130884 0.111755 0.141213 0.08385 0.079269
34a1 43 0.128261 0.109375 0.13594 0.083931 0.071109
34a1 53 0.12997 0.126922 0.142644 0.100092 0.07645
34a1 59 0.137624 0.106901 0.140647 0.08392 0.074337
34a1 67 0.138161 0.108241 0.136687 0.090009 0.081168
34a1 69 0.127983 0.12296 0.146421 0.100218 0.072738
34a1 77 0.130561 0.122929 0.148044 0.098305 0.080768
34a1 83 0.130011 0.103015 0.141567 0.085853 0.077536
34a1 89 0.132737 0.115162 0.150665 0.0967 0.08984
34a1 93 0.127371 0.120887 0.147903 0.100794 0.071056
34a1 101 0.124347 0.118514 0.136806 0.10392 0.079833
34a1 115 0.13509 0.116645 0.140947 0.089575 0.075199
34a1 117 0.130015 0.124739 0.140575 0.104611 0.078912
34a1 123 0.135871 0.107328 0.133703 0.093233 0.07586
SELMER GROUPS 15
E n0 225 256 289 324 361
34a1 1 0.120486 0.06472 0.062951 0.070443 0.055249
34a1 13 0.120107 0.0621 0.0692 0.077468 0.055683
34a1 19 0.077919 0.059063 0.053673 0.070172 0.038649
34a1 21 0.118898 0.062197 0.06578 0.071127 0.056771
34a1 33 0.118185 0.059254 0.065998 0.072214 0.053245
34a1 35 0.092942 0.058685 0.050621 0.063069 0.039964
34a1 43 0.097305 0.057743 0.053748 0.069417 0.0381
34a1 53 0.120493 0.060345 0.065208 0.071921 0.049442
34a1 59 0.097042 0.061403 0.047947 0.065448 0.040482
34a1 67 0.09381 0.057656 0.066198 0.064485 0.038135
34a1 69 0.121028 0.065232 0.066193 0.066645 0.052697
34a1 77 0.120666 0.065448 0.067139 0.072256 0.052677
34a1 83 0.098931 0.058119 0.049453 0.067301 0.040662
34a1 89 0.117926 0.059319 0.062727 0.072717 0.055294
34a1 93 0.115051 0.060207 0.061973 0.076746 0.058861
34a1 101 0.123836 0.063449 0.070854 0.072985 0.058416
34a1 115 0.099212 0.057176 0.049053 0.065697 0.037937
34a1 117 0.120381 0.061819 0.067332 0.07149 0.054083
34a1 123 0.097763 0.057011 0.049838 0.06636 0.038657
4.4. A Graphical Example. We plot some graph of the data for E = 11a1, n0 =
1, k = 1. In this graph on the x−axis we plot x 7n (M) up toM = 10 . Dots above at0
the beginning belong to the graph of the function s1,1(x1(M))/x1(M), dots below
1.005
at the beginning belong to the graph of the function 0.295669 (log log(x1(M)))log(x1(M)) .
References
[1] J. A. Antoniadis, M. Bungert and G. Frey, Properties of Twist of Elliptic Curves, J. Reine
Angew. Math., 405 (1990), 1-28,
[2] B. Birch and H. P. F. Swinnerton-Dyer, Notes on Elliptic Curves II, J. Reine Angew. Math.,
218 (1965), 79-108,
[3] J. W. S. Cassels, Arithmetic on Curves Genus 1, VIII On Conjectures of Birch and
Swinnerton-Dyer, J. Reine Angew. Math., 217 (1965), 180-199,
[4] J. E. Cremona, Algorithms for Modular Elliptic Curves, 2nd Edition, Cambridge Univ. Press,
Cambridge, 1997,
[5] G. Frey, Construction and Arithmetical Applications of Modular Forms of Low Weight, CRM
Proceedings & Lecture Notes Amer. Math. Soc, 4, (1994), 1-21,
[6] V. Kolyvagin, Finiteness of E(Q) and ShaE(Q) for a class of Weil curves, Math. USSR, Izv.,
32 (1989), 523-541,
[7] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,
J. Symbolic Comput., 24 (1997), (3-4):235-265,
[8] A. P. Ogg, Rational Points of Finite Order on Elliptic Curves, Invent. Math., 9 (1971),
105-111,
[9] J. P. Serre, Local Fields, Volume 67 of Graduate Texts in Mathematics, Springer-Verlag, New
York, 1979,
[10] G. Shimura, On Modular Forms of half-integral weight, Annals of Math., 97 (1973), 440-481,
[11] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986, ISBN 0-387-96203-
4,
[12] J. L. Waldspurger, Sur les Coefficients de Fourier des Formes Modulaires de Poids Demi-
Entier, J. Math. Pures et Appl., 60 (1981), 375-484.
16 ILKER INAM
Received: 27 September, 2010 and in revised form 24 January 2011.
Uludag University, Faculty of Art and Science, Department of Mathematics, Gorukle,
Bursa-Turkey
E-mail address: inam@uludag.edu.tr, ilker.inam@gmail.com