Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
A Statistical Analysis of 2-Selmer Groups forElliptic Curves with Prescribed Torsion
A. Rollick1 J. Weigandt2
1Department of MathematicsJohn Carroll University
2Department of MathematicsPurdue University
Summer Undergraduate MathematicalSciences Research Institute 2007
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Outline
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8The 2-Selmer Group
ComputationsmwrankAlgorithm
Distribution of 2-Selmer RanksPoisson Distribution?Generating Functions
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
Introduction
I We are attempting to find Mordell-Weil rank r of ellipticcurves
I An upper bound for the rank is s, the rank of the 2-Selmergroup of the elliptic curve
I E(Q) may be infinite group
I Instead considerE(Q)
2E(Q), which is a finite group
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
Connecting HomomorphismLet E be a curve such that:
I is defined by Y 2 = X 3 + A X + B where A and B ∈ ZI X 3 + A X + B has three distinct rational roots: e1, e2, e3
Then we have the “connecting homomorphism”:
δE :E(Q)
2E(Q)→ Q×
(Q×)2 ×Q×
(Q×)2 , P = (X ,Y ) 7→ (X −e1,X −e2)
δE is injective and its image lies in a finite group
G =
{(d1, d2)
∣∣∣∣ di = ±`ai11 · · · `
airr , where
`j divides −16(4 A3 + 27 B2)
}
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
Counting Rational Points
For (d1,d2) ∈ G consider the curve
Cd : d1u2 − d2v2 = e2 − e1, d1u2 − d1d2w2 = e3 − e1.
Then the connecting homomorphism implies the isomorphism
E(Q)
2 E(Q)'{
d ∈ G∣∣∣∣ Cd (Q) 6= ∅
}' Z2
r+2.
That is, if (u, v ,w) ∈ Cd (Q) then (d1 u2 + e1, d1 d2 u v w) inE(Q). Conversely, if P ∈ E(Q) then there is a point in Cd (Q) ford = δE (P).
To compute the rank r we count d ∈ G such that Cd (Q) 6= ∅.
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
Motivation
I Identify points inE(Q)
2 E(Q)with homogeneous spaces.
I Need to find rational points on Cd for d ∈ G.
I How do we do this?
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
2-Selmer and Shafarevich-Tate Groups
E(Q)
2 E(Q)
δE−−−−→ Sel(2)(E/Q) −−−−→ III(E/Q)[2]
where we define the 2-Selmer and Shafarevich-Tate groups
Sel(2)(E/Q) =
{d ∈ G
∣∣∣∣ Cd (R) 6= ∅ andCd (Qp) 6= ∅ for all primes p
}
III(E/Q)[2] =
d ∈ G
∣∣∣∣∣∣Cd (R) 6= ∅ and
Cd (Qp) 6= ∅ for all primes pbut Cd (Q) = ∅
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
The 2-Selmer Group
2-Selmer and Shafarevich-Tate Groups
I
∣∣∣∣ E(Q)
2E(Q)
∣∣∣∣ = 2r+2
I
∣∣∣Sel(2)(E/Q)∣∣∣ = 2s+2
I∣∣III(E/Q)[2]
∣∣ = 2s−r
I 2-Selmer group is easy to compute, Shafarevich-Tategroup is hard to compute
I Hopefully r = s, i.e., the Shafarevich-Tate group is trivialI We use mwrank
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
mwrank
John Cremonahttp://www.maths.nott.ac.uk/personal/jec/mwrank/index.html
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
mwrank
Searching For a Curve with Rank r ≥ 3
To search for an elliptic curve defined over Q with torsionsubgroup Z2 × Z8 and rank r ≥ 3, we use the followingalgorithm:
#1. Generate a list of candidate curves
#2. Compute the ranks of the 2-Selmer groups of these curves.
#3. Compute the Mordell-Weil ranks of the curves with2-Selmer ranks s ≥ 3.
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Classification of Curves with Torsion Z2 × Z8
E is an elliptic curve with torsion subgroup Z2 × Z8 if and only ifthere exist integers a and b such that E is birationallyequivalent to
y2 = (1− x2)(1− k2x2), where k =a4 − 6a2b2 + b4
(a2 + b2)2 .
Using the maps (a,b) 7→ (−a,b) and (a,b) 7→ (a− b,a + b), wemay choose a and b such that 0 < (1 +
√2)a < b.
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Generating Candidate Curves#1. INPUT: Bound N#2. For integers a and b satisfying 0 < (1 +
√2)a < b ≤ N
a. Define
p = a4 − 6a2b2 + b4
q = (a2 + b2)2
A = −27(p4 + 14p2q2 + q4)
B = −54(p6 − 33p4q2 − 33p2q4 + q6)
b. Record the elliptic curve Y 2 = X 3 + AX + B to a list.
#3. OUTPUT: List of elliptic curves
I N = 5 000 took 10 minutes to generate 3 148 208 curvesI Divide into 256 files (12 300 curves per file) for parallel
processing
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Redhawk at Miami University
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Bound N 1 000 2 000 3 000 4 000 5 000
s = 0 19 309(15.32%)
75 384(14.96%)
167 581(14.79%)
296 135(14.70%)
461 127(14.65%)
s = 1 45 807(36.35%)
179 361(35.59%)
401 351(35.41%)
711 392(35.31%)
1 110 462(35.27%)
s = 2 40 044(31.75%)
161 031(31.96%)
362 152(31.95%)
643 340(31.93%)
1 004 658(31.91%)
s = 3 16 933(13.44%)
70 481(13.99%)
160 695(14.18%)
287 682(14.28%)
450 939(14.32%)
s = 4 3 550(2.82%)
15 845(3.14%)
36 956(3.26%)
67 289(3.34%)
106 791(3.39%)
s = 5 338(0.27%)
1 707(0.34%)
4 370(0.39%)
8 208(0.41%)
13 371(0.42%)
s = 6 22(0.02%)
112(0.02%)
256(0.02%)
509(0.03%)
839(0.03%)
s = 7 0(0.00%)
2(0.00%)
4(0.00%)
8(0.00%)
21(0.00%)
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Computation of Mordell-Weil Ranks
I Data for N = 1 000 and s = 3
I 16 933 curves in this list
I 12 of these curves are known to have r = 3
I Can we find more?
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
New Curves we found
Parameter t Rank r1984 3
101299 386
333 31265 2 ≤ r ≤ 32192 2 ≤ r ≤ 39
296 2 ≤ r ≤ 365
337 2 ≤ r ≤ 3
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Radon at Purdue University
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Algorithm
Q: What do we do while mwrank is running?
A: Consider 2-Selmer ranks,of course!
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Poisson Distribution?
Histogram of Ranks of 2-Selmer Groups for N = 5 000
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Poisson Distribution?
Is this Poisson?
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Poisson Distribution?
Poisson Distributions
I Observed: O(s) with average s̄ =
∑m(N)s=0 s ·O(s)∑m(N)
s=0 O(s)
I Expected: E(s) =
m(N)∑m=0
O(m)
· λs
s!e−λ with λ = s̄
I Chi-square distribution: χ2 =
m(N)∑s=0
[O(s)− E(s)]2
E(s)
I Compare with value of χ2α,df where α = 5% and
df = m(N)− 1. If χ2 ≤ χ2α,df, accept hypothesis.
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Poisson Distribution?
Chi-Square Distribution of 2-Selmer Ranks
Bound N m(N) s̄ χ2 χ2α, df
1 000 6 1.529456 7 700.072 11.070
2 000 7 1.558704 29 761.771 12.592
3 000 7 1.569643 65 653.675 12.592
4 000 7 1.575738 115 008.433 12.592
5 000 7 1.579246 177 788.496 12.592
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Poisson Distribution?
Bound N 1 000 2 000 3 000 4 000 5 000
s = 0 19 309(15.32%)
75 384(14.96%)
167 581(14.79%)
296 135(14.70%)
461 127(14.65%)
s = 1 45 807(36.35%)
179 361(35.59%)
401 351(35.41%)
711 392(35.31%)
1 110 462(35.27%)
s = 2 40 044(31.75%)
161 031(31.96%)
362 152(31.95%)
643 340(31.93%)
1 004 658(31.91%)
s = 3 16 933(13.44%)
70 481(13.99%)
160 695(14.18%)
287 682(14.28%)
450 939(14.32%)
s = 4 3 550(2.82%)
15 845(3.14%)
36 956(3.26%)
67 289(3.34%)
106 791(3.39%)
s = 5 338(0.27%)
1 707(0.34%)
4 370(0.39%)
8 208(0.41%)
13 371(0.42%)
s = 6 22(0.02%)
112(0.02%)
256(0.02%)
509(0.03%)
839(0.03%)
s = 7 0(0.00%)
2(0.00%)
4(0.00%)
8(0.00%)
21(0.00%)
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Generating Functions
-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25
0.25
0.5
0.75
1
fsel(z) ≈ 0.146 + 0.353 z + 0.319 z2 + 0.143 z3 + · · ·
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks
Family of Elliptic Curves with E(Q)tors ' Z2 × Z8 Computations Distribution of 2-Selmer Ranks
Generating Functions
Acknowledgments and ReferencesI SUMSRI and Miami UniversityI Residential Computing at Miami UniversityI Rosen Center for Advanced Computing at PurdueI Dr. Goins and Maria SalcedoI Dr. Waikar and Ashley SwandbyI NSF and NSA
References1. Edray Goins. SUMSRI Number Theory Notes. In preparation, 2007.
2. John Cremona. mwrank and related programs for elliptic curves over Q.http://www.maths.nott.ac.uk/personal/jec/mwrank/.
3. Joseph H. Silverman. The Arithmetic of Elliptic Curves,Springer-Verlag, New York-Berlin, 1986.
Rollick, Weigandt SUMSRI, Miami University
2-Selmer Ranks