a statistical analysis of 2-selmer groups for elliptic...

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


Outline

Family of Elliptic Curves with E(Q)tors ' Z2 × Z8The 2-Selmer Group

ComputationsmwrankAlgorithm

Distribution of 2-Selmer RanksPoisson Distribution?Generating Functions


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


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)

}


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= ∅.


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?


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) = ∅


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


2-Selmer Ranks


mwrank

John Cremonahttp://www.maths.nott.ac.uk/personal/jec/mwrank/index.html


2-Selmer Ranks

http://www.maths.nott.ac.uk/personal/jec/mwrank/index.html


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.


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.


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


2-Selmer Ranks


Algorithm

Redhawk at Miami University


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%)


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?


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


2-Selmer Ranks


Algorithm

Radon at Purdue University


2-Selmer Ranks


Algorithm

Q: What do we do while mwrank is running?

A: Consider 2-Selmer ranks,of course!


2-Selmer Ranks


Poisson Distribution?

Histogram of Ranks of 2-Selmer Groups for N = 5 000


2-Selmer Ranks



Is this Poisson?


2-Selmer Ranks



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.


2-Selmer Ranks



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


2-Selmer Ranks



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%)


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 + · · ·


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.


2-Selmer Ranks

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