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Primes dividing the ECHO sequence Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons Wake Forest University July 30, 2015 Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 1/27 Outline

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Primes dividing the ECHO sequence

Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons

Wake Forest University

July 30, 2015

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 1/27

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Outline

Sequence and Elliptic Curves Numerical Approximations Galois Toolbox Calculating the Fraction Conclusion

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 2/27

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Introduction

Question What is the density of primes p such that p divides some 2n + 1 term for n ≥ 0? Answer 17 24 (Hasse). Question What is the density of primes p such that p divides some Ln, a term of the Lucas sequence? Answer 2 3 (Lagarias).

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 3/27

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Introduction

Question What is the density of primes p such that p divides some 2n + 1 term for n ≥ 0? Answer 17 24 (Hasse). Question What is the density of primes p such that p divides some Ln, a term of the Lucas sequence? Answer 2 3 (Lagarias).

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 3/27

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Introduction

Question What is the density of primes p such that p divides some 2n + 1 term for n ≥ 0? Answer 17 24 (Hasse). Question What is the density of primes p such that p divides some Ln, a term of the Lucas sequence? Answer 2 3 (Lagarias).

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 3/27

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Sequences, Elliptic Curves, and Galois Theory

Lagarias and Hasse derived number fields with behaviors dependent entirely on whether p is a “good prime or not. They then calculated the density using the Chebotarev density theorem. We do the same by analyzing Galois groups attached to elliptic curves.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 4/27

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Significance

Theorem (Jones and Rouse) The Somos-4 sequence is defined by a0 = a1 = a2 = a3 = 1 and further recursively defined by anan−4 = an−1an−3 + a2

n−2.

The density of primes dividing a term of this sequence is 11 21. Proposition (Connection to Elliptic Curves (Jones and Rouse)) Let E : y2 + y = x3 − x and P = (0, 0) be an elliptic curve and

  • point. Then

(2n − 3)P =

  • a2

n − an−1an+1

a2

n

, a2

n−1an+2 − 2an−1anan+1

a3

n

  • .

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 5/27

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The ECHO Sequence

The ECHO sequence is defined by b0 = 1, b1 = 2, b2 = 1, b3 = −3, and for n > 3, bn =   

bn−1bn−3−b2

n−2

bn−4

if n ≡ 2 (mod 3),

bn−1bn−3−3b2

n−2

bn−4

if n ≡ 2 (mod 3). The next few terms are − 7, −17, 2, 101, 247, 571, −1669, − 13766, −43101, −205897, 1640929, 8217293, 101727662, 173114917, −5439590147, −70987557871, ... Fact: bn ∈ Z ∀n ≥ 0.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 6/27

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The ECHO Sequence

The ECHO sequence is defined by b0 = 1, b1 = 2, b2 = 1, b3 = −3, and for n > 3, bn =   

bn−1bn−3−b2

n−2

bn−4

if n ≡ 2 (mod 3),

bn−1bn−3−3b2

n−2

bn−4

if n ≡ 2 (mod 3). The next few terms are − 7, −17, 2, 101, 247, 571, −1669, − 13766, −43101, −205897, 1640929, 8217293, 101727662, 173114917, −5439590147, −70987557871, ... Fact: bn ∈ Z ∀n ≥ 0.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 6/27

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The ECHO Sequence

The ECHO sequence is defined by b0 = 1, b1 = 2, b2 = 1, b3 = −3, and for n > 3, bn =   

bn−1bn−3−b2

n−2

bn−4

if n ≡ 2 (mod 3),

bn−1bn−3−3b2

n−2

bn−4

if n ≡ 2 (mod 3). The next few terms are − 7, −17, 2, 101, 247, 571, −1669, − 13766, −43101, −205897, 1640929, 8217293, 101727662, 173114917, −5439590147, −70987557871, ... Fact: bn ∈ Z ∀n ≥ 0.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 6/27

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The ECHO Sequence

The ECHO sequence is defined by b0 = 1, b1 = 2, b2 = 1, b3 = −3, and for n > 3, bn =   

bn−1bn−3−b2

n−2

bn−4

if n ≡ 2 (mod 3),

bn−1bn−3−3b2

n−2

bn−4

if n ≡ 2 (mod 3). The next few terms are − 7, −17, 2, 101, 247, 571, −1669, − 13766, −43101, −205897, 1640929, 8217293, 101727662, 173114917, −5439590147, −70987557871, ... Fact: bn ∈ Z ∀n ≥ 0.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 6/27

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Elliptic Curves

We want to know more about the number of primes dividing the sequence. It turns out that we can relate this problem to elliptic curves. Generally speaking, a normal elliptic curve E is a polynomial

  • f the form y2 = x3 + Ax + B, where A, B are in a field F.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 7/27

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Elliptic Curves

We want to know more about the number of primes dividing the sequence. It turns out that we can relate this problem to elliptic curves. Generally speaking, a normal elliptic curve E is a polynomial

  • f the form y2 = x3 + Ax + B, where A, B are in a field F.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 7/27

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Elliptic Curves (Examples)

y2 = x3 − x

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 8/27

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Elliptic Curves (Examples)

y2 = x3 − x + 1

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 9/27

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Elliptic Curves (Group Law)

It turns out that in most cases, one can turn the curve into a group: if P and Q are two points on the curve, one can define the operation for P add Q: take the line intersecting both P and Q: it will intersect the curve at another point, say R. Then reflect that point over the y axis, and call this point P + Q.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 10/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

The elliptic curve we considered for this project was E : y2 + y = x3 − 3x + 4, and is pictured below: A point on this curve is P = (4, 7).

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 11/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

The elliptic curve we considered for this project was E : y2 + y = x3 − 3x + 4, and is pictured below: A point on this curve is P = (4, 7).

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 11/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

It turns out that (2n + 3)P = g(n) b2

n

, f (n) b3

n

  • where

g(n) = 2b2

n − bn−3bn+3

and f (n) =      b3

n + b2 n−1bn+2

if n ≡ 0 (mod 3), b3

n + 9b2 n−1bn+2

if n ≡ 1 (mod 3), b3

n + 3b2 n−1bn+2

if n ≡ 2 (mod 3), and this point is in reduced form.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 12/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

Since (2n + 3)P = g(n) b2

n

, f (n) b3

n

  • ,

in projective coordinates we have (2n + 3)P = (bng(n) : f (n) : b3

n).

Since the point is in reduced form, gcd(bn, f (n)) = 1. One can reduce P modulo p by modding out the projective coordinates of P by p. This implies that p|bn for some n ≥ 0 if and only if P reduced mod p has odd order. So, the fraction of primes dividing the sequence is equivalent to the fraction of primes modulo which P has odd order.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 13/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

Since (2n + 3)P = g(n) b2

n

, f (n) b3

n

  • ,

in projective coordinates we have (2n + 3)P = (bng(n) : f (n) : b3

n).

Since the point is in reduced form, gcd(bn, f (n)) = 1. One can reduce P modulo p by modding out the projective coordinates of P by p. This implies that p|bn for some n ≥ 0 if and only if P reduced mod p has odd order. So, the fraction of primes dividing the sequence is equivalent to the fraction of primes modulo which P has odd order.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 13/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

Since (2n + 3)P = g(n) b2

n

, f (n) b3

n

  • ,

in projective coordinates we have (2n + 3)P = (bng(n) : f (n) : b3

n).

Since the point is in reduced form, gcd(bn, f (n)) = 1. One can reduce P modulo p by modding out the projective coordinates of P by p. This implies that p|bn for some n ≥ 0 if and only if P reduced mod p has odd order. So, the fraction of primes dividing the sequence is equivalent to the fraction of primes modulo which P has odd order.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 13/27

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The Elliptic Curve y 2 + y = x3 − 3x + 4

Since (2n + 3)P = g(n) b2

n

, f (n) b3

n

  • ,

in projective coordinates we have (2n + 3)P = (bng(n) : f (n) : b3

n).

Since the point is in reduced form, gcd(bn, f (n)) = 1. One can reduce P modulo p by modding out the projective coordinates of P by p. This implies that p|bn for some n ≥ 0 if and only if P reduced mod p has odd order. So, the fraction of primes dividing the sequence is equivalent to the fraction of primes modulo which P has odd order.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 13/27

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Numerical Computation of the Fraction

Consider the fractions of π(x)/π(x) x π(x) π(x)

π(x) π(x)

10 3 4 0.75 102 13 25 0.52 103 91 168 0.541666667 104 636 1229 0.517493897 105 5118 9592 0.533569641 106 41856 78498 0.533211037 107 354158 664579 0.532905794 108 3069170 5761455 0.532707450 109 27092923 50847534 0.532826685 1010 242426819 455052511 0.532744712 1011 2193850226 4118054813 0.532739443

179 336 ≈ 0.532738095

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 14/27

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Understanding Gal(Q(E[N])/Q): Galois Representations

Fact: E[N] ∼ = (Z/NZ)2. We adjoin to Q the coordinates of the N-torsion points of E, and use the action of the Galois group on the torsion : Definition ρN : Gal(Q(E[N])/Q) − → Aut(E[N]) = GL2(Z/NZ) This map is injective! Some uses: determine the curve up to isogeny (Faltings), FLT

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 15/27

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Understanding Gal(Q(E[N])/Q): Galois Representations

Fact: E[N] ∼ = (Z/NZ)2. We adjoin to Q the coordinates of the N-torsion points of E, and use the action of the Galois group on the torsion : Definition ρN : Gal(Q(E[N])/Q) − → Aut(E[N]) = GL2(Z/NZ) This map is injective! Some uses: determine the curve up to isogeny (Faltings), FLT

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 15/27

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Understanding Gal(Q(E[N])/Q): Galois Representations

Fact: E[N] ∼ = (Z/NZ)2. We adjoin to Q the coordinates of the N-torsion points of E, and use the action of the Galois group on the torsion : Definition ρN : Gal(Q(E[N])/Q) − → Aut(E[N]) = GL2(Z/NZ) This map is injective! Some uses: determine the curve up to isogeny (Faltings), FLT

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 15/27

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Arboreal Representations

Takes into account arithmetic of non-torsion points. Given a curve E and a rational point P on E. Fix an N-division pt. of P, β. Beefed Up Galois Representation: Definition ωN : Gal(Q([N]−1P)/Q) − → AGL2(Z/NZ) AGL2(Z/NZ) := (Z/NZ)2 ⋊ GL2(Z/NZ) σ − → (σ(β) − β, ρN(σ))

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 16/27

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Arboreal Representations

Takes into account arithmetic of non-torsion points. Given a curve E and a rational point P on E. Fix an N-division pt. of P, β. Beefed Up Galois Representation: Definition ωN : Gal(Q([N]−1P)/Q) − → AGL2(Z/NZ) AGL2(Z/NZ) := (Z/NZ)2 ⋊ GL2(Z/NZ) σ − → (σ(β) − β, ρN(σ))

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 16/27

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Arboreal Representations

Takes into account arithmetic of non-torsion points. Given a curve E and a rational point P on E. Fix an N-division pt. of P, β. Beefed Up Galois Representation: Definition ωN : Gal(Q([N]−1P)/Q) − → AGL2(Z/NZ) AGL2(Z/NZ) := (Z/NZ)2 ⋊ GL2(Z/NZ) σ − → (σ(β) − β, ρN(σ))

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 16/27

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AGL2(Z/2kZ) and Primes dividing ECHO

Note a ∈ G has odd order iff for all k ∈ Z there is βk ∈ G such that 2kβk = a Form the number field Kk := Q([2k]−1P) by adjoining the coordinates of all such βk to Q Want the primes p unramified such that σp ∈ Gal(Kk/Q) fixes some βk : 2kβk = P. If σp fixes βk then βk ∈ E(Fp) as desired.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 17/27

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AGL2(Z/2kZ) and Primes dividing ECHO

Recall the homomorphism ωk : Gal(Kk/Q) → AGL2(Z/2kZ) := (Z/2kZ)2 ⋊ GL2(Z/2kZ) which is given by ωk(σp) = ( v, M). Definition The action of AGL2(Z/2kZ) on (Z/2kZ)2 is given by ( v, M)( x) = M x + v.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 18/27

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AGL2(Z/2kZ) and Primes dividing ECHO

Theorem There exists a fixed point βk of σp iff (M − I) x = v for some x. This is because σp has a fixed point iff M x + v = x for some x, or equivalently (M − I) x = v, and any such fixed point for σp is a βk. Now we need only understand the image of ωk in order to compute the fraction.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 19/27

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Kinetic Subgroups of AGL2(Z/2kZ)

Definition (Kinetic) Let a subgroup G ⊂ AGL2(Z/2kZ) for k ≥ 2 be called kinetic if both the maps are surjective: pr : G ։ GL2(Z/2kZ) φ : G ։ AGL2(Z/2Z). Proposition The image of ωk must be kinetic.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 20/27

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Kinetic Subgroups of AGL2(Z/2kZ)

Definition (Kinetic) Let a subgroup G ⊂ AGL2(Z/2kZ) for k ≥ 2 be called kinetic if both the maps are surjective: pr : G ։ GL2(Z/2kZ) φ : G ։ AGL2(Z/2Z). Proposition The image of ωk must be kinetic.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 20/27

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Kinetic Subgroups of AGL2(Z/2kZ)

Proposition For all k ≥ 2, there is one proper subgroup (up to conjugacy) Hk ⊂ AGL2(Z/2kZ) this is kinetic. Proposition For our curve, the image ωk : Gal(Kk/Q) → AGL2(Z/2kZ) is Hk.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 21/27

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The fraction at the limit

Recall that by the Chebotarev density theorem lim

x→∞

  • {p prime, unramified in Kk : p ≤ x,

Kk/Q

p

  • ⊆ S}
  • π(x)

= |S| | Gal(Kk/Q)| where S is a union of conjugacy classes in Gal(Kk/Q). We need to evaluate this fraction for all k ∈ Z.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 22/27

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The fraction at the limit

We make two choices for S:

1

Elements of Gal(Kk/Q) such that for all ( v, M) ∈ Hk, v is in the column space of M − I.

2

same set except ( v, M) such that det(M − I) ≡ 0 mod (2k)

We observe that the set of elements ( v, M ∈ Hk) we desire for S sits between those two sets but at the limit as k → ∞ the sizes of both sets over | Gal(Kk/Q)| are equal.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 23/27

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Theoretical Computation of the Fraction

Thus our final job was to compute the limit lim

k→∞

|{( v, M) ∈ Hk| v ∈ im(M − I)}| |Hk| =

  • limk→∞

π(x) π(x)

  • .

Our process: partition the numerator based on its reduction mod 4, because we understand the image at k = 2 very well.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 24/27

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Theoretical Computation of the Fraction

Thus our final job was to compute the limit lim

k→∞

|{( v, M) ∈ Hk| v ∈ im(M − I)}| |Hk| =

  • limk→∞

π(x) π(x)

  • .

Our process: partition the numerator based on its reduction mod 4, because we understand the image at k = 2 very well.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 24/27

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SLIDE 41

The Answer

179 336

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 25/27

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Future Work

Are there infinitely many curves E/Q for which there is a point P ∈ E(Q) so that im(ωk) = Hk? E : y2 + axy + by = x3 + bx2 P = (0, 0) Compute fa,b, a two-parameter, degree 4 polynomial.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 26/27

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Summary of Results

Theorem The density of primes dividing the ECHO sequence is the following: lim

x→∞

π′(x) π(x) = 179 336 ≈ 0.532738095. Theorem Let E be an elliptic curve over Q and let P be a rational point on

  • E. Suppose further that the image of the classical Galois

representation is surjective, and that P has no rational 2-division

  • points. Then there are only two possibilities for the image of the

2-adic arboreal representation up to conjugacy. As a consequence, the density of primes p for which the reduction of P modulo p has

  • dd order is either 11

21 or 179 336.

Block Gorman, Genao, Hwang, Kantor, Parsons Primes dividing the ECHO sequence 27/27

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Acknowledgements

We would like to thank the NSF for providing our grant, as well as MAGMA, PARI/GP, and Jeremy Rouse for his guidance and support.

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