Sporadic torsion David Zureick-Brown Anastassia Etropolski (Emory - - PowerPoint PPT Presentation

Sporadic torsion

David Zureick-Brown Anastassia Etropolski (Emory University) Jackson Morrow (Emory University)

Emory University Slides available at http://www.mathcs.emory.edu/~dzb/slides/

SERMON XXIX, Harrisonburg, VA April 2-3, 2016

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 1 / 14

Mazur’s Theorem

Theorem (Mazur, 1978)

Let E/Q be an elliptic curve. Then E(Q)tors is isomorphic to one of the following groups. Z/NZ, for 1 ≤ N ≤ 10 or N = 12, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 4.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 2 / 14

Mazur’s Theorem

Theorem (Mazur, 1978)

Let E/Q be an elliptic curve. Then E(Q)tors is isomorphic to one of the following groups. Z/NZ, for 1 ≤ N ≤ 10 or N = 12, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 4. More precisely, let Y1(N) be the curve paramaterizing (E, P), where P is a point of exact order N on E, and let

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 2 / 14

Mazur’s Theorem

Theorem (Mazur, 1978)

Let E/Q be an elliptic curve. Then E(Q)tors is isomorphic to one of the following groups. Z/NZ, for 1 ≤ N ≤ 10 or N = 12, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 4. More precisely, let Y1(N) be the curve paramaterizing (E, P), where P is a point of exact order N on E, and let Y1(M, N) (with M | N) be the curve paramaterizing E/K such that E(K)tors contains Z/MZ ⊕ Z/NZ.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 2 / 14

Mazur’s Theorem

Theorem (Mazur, 1978)

Let E/Q be an elliptic curve. Then E(Q)tors is isomorphic to one of the following groups. Z/NZ, for 1 ≤ N ≤ 10 or N = 12, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 4. More precisely, let Y1(N) be the curve paramaterizing (E, P), where P is a point of exact order N on E, and let Y1(M, N) (with M | N) be the curve paramaterizing E/K such that E(K)tors contains Z/MZ ⊕ Z/NZ. Then Y1(N)(Q) = ∅ and Y1(2, 2N)(Q) = ∅ iff N are as above.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 2 / 14

Modular curves

Example (N = 9)

E(K) ∼ = Z/9Z if and only if there exists t ∈ K such that E is isomorphic to y2 + (t − rt + 1)xy + (rt − r2t)y = x3 + (rt − r2t)x2 where r is t2 − t + 1. The torsion point is (0, 0).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 3 / 14

Modular curves

Example (N = 9)

E(K) ∼ = Z/9Z if and only if there exists t ∈ K such that E is isomorphic to y2 + (t − rt + 1)xy + (rt − r2t)y = x3 + (rt − r2t)x2 where r is t2 − t + 1. The torsion point is (0, 0).

Example (N = 11)

E(K) ∼ = Z/11Z correspond to a, b ∈ K such that a2 + (b2 + 1)a + b; in which case E is isomorphic to y2 + (s − rs + 1)xy + (rs − r2s)y = x3 + (rs − r2s)x2 where r is ba + 1 and s is −b + 1.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 3 / 14

Rational Points on X1(N) and X1(2, 2N)

Let X1(N) and X1(M, N) be the smooth compactifications of Y1(N) and Y1(M, N).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 4 / 14

Rational Points on X1(N) and X1(2, 2N)

Let X1(N) and X1(M, N) be the smooth compactifications of Y1(N) and Y1(M, N). We can restate the results of Mazur’s Theorem as follows.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 4 / 14

Rational Points on X1(N) and X1(2, 2N)

Let X1(N) and X1(M, N) be the smooth compactifications of Y1(N) and Y1(M, N). We can restate the results of Mazur’s Theorem as follows. X1(N) and X1(2, 2N) have genus 0 for exactly the N appearing in Mazur’s Theorem. (So in particular, there are infinitely many E/Q with such torsion structure.)

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 4 / 14

Rational Points on X1(N) and X1(2, 2N)

Let X1(N) and X1(M, N) be the smooth compactifications of Y1(N) and Y1(M, N). We can restate the results of Mazur’s Theorem as follows. X1(N) and X1(2, 2N) have genus 0 for exactly the N appearing in Mazur’s Theorem. (So in particular, there are infinitely many E/Q with such torsion structure.) If g(X1(N)) (resp. g(X1(2, 2N))) is greater than 0, then X1(N)(Q) (resp. X1(2, 2N)(Q)) consists only of cusps.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 4 / 14

Rational Points on X1(N) and X1(2, 2N)

Let X1(N) and X1(M, N) be the smooth compactifications of Y1(N) and Y1(M, N). We can restate the results of Mazur’s Theorem as follows. X1(N) and X1(2, 2N) have genus 0 for exactly the N appearing in Mazur’s Theorem. (So in particular, there are infinitely many E/Q with such torsion structure.) If g(X1(N)) (resp. g(X1(2, 2N))) is greater than 0, then X1(N)(Q) (resp. X1(2, 2N)(Q)) consists only of cusps. So, in a sense, the simplest thing that could happen does happen for these modular curves.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 4 / 14

Higher Degree Torsion Points

Theorem (Merel, 1996)

For every integer d ≥ 1, there is a constant N(d) such that for all K/Q of degree at most d and all E/K, #E(K)tors ≤ N(d).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 5 / 14

Higher Degree Torsion Points

Theorem (Merel, 1996)

For every integer d ≥ 1, there is a constant N(d) such that for all K/Q of degree at most d and all E/K, #E(K)tors ≤ N(d).

Problem

Fix d ≥ 1. Classify all groups which can occur as E(K)tors for K/Q of degree d. Which of these occur infinitely often?

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 5 / 14

The Quadratic Case

Theorem (Kamienny-Kenku-Momose, 1980’s)

Let E be an elliptic curve over a quadratic number field K. Then E(K)tors is one of the following groups. Z/NZ, for 1 ≤ N ≤ 16 or N = 18, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 6, Z/3Z ⊕ Z/3NZ, for 1 ≤ N ≤ 2, or Z/4Z ⊕ Z/4Z.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 6 / 14

The Quadratic Case

Theorem (Kamienny-Kenku-Momose, 1980’s)

Let E be an elliptic curve over a quadratic number field K. Then E(K)tors is one of the following groups. Z/NZ, for 1 ≤ N ≤ 16 or N = 18, Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 6, Z/3Z ⊕ Z/3NZ, for 1 ≤ N ≤ 2, or Z/4Z ⊕ Z/4Z. In particular, the corresponding curves X1(M, N) all have g ≤ 2, which guarantees that they have infinitely many quadratic points.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 6 / 14

Modular curves

Example (N = 9)

E(K) ∼ = Z/9Z if and only if there exists t ∈ K such that E is isomorphic to y2 + (t − rt + 1)xy + (rt − r2t)y = x3 + (rt − r2t)x2 where r is t2 − t + 1. The torsion point is (0, 0).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 7 / 14

Modular curves

Example (N = 9)

E(K) ∼ = Z/9Z if and only if there exists t ∈ K such that E is isomorphic to y2 + (t − rt + 1)xy + (rt − r2t)y = x3 + (rt − r2t)x2 where r is t2 − t + 1. The torsion point is (0, 0).

Example (N = 11)

E(K) ∼ = Z/11Z correspond to a, b ∈ K such that a2 + (b2 + 1)a + b; in which case E is isomorphic to y2 + (s − rs + 1)xy + (rs − r2s)y = x3 + (rs − r2s)x2 where r is ba + 1 and s is −b + 1.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 7 / 14

Expected K-Rational Points

Let X/Q be a curve. If X admits a degree d = [K : Q] map to P1

Q, then X(K) is infinite.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 8 / 14

Expected K-Rational Points

Let X/Q be a curve. If X admits a degree d = [K : Q] map to P1

Q, then X(K) is infinite.

More precisely, if D is a divisor of degree d on X and dim |D| ≥ 1, then D paramaterizes an infinite family of effective degree d divisors.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 8 / 14

Expected K-Rational Points

Let X/Q be a curve. If X admits a degree d = [K : Q] map to P1

Q, then X(K) is infinite.

More precisely, if D is a divisor of degree d on X and dim |D| ≥ 1, then D paramaterizes an infinite family of effective degree d divisors.

Question

If Y1(M, N)(K) = ∅, are all of the points coming from the existence of such divisors?

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 8 / 14

Expected K-Rational Points

Let X/Q be a curve. If X admits a degree d = [K : Q] map to P1

Q, then X(K) is infinite.

More precisely, if D is a divisor of degree d on X and dim |D| ≥ 1, then D paramaterizes an infinite family of effective degree d divisors.

Question

If Y1(M, N)(K) = ∅, are all of the points coming from the existence of such divisors? If not, we call these outliers sporadic points.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 8 / 14

Sporadic Cubic Points

Theorem (Jeon-Kim-Schweizer, 2004)

Let E be an elliptic curve over a cubic number field K. Then the subgroups which arise as E(K)tors infinitely often are exactly the following. Z/NZ, for 1 ≤ N ≤ 20, N = 17, 19, or Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 7.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 9 / 14

Sporadic Cubic Points

Theorem (Jeon-Kim-Schweizer, 2004)

Let E be an elliptic curve over a cubic number field K. Then the subgroups which arise as E(K)tors infinitely often are exactly the following. Z/NZ, for 1 ≤ N ≤ 20, N = 17, 19, or Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 7.

Theorem (Najman, 2014)

There is an elliptic curve E/Q whose torsion subgroup over a cubic field is Z/21Z.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 9 / 14

Sporadic Cubic Points

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 10 / 14

Classification of Cubic Torsion

Theorem (Etropolski–Morrow–ZB, Derickx)

The only torsion subgroups which appear for an elliptic curve over a cubic field are Z/NZ, for 1 ≤ N ≤ 21, N = 17, 19, and Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 7.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 11 / 14

Classification of Cubic Torsion

Theorem (Etropolski–Morrow–ZB, Derickx)

The only torsion subgroups which appear for an elliptic curve over a cubic field are Z/NZ, for 1 ≤ N ≤ 21, N = 17, 19, and Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 7. In other words, there is only one cubic sporadic point.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 11 / 14

Classification of Cubic Torsion

Theorem (Etropolski–Morrow–ZB, Derickx)

The only torsion subgroups which appear for an elliptic curve over a cubic field are Z/NZ, for 1 ≤ N ≤ 21, N = 17, 19, and Z/2Z ⊕ Z/2NZ, for 1 ≤ N ≤ 7. In other words, there is only one cubic sporadic point.

Remark

Parent showed that the largest prime that can divide E(K)tors in the cubic case is p = 13.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 11 / 14

Get lucky

Let X be either of X1(N) or X1(2, 2N).

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Get lucky

Let X be either of X1(N) or X1(2, 2N). For almost all N we need to consider, rk JX(Q) = 0.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 12 / 14

Get lucky

Let X be either of X1(N) or X1(2, 2N). For almost all N we need to consider, rk JX(Q) = 0.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 12 / 14

The Mordell-Weil Sieve

Let X (d) := X d/Sd denote the dth symmetric power of X. Note that degree d points of X are Q-points of X (d).

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The Mordell-Weil Sieve

Let X (d) := X d/Sd denote the dth symmetric power of X. Note that degree d points of X are Q-points of X (d). For a finite set S of primes of good reduction, we have the following commutative diagram. X (d)(Q)

• J(Q)

α

• p∈S

X(Fq)

β p∈S

J(Fp)

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 13 / 14

The Mordell-Weil Sieve

Let X (d) := X d/Sd denote the dth symmetric power of X. Note that degree d points of X are Q-points of X (d). For a finite set S of primes of good reduction, we have the following commutative diagram. X (d)(Q)

• J(Q)

α

• p∈S

X(Fq)

β p∈S

J(Fp) We want to choose S so that, once we remove any known rational points, the images of α and β are disjoint.

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An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10.

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An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33). g(X0(33)) = 2 and J0(33)(Q) ≃ Z/10 × Z/10 = D1, D2.

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33). g(X0(33)) = 2 and J0(33)(Q) ≃ Z/10 × Z/10 = D1, D2. Write P − 3Q = mD1 + nD2 in J0(33).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33). g(X0(33)) = 2 and J0(33)(Q) ≃ Z/10 × Z/10 = D1, D2. Write P − 3Q = mD1 + nD2 in J0(33). mod 7: (m, n) is either (0, 3), (2, 2), (5, 8), or (7, 7).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33). g(X0(33)) = 2 and J0(33)(Q) ≃ Z/10 × Z/10 = D1, D2. Write P − 3Q = mD1 + nD2 in J0(33). mod 7: (m, n) is either (0, 3), (2, 2), (5, 8), or (7, 7). mod 13: (m, n) is either (1, 1), (1, 4), (3, 3), (4, 7), (6, 6), (6, 9) (8, 8), or (9, 2).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14

An Example

Let Y = X1(33). Then g(Y ) = 21 and γY = 10. JY (Q) contains a subgroup of order 2 · 3 · 5 · 11 · 61 · 421. We sieve using the map f : X1(33) → X0(33). g(X0(33)) = 2 and J0(33)(Q) ≃ Z/10 × Z/10 = D1, D2. Write P − 3Q = mD1 + nD2 in J0(33). mod 7: (m, n) is either (0, 3), (2, 2), (5, 8), or (7, 7). mod 13: (m, n) is either (1, 1), (1, 4), (3, 3), (4, 7), (6, 6), (6, 9) (8, 8), or (9, 2).

David Zureick-Brown (Emory University) Sporadic torsion April 2-3, 2016 14 / 14