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The geometric average size of Selmer groups over function fields

Aaron Landesman

Stanford University

Number Theory, Arithmetic Geometry, and Computation II Baltimore, MD

Slides available at http://www.web.stanford.edu/~aaronlan/slides/

Ranks of elliptic curves

Theorem (Mordell-Weil)

Let E be an elliptic curve over a global field K (such as Q or Fq(t)). Then the group of K-rational points E(K) is a finitely generated abelian group. For E an elliptic curve over K, write E(K) ≃ Zr ⊕ T for T a finite group. Then, r is the rank of E.

Question

What is the average rank of an elliptic curve?

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Motivation

Conjecture (Minimalist Conjecture)

The average rank of elliptic curves is 1/2. Moreover,

• 50% of curves have rank 0,
• 50% have rank 1,
• 0% have rank more than 1.

Goal

Explain why 0% of elliptic curves have rank more than 1, in a certain sense.

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Definition of Selmer group

Let K = Fq(t), and let v index the closed points of P1

• Fq. For E an elliptic

curve over K, the multiplication by n exact sequence E[n] E E

×n

induces the sequences on ´ etale cohomology

E(K)/nE(K) H1(Spec K, E[n]) H1(Spec K, E)[n] ∏v∈P1

Fq E(Kv)/nEv(Kv)

∏v H1(Spec Kv, Ev[n]) ∏v H1(Spec Kv, Ev)[n] 0.

β α

Definition

The n-Selmer group of E is Seln(E) := ker α.

Aaron Landesman The geometric average size of Selmer groups over function fields 4 / 14

Selmer group and rank

E(K)/nE(K) H1(Spec K, E[n]) H1(Spec K, E)[n] ∏v∈P1

Fq E(Kv)/nEv(Kv)

∏v H1(Spec Kv, Ev[n]) ∏v H1(Spec Kv, Ev)[n] 0.

β α

Lemma

There is an injection E(K)/nE(K) → Seln(E).

Proof.

E(K)/nE(K) = ker β ⊂ ker α = Seln(E).

Corollary

The Z/n rank of Seln(E) is an upper bound for the rank of E.

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Average size of Selmer groups

Say E/Fq(t) is in minimal Weierstrass form given by y2z = x3 + A(s, t)xz2 + B(s, t)z3, (so char Fq > 3,) where there exists d so that A(s, t) and B(s, t) are homogeneous polynomials in Fq[s, t] of degrees 4d and 6d. The height of E is h(E) := d.

Definition

The average size of the n-Selmer group of height up to d is Average≤d(# Seln /Fq(t)) := ∑E/Fq(t),h(E)≤d # Seln(E) #{E/Fq(t): h(E) ≤ d} , where the sum runs over isomorphism classes of elliptic curves E/Fq(t), having h(E) ≤ d.

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Conjecture on the average size of Selmer groups

Conjecture (Bhargava–Shankar and Poonen–Rains)

When all elliptic curves are ordered by height, lim

q→∞ lim d→∞ Average≤d(# Seln /Fq(t)) = ∑ s|n

s.

Remark

• An analogous statement over Q (without a limit in q) was shown for

n = 2, 3, 4, 5 by Bhargava and Shankar.

• The upper bound was shown for n = 3 over Fq(t) by de Jong.
• This was shown for n = 2 more generally over function fields by Ho,

Le Hung, and Ngo.

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Application of conjecture to ranks

Assuming the conjecture, 100% of elliptic curves have rank at most 1:

Corollary (Assuming conjecture)

Let P≤d

q

denote the proportion of elliptic curves of rank ≥ 2 over Fq(t) of height up to d. If the conjecture were true, lim

q→∞ lim d→∞ P≤d q

= 0.

Proof.

Take n prime. Since nrk E ≤ # Seln(E), in the limit we have n2P≤d

q

= n2 Average(δrk(E)≥2) ≤ Average(nrk E) ≤ Average (# Seln(E)) = ∑

s|n

s = n + 1. Since, n2P≤d

q

≤ n + 1, taking n large shows P≤d

q

→ 0.

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Main result

We can try to approach the conjecture by reversing the limits. Conjecture: lim

q→∞ lim d→∞

∑E/Fq,h(E)≤d # Seln(E) # {E : h(E) ≤ d} = ∑

s|n

s. Limits reversed: lim

d→∞ lim q→∞

∑E/Fq,h(E)≤d # Seln(E) # {E : h(E) ≤ d} = ∑

s|n

s.

Theorem (L.)

For n ≥ 1 and d ≥ 2, lim

q→∞ gcd(q,2n)=1

Average≤d(# Seln /Fq(t)) = ∑

s|n

s.

Aaron Landesman The geometric average size of Selmer groups over function fields 9 / 14

Application to ranks

Theorem (L.)

For n ≥ 1 and d ≥ 2, lim

q→∞ gcd(q,2n)=1

Average≤d(# Seln /Fq(t)) = ∑

s|n

s. Analogously to the corollary to the conjecture, 100% of elliptic curves of height up to d have rank at most 1 in the large q limit:

Corollary

If P≤d

q

denotes the proportion of elliptic curve of rank ≥ 2 over Fq(t) of height up to d, for d ≥ 2, lim

q→∞ gcd(q,2n)=1

P≤d

q

= 0.

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Three heuristics for the average size of Selmer groups

Question

Why is the average size of the n-Selmer group ∑s|n s?

• In the known cases over Q, the proof connects the average size to

Tamagawa number τ(PGLs) = s, and the average size is ∑s|n τ(PGLs)

• We show, via a monodromy computation, that the average size is the

number of orbits of a certain orthogonal group. If n is prime, these

• rbits are the 0 orbit and the n level sets of the associated quadratic

form.

• The average size is the number of balanced rank s projective bundles
• ver P1 for s | n, which are of the form

Proj P1 Sym• (O⊕a ⊕ O(−1)⊕s−a) for 1 ≤ a ≤ s. Altogether, there are ∑s|n s such bundles as s ranges over the divisors of n.

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Higher moments of Selmer groups

Let Average≤d((# Seln)m/Fq(t)) denote the average size of Seln(E)m as E varies through elliptic curves over Fq(t) of height ≤ d.

Conjecture (Poonen-Rains)

For ℓ prime and m ≥ 1, lim

q→∞ lim d→∞ Average≤d((# Selℓ)m/Fq(t)) = (1 + ℓ)(1 + ℓ2) · · · (1 + ℓm).

Using a more involved monodromy computation, we can prove:

Theorem (Feng-L)

Let ℓ be prime and m ≥ 1. For d ≥ max(2, m+4

6 ),

lim

q→∞ gcd(q,2n)=1

Average≤d((# Selℓ)m/Fq(t)) = (1 + ℓ)(1 + ℓ2) · · · (1 + ℓm).

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Quadratic twists

Definition

For E a fixed elliptic curve over Fq(t) defined by y2z = x3 + A(s, t)xz2 + B(s, t)z3, one can define the quadratic twist family of degree d as those elliptic curves of the form f (s, t)y2z = x3 + A(s, t)xz2 + B(s, t)z3, for f (s, t) ∈ k[s, t] varying over square-free homogeneous polynomial of degree d.

Remark (Average sizes, with a twist!)

Sun Woo Park and Niudun Wang proved that the average size of n-Selmer groups in certain quadratic twist families is ∑s|n s, using a similar method.

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Proof overview

Theorem (L)

For n ≥ 1 and d ≥ 2, lim

q→∞ gcd(q,2n)=1

Average≤d(# Seln /Fq(t)) = ∑

s|n

s. Proof overview: (1) Construct a space Seld

n,k parameterizing n-Selmer elements of elliptic

curves of height d over k. (2) By Lang-Weil, the average size of the n-Selmer group is the number

• f components of Seld

n,k

(3) Compute the number of components of Seld

n,k by viewing it as a finite

cover of the moduli of height d elliptic curves, and computing the monodromy.

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Proof sketch

For k a finite field, construct a space Seld

n,k parameterizing pairs (E, X),

where E is an elliptic curve over k(t) and X is an n-Selmer element of E. Letting W d

k denote a parameter space for Weierstrass equations of elliptic

curves E/k(t) of height d. There is a projection map π : Seld

n,k → W d k

(E, X) → [E]. The key property is π−1([E])(k) = Seln(E). The total number of Selmer elements over varying elliptic curves is Seld

n,k(k), so we are reduced to computing

#Seld

n,k(k′)

#W d

k (k′)

for large finite extensions k′ of k.

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Proof sketch, continued

We want to compute #Seld

n,k(k′)

#W d

k (k′) .

Theorem (Lang-Weil)

For X a finite type space over Fp with r geometrically irreducible components, limq→∞ X(Fq) = rqdim X + O(qdim X−1/2). So, #Seld

n,k(k′)

#W d

k (k′) = #components of Seld n,k

#components of W d

k

= #components of Seld

n,k

1 = #components of Seld

n,k.

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Proof sketch, continued

To complete the proof, we want to show #components of Seld

n,k = ∑ s|n

s. Let W ◦d

k ⊂ W d k be the dense open parameterizing smooth Weierstrass

• models. Set up the fiber square

Sel◦d

n,k

Seld

n,k

W ◦d

k

W d

k . π◦ π

The resulting map π◦ is finite ´

• etale. Hence, we obtain a monodromy

representation (or Galois representation) ρd

k(n) : π´ et 1 (W ◦d k) → GL(V d n,k).

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Proof sketch, continued

Recall we are trying to compute #components of Sel◦d

n,k, which is a finite

´ etale cover of W ◦d

k with Galois representation

ρd

k(n) : π´ et 1 (W ◦d k) → GL(V d n,k).

Therefore, the number of components is the number of orbits of im ρd

k(n).

Theorem

For n prime, there is a quadratic form qd

n on V d n,k so that, up to index 2,

im ρd

k(n) = O(qd n ).

For n is prime, there are n + 1 orbits of O(qd

n ), corresponding to the n

level sets of qd

n , along with the 0 vector. We find that for n prime,

#components of Sel◦d

n,k = #orbits of O(qd n ) = n + 1 = ∑ s|n

s.

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Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces

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Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 Vakil n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 Vakil n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces Ho-Le Hung-Ngo

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 Vakil n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces Ho-Le Hung-Ngo de Jong, 3-Selmer

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). n = 6 n = 5 n = 4 n = 3 n = 2 Vakil n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces Ho-Le Hung-Ngo de Jong, 3-Selmer Lines on del Pezzo 1’s

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Selmer Spaces

For given n and d over a fixed finite field k, we have a space Seld

n,k

parameterizing n-Selmer elements for height d elliptic curves over k(t). . . . n = 6 n = 5 n = 4 n = 3 n = 2 Vakil n = 1 d = 1 d = 2 d = 3 d = 4 d = 5 d = 6

Table: In which directions have Selmer spaces been investigated?

Elliptic Surfaces Ho-Le Hung-Ngo de Jong, 3-Selmer Lines on del Pezzo 1’s Geometric average n-Selmer (H0 stability)

Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14

Defining the n-Selmer space

The k-points of the n-Selmer space are not exactly n-Selmer elements. Really, the fiber over an elliptic curve E is H1(P1

k, E 0[n]) for E 0 the

identity component of the N´ eron model of E. But, we can show # Seln(E) ≤ #H0 P1

k, E [n]

· #H1(P1

k, E 0[n])

with equality if E [n] = 0. To construct the Selmer space, let UW d

k be the

universal family of Weierstrass models over W d

k . We have projections

UW d

k f

− → P1 × W d

k g

− → W d

k .

Then, Seld

n,k := R1g∗(R1f∗µn).

To identify the fibers as above, use that R1f∗µn ≃ E 0[n].

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