The Geometric Distribution of Ranks and Selmer Groups of Elliptic - - PowerPoint PPT Presentation

The Geometric Distribution of Ranks and Selmer Groups

• f Elliptic Curves over Function Fields

Aaron Landesman (Stanford) Tony Feng (MIT) Eric Rains (Caltech) Special Session on Geometry and Topology in Arithmetic Madison, WI

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?

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 this and related conjectures hold for elliptic curves over Fq(t), in a q → ∞ limit.

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.

α

Definition

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

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.

Conjecture on the average size of Selmer groups

Conjecture (Bhargava–Shankar and Poonen–Rains)

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.

Conjectures on the distribution 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).

Conjecture (Poonen–Rains)

For ℓ a prime, as E ranges over elliptic curves over Fq(t), Prob (dimFℓ Selℓ(E) = v) =

• ∏

j≥0

1 1 + ℓ−j

v

∏

j=1

ℓ ℓj − 1

• .

The Poonen–Rains and BKLPR model

Let O2r denote the orthogonal group associated to the quadratic form q2r = x1x2 + x3x4 + · · · + x2r−1x2r over Fℓ. Choose two random r-dimensional isotropic subspaces V and W . I.e., q2r|V = q2r|W = 0.

Model (Poonen–Rains)

The ℓ-Selmer group of an elliptic curve E is modeled as V ∩ W . So, Prob(dim Selℓ(E) = α) = lim

r→∞ Prob(dim V ∩ W = α).

The distribution of the rank of an elliptic curve is modeled as rk(E) =

• if dim V ∩ W is even

1 if dim V ∩ W is odd.

Remark

Bhargava, Kane, Lenstra, Poonen, and Rains have a similar but more sophisticated model applying for composite n.

Main result

Definition

Let (rkBKLPR, SelBKLPR

n

) denote the prediction of Bhargava, Kane, Lenstra, Poonen, and Rains for the joint distribution of ranks and n-Selmer groups. Let (rk, Seln)d

Fq denote the joint distribution of ranks and n-Selmer groups

• f elliptic curves of height ≤ d over Fq(t).

Theorem (Feng-L-Rains)

For any integer n ≥ 1, we have (rkBKLPR, SelBKLPR

n

) = lim

d→∞

   lim sup

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

   = lim

d→∞

  lim inf

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

  .

Reversed limits

Observe that we first take a large q limit, whereas BKLPR first takes a large height limit. Original Conjecture: lim

q→∞ lim d→∞

ProbE:h(E)≤d(Seln(E) ≃ G) # {E : h(E) ≤ d} = Prob((rkBKLPR, SelBKLPR

n

) = G). Limits reversed: lim

d→∞ lim q→∞

ProbE:h(E)≤d(Seln(E) ≃ G) # {E : h(E) ≤ d} = Prob((rkBKLPR, SelBKLPR

n

) = G).

Remark

Even though the limits are reversed, our results provide some of the first direct evidence for the connection between the arithmetic of elliptic curves and the complete conjectures of BKLPR.

Consequences

Theorem (Feng-L-Rains)

(rkBKLPR, SelBKLPR

n

) = lim

d→∞

   lim sup

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

   = lim

d→∞

  lim inf

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

  .

Corollary

In the large q limit, the minimalist conjecture holds, meaning 50% of elliptic curves have rank 0 and 50% have rank 1. Similarly, in the large q limit, the Poonen–Rains conjectures on average sizes, moments, and distributions of Selmer groups hold.

Theorem (Feng-L-Rains)

(rkBKLPR, SelBKLPR

n

) = lim

d→∞

   lim sup

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

   = lim

d→∞

  lim inf

q→∞ gcd(q,6n)=1

(rk, Seln)d

Fq

  .

Remark

Similar results hold when working with quadratic twists of a fixed elliptic curve instead of all elliptic curves, as we just learned from Niudun Wang!

Remark

We write lim sup and lim inf because when d is even and n > 2, the limit does not exist. But the lim sup and lim inf tend to each other after further taking the height d → ∞.

Proof overview

The proof proceeds as follows: (1) Compare the actual distribution to a different model, which we call the “random kernel” model. (2) Show the random kernel model agrees with the BKLPR model in the large height limit.

Model (Random kernel model)

Here is the random kernel model for elliptic curves of height d over Fq(t): Let g ∈ O12d−4(Z/nZ) be a random element of the orthogonal group. Then, in the large q limit, Seln(E) ≃ ker(g − 1). Further rk(E) =

• if g ∈ SO12d−4(Z/nZ)

1 if g / ∈ SO12d−4(Z/nZ)

Proof overview, continued

The proof proceeds as follows: (1) Compare the actual distribution to a different model, which we call the “random kernel” model. (2) Show the random kernel model agrees with the BKLPR model in the large height limit. We check (2) directly for prime n, and deduce it for composite n by showing the two models satisfy the same Markov property. To check (1), we relate the distribution of Selmer groups to ker(g − id), for g a random element of an orthogonal group as follows: (A) Create a space parameterizing Selmer elements and covering the space of elliptic curves (B) Show the monodromy group of this cover is approximately an

• rthogonal group.

(C) Use equidistribution of Frobenius to connect the actual distribution to

• ur random kernel distribution.

Step (A): Create a space parameterizing Selmer elements

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).

Step (B): Compute the monodromy group

We have constructed a space Seld

n,k whose k points parameterize Selmer

• elements. 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

n,k : π´ et 1 (W ◦d k) → GL(V d n,k).

Theorem

For n prime, there is a quadratic form Qd

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

im ρd

n,k = O(Qd n ).

Step (C): Connect the actual distribution to the random kernel model

We have now constructed a monodromy representation. ρd

n,k : π´ et 1 (W ◦d k) → GL(V d n,k).

Theorem

For n prime, there is a quadratic form Qd

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

im ρd

n,k = O(Qd n ).

We can connect this to the random kernel model as follows: (I) The n-Selmer group of an elliptic curve [E] is realized as the Fq points of a fiber of π : Seld

n,k → W d k . But, Fq points are the same as

points fixed by Frobenius, i.e., ker(ρd

n,k(Frob) − id).

(II) Equidistribution of ρd

n,k(Frob) in the monodromy group realizes the

Selmer group as ker(g − id), for g a random element of the monodromy group im ρd

n,k. This is our random kernel model.

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?

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

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

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

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

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

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)

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 = E 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].