Counting points, counting fields, and heights on stacks. David - - PowerPoint PPT Presentation

Counting points, counting fields, and heights on stacks.

David Zureick-Brown

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

JMM Special Session on Arithmetic Statistics January 18, 2019

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Batyrev–Manin–Malle

Let K be a number field or function field of a curve. Let X ֒ → PN

K be a projective variety.

Conjecture (Batyrev–Manin)

There exists a nonempty open subscheme U ⊂ X and constants a, b, c such that NU(B) ∼ cBa (log B)b . Let G ⊂ Sn be a transitive subgroup.

Conjecture (Malle)

There exists constants a, b, c such that NG,K(B) ∼ cBa (log B)b .

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Bat–Man for stacks

1 Let X be a proper Artin stack with finite diagonal. 2 Let V ∈ Vect X be a (Northcott) vector bundle.

Conjecture (Ellenberg–Satriano–ZB)

There exists a non-empty open substack U ⊂ X and constants a, b, c such that NU ,V (B) ∼ cBa (log B)b .

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Why bother?

BG = [Spec Z/G] BG(K) ↔ L ⊃ K with Gal(L/K) ∼ = G

Question

Is there an intrinsic notion of height on BG?

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99 problems

There does not exist an embedding X ֒ → PN. The coarse space of BG is a point.

1 Vect BG ∼

= Rep G ⇒

2 Pic BG is torsion ⇒ 3 htV cannot be additive David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 5 / 13

99 problems

Let R be a DVR with fraction field K.

Problem

X (R) → X (K) is not surjective Spec L

• P
• ´

et

• ⋆
• Spec K

T BG

One must deal with non-tame Artin stacks.

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Geometric heights

Let K = k(C), where C is a smooth proper curve over k. Let X be a proper variety over C. Let L ∈ Pic X. Let x ∈ X(K), with extension x : C → X X

π

• Spec k(C)
• x
• C

x

• htL(x) := deg x∗L

(Also true for varieties over number fields if L is metrizied and deg is the Arakelov degree.)

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Tuning stacks

Let X be a proper Artin stack with finite diagonal over C (either a smooth proper curve over a field or Spec OK, with function field K). Let x ∈ X(K).

Theorem

There exists a stack C an a commutative diagram Spec K

• x
• C

x

• π
• X

p

• C

such that π is a birational moduli space morphism. We call such a C a tuning stack for x, and we call a terminal such C a “universal” tuning stack.

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Bµ2

Spec k(H)

• H
• ⋆
• Spec k(t)
• C

x

• π
• Bµ2

P1 htL(x) = −(g + 1)

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BG redux

Vect BG ∼ = Rep G p : ⋆ → BG Let V = p∗O⋆ (corresponds to the regular representation of G) Let x ∈ BG(Q) be a rational point, corresponding to a G-extension L ⊃ Q.

Proposition

htV (x) = ∆L 2

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BG redux: H ⊂ G

Vect BG ∼ = Rep G p : BH → BG Let V = p∗OBH (corresponds to the permutation representation of G on G/H) Let x ∈ BG(Q) be a rational point, corresponding to a G-extension L ⊃ Q.

Proposition

htV (x) = ∆LH 2

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Bat–Man for stacks

1 Let X be a proper Artin stack with finite diagonal. 2 Let V ∈ Vect X be a (Northcott) vector bundle.

Conjecture (Ellenberg–Satriano–ZB)

There exists a non-empty open substack U ⊂ X and constants a, b, c such that NU ,V (B) ∼ cBa (log B)b .

David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 12 / 13

Thanks

Thank you!

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