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 David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 1 / 13
Batyrev–Manin–Malle Let K be a number field or function field of a curve. → P N Let X ֒ K be a projective variety. Conjecture (Batyrev–Manin) There exists a nonempty open subscheme U ⊂ X and constants a , b , c such that N U ( B ) ∼ cB a (log B ) b . Let G ⊂ S n be a transitive subgroup. Conjecture (Malle) There exists constants a , b , c such that N G , K ( B ) ∼ cB a (log B ) b . David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 2 / 13
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 N U , V ( B ) ∼ cB a (log B ) b . David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 3 / 13
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 ? David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 4 / 13
99 problems → P N . There does not exist an embedding X ֒ The coarse space of BG is a point. 1 Vect BG ∼ = Rep G ⇒ 2 Pic BG is torsion ⇒ 3 ht V 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 � T � BG Spec K One must deal with non-tame Artin stacks. David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 6 / 13
� � � � 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 x π x Spec k ( C ) C ht L ( x ) := deg x ∗ L (Also true for varieties over number fields if L is metrizied and deg is the Arakelov degree.) David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 7 / 13
� � � � � � Tuning stacks Let X be a proper Artin stack with finite diagonal over C (either a smooth proper curve over a field or Spec O K , with function field K ). Let x ∈ X ( K ). Theorem There exists a stack C an a commutative diagram x x Spec K X C π 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. David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 8 / 13
� � � � � � � � � B µ 2 Spec k ( H ) H ⋆ x Spec k ( t ) C B µ 2 π P 1 ht L ( x ) = − ( g + 1) David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 9 / 13
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 ht V ( x ) = ∆ L 2 David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 10 / 13
BG redux: H ⊂ G Vect BG ∼ = Rep G p : BH → BG Let V = p ∗ O BH (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 ht V ( x ) = ∆ L H 2 David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 11 / 13
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 N U , V ( B ) ∼ cB a (log B ) b . David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 12 / 13
Thanks Thank you! David Zureick-Brown (Emory University) Stacky Batman January 18, 2019 13 / 13
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