Computing zeta functions of nondegenerate hypersurfaces in toric - PowerPoint PPT Presentation

Computing zeta functions of nondegenerate hypersurfaces in toric varieties Edgar Costa (Massachusetts Institute of Technology) July 16th, 2018 Presented at ANTS XIII Joint work with David Harvey (UNSW) and Kiran Kedlaya (UCSD) Slides available at edgarcosta.org under Research

The zeta function problem Problem • Arithmetic statistics (Sato–Tate, Lang–Trotter, etc) A for an abelian variety • • L-functions and their special values • Some applications include: • In practice, this only works for very few classes of varieties X • Theoretically, this is “trivial”, the geometry of X gives us • Other geometric invariants i Consider: • F q finite field of characteristic p • X a smooth variety over F q   ∑  ∈ Q ( t ) ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X .

The zeta function problem Problem • Arithmetic statistics (Sato–Tate, Lang–Trotter, etc) A for an abelian variety • • L-functions and their special values • Some applications include: • In practice, this only works for very few classes of varieties • Other geometric invariants Consider: i • F q finite field of characteristic p • X a smooth variety over F q   ∑  ∈ Q ( t ) ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X . • Theoretically, this is “trivial”, the geometry of X gives us deg ζ X

The zeta function problem i • Arithmetic statistics (Sato–Tate, Lang–Trotter, etc) • L-functions and their special values • Some applications include: • In practice, this only works for very few classes of varieties Problem • Other geometric invariants Consider: • F q finite field of characteristic p • X a smooth variety over F q   ∑  ∈ Q ( t ) ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X . • Theoretically, this is “trivial”, the geometry of X gives us deg ζ X • End( A ) for an abelian variety

The zeta function problem Problem balance between practicality and generality . Today i A quasi-linear in p algorithm for hypersurfaces in toric varieties. Consider: • F q finite field of characteristic p • X a smooth variety over F q   ∑  ∈ Q ( t ) ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X . New p -adic method to compute ζ X ( t ) that achieves a striking

The zeta function problem Problem balance between practicality and generality . Today i A quasi-linear in p algorithm for hypersurfaces in toric varieties. Consider: • F q finite field of characteristic p • X a smooth variety over F q   ∑  ∈ Q ( t ) ζ X ( t ) := exp # X ( F q i ) t i i ≥ 1 Compute ζ X from an explicit description of X . New p -adic method to compute ζ X ( t ) that achieves a striking

Hypersurfaces in toric varieties

n , Toy example, the Projective space Then we have • Idea: generalize is the standard simplex. where R d • We can think of P d P n P d 0 d P and the graded ring 1 variables homogeneous polynomials of degree d in n P d • For example, consider to be any polytope. • There are many ways to define P n

n , Toy example, the Projective space • We can think of P d • Idea: generalize is the standard simplex. where R d to be any polytope. and the graded ring • For example, consider • There are many ways to define P n P d := homogeneous polynomials of degree d in n + 1 variables ⊕ P := P d . d ≥ 0 Then we have P n := Proj P

Toy example, the Projective space • For example, consider and the graded ring • There are many ways to define P n P d := homogeneous polynomials of degree d in n + 1 variables ⊕ P := P d . d ≥ 0 Then we have P n := Proj P • We can think of P d := R [ d ∆ ∩ Z n ] , where ∆ is the standard simplex. • Idea: generalize ∆ to be any polytope.

X f is an hypersurface in the toric variety P f P X f P n d R x P d P d 0 d Toric hypersurfaces • To Newton polytope of f = convex hull of the support of f • we associate a graded ring and a projective variety. c α x α ∈ R [ x ± ∑ 1 , . . . , x ± • f = n ] a Laurent polynomial α ∈ Z n • f defines an hypersurface in the torus Spec( R [ x ± 1 , . . . , x ± n ])

X f is an hypersurface in the toric variety d f P X f P n d R x P d P d 0 Toric hypersurfaces we associate a graded ring and a projective variety. • To P c α x α ∈ R [ x ± ∑ 1 , . . . , x ± • f = n ] a Laurent polynomial α ∈ Z n • f defines an hypersurface in the torus Spec( R [ x ± 1 , . . . , x ± n ]) • ∆ := Newton polytope of f = convex hull of the support of f

X f is an hypersurface in the toric variety d f P X f P n d R x P d P d 0 Toric hypersurfaces P c α x α ∈ R [ x ± ∑ 1 , . . . , x ± • f = n ] a Laurent polynomial α ∈ Z n • f defines an hypersurface in the torus Spec( R [ x ± 1 , . . . , x ± n ]) • ∆ := Newton polytope of f = convex hull of the support of f • To ∆ we associate a graded ring and a projective variety.

Toric hypersurfaces c α x α ∈ R [ x ± ∑ 1 , . . . , x ± • f = n ] a Laurent polynomial α ∈ Z n • f defines an hypersurface in the torus Spec( R [ x ± 1 , . . . , x ± n ]) • ∆ := Newton polytope of f = convex hull of the support of f • To ∆ we associate a graded ring and a projective variety. P d := R [ x α : α ∈ d ∆ ∩ Z n ] ⊕ P ∆ := P d , d ≥ 0 P ∆ := Proj P ∆ X f := Proj P ∆ / ( f ) ⊂ P ∆ X f is an hypersurface in the toric variety P ∆

Toric hypersurfaces are everywhere Quartic K3 surface • in 95 weighed projective spaces (Reid’s list); 3 , as a quartic surface; • in K3 surfaces can arise as hypersurfaces: (The examples above are hypersurfaces in a weighted proj. spaces) Degree 2 K3 surface • in 4319 toric varieties. Resulting hypersurface Odd hyperelliptic curve of genus g Vertices of ∆ 0 , e 1 , . . . , e n Hypersurface in P n 0 , ( 2 g + 1 ) e 1 , 2 e 2 0 , ae 1 , be 2 C a , b -curve 0 , 4 e 1 , 4 e 2 , 4 e 3 0 , 2 e 1 , 6 e 2 , 6 e 3

Toric hypersurfaces are everywhere Quartic K3 surface • in 95 weighed projective spaces (Reid’s list); 3 , as a quartic surface; • in K3 surfaces can arise as hypersurfaces: (The examples above are hypersurfaces in a weighted proj. spaces) Degree 2 K3 surface • in 4319 toric varieties. Resulting hypersurface Odd hyperelliptic curve of genus g Vertices of ∆ 0 , e 1 , . . . , e n Hypersurface in P n 0 , ( 2 g + 1 ) e 1 , 2 e 2 0 , ae 1 , be 2 C a , b -curve 0 , 4 e 1 , 4 e 2 , 4 e 3 0 , 2 e 1 , 6 e 2 , 6 e 3

Toric hypersurfaces are everywhere Quartic K3 surface • in 95 weighed projective spaces (Reid’s list); K3 surfaces can arise as hypersurfaces: (The examples above are hypersurfaces in a weighted proj. spaces) Degree 2 K3 surface • in 4319 toric varieties. Odd hyperelliptic curve of genus g Resulting hypersurface Vertices of ∆ 0 , e 1 , . . . , e n Hypersurface in P n 0 , ( 2 g + 1 ) e 1 , 2 e 2 0 , ae 1 , be 2 C a , b -curve 0 , 4 e 1 , 4 e 2 , 4 e 3 0 , 2 e 1 , 6 e 2 , 6 e 3 • in P 3 , as a quartic surface;

Toric hypersurfaces are everywhere Quartic K3 surface • in 95 weighed projective spaces (Reid’s list); K3 surfaces can arise as hypersurfaces: (The examples above are hypersurfaces in a weighted proj. spaces) Degree 2 K3 surface • in 4319 toric varieties. Odd hyperelliptic curve of genus g Resulting hypersurface Vertices of ∆ 0 , e 1 , . . . , e n Hypersurface in P n 0 , ( 2 g + 1 ) e 1 , 2 e 2 0 , ae 1 , be 2 C a , b -curve 0 , 4 e 1 , 4 e 2 , 4 e 3 0 , 2 e 1 , 6 e 2 , 6 e 3 • in P 3 , as a quartic surface;

Toric hypersurfaces are everywhere Quartic K3 surface • in 95 weighed projective spaces (Reid’s list); K3 surfaces can arise as hypersurfaces: (The examples above are hypersurfaces in a weighted proj. spaces) Degree 2 K3 surface • in 4319 toric varieties. Odd hyperelliptic curve of genus g Resulting hypersurface Vertices of ∆ 0 , e 1 , . . . , e n Hypersurface in P n 0 , ( 2 g + 1 ) e 1 , 2 e 2 0 , ae 1 , be 2 C a , b -curve 0 , 4 e 1 , 4 e 2 , 4 e 3 0 , 2 e 1 , 6 e 2 , 6 e 3 • in P 3 , as a quartic surface;

Keeping our eyes on the prize efficiently compute We will need a bit more, we will need nondegeneracy . But under what assumptions on X ? Is smoothness enough? X i Given A generic condition over an infinite field and a fixed c α x α ∈ F q [ x ± ∑ 1 , . . . , x ± f = n ] α ∈ Z n   ∑ ζ X ( t ) := exp # X ( F q i ) t i  i ≥ 1 = det( 1 − q − 1 t Frob | PH † , n − 1 ( ) ) ( − 1 ) n ζ P ∆ ( t ) , where X := Proj P ∆ / ( f ) ⊂ P ∆ .

Keeping our eyes on the prize efficiently compute We will need a bit more, we will need nondegeneracy . But under what assumptions on X ? Is smoothness enough? X i Given A generic condition over an infinite field and a fixed c α x α ∈ F q [ x ± ∑ 1 , . . . , x ± f = n ] α ∈ Z n   ∑ ζ X ( t ) := exp # X ( F q i ) t i  i ≥ 1 = det( 1 − q − 1 t Frob | PH † , n − 1 ( ) ) ( − 1 ) n ζ P ∆ ( t ) , where X := Proj P ∆ / ( f ) ⊂ P ∆ .