Computing zeta functions of nondegenerate toric hypersurfaces

Transcription

1 Computing zeta functions of nondegenerate toric hypersurfaces Edgar Costa (Dartmouth College) December 4th, 2017 Presented at Boston University Number Theory Seminar Joint work with David Harvey (UNSW) and Kiran Kedlaya (UCSD) Slides available at edgarcosta.org under Research 1 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

2 Motivation

3 Riemann zeta function ζ(s) = s s s s s s = s s s One of the most famous examples of a global zeta function Together with the functional equation ξ(s) := π s/2 Γ(s/2)ζ(s) = ξ(1 s) encodes a lot of the arithmetic information of Z. e.g.: Zeros of ζ(s) precise prime distribution ζ(s) still keeps secret many of its properties 2 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

4 Hasse Weil zeta functions Hasse and Weil generalized an analog of ζ(s) for algebraic varieties ζ X (s) := p ζ Xp (p s ) 3 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

5 Hasse Weil zeta functions Hasse and Weil generalized an analog of ζ(s) for algebraic varieties ζ X (s) := p ζ Xp (p s ) If X p := X mod p is smooth, then ζ Xp (t) := exp i 0 #X p (F p i) ti i Q(t) 3 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

6 Hasse Weil zeta functions Hasse and Weil generalized an analog of ζ(s) for algebraic varieties ζ X (s) := p ζ Xp (p s ) If X p := X mod p is smooth, then ζ Xp (t) := exp i 0 #X p (F p i) ti i Q(t) Example: X = { }, a point, then ζ { } (s) = ζ(s) 3 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

7 Hasse Weil zeta functions Hasse and Weil generalized an analog of ζ(s) for algebraic varieties ζ X (s) := p ζ Xp (p s ) If X p := X mod p is smooth, then ζ Xp (t) := exp i 0 #X p (F p i) ti i Q(t) Example: X = { }, a point, then ζ { } (s) = ζ(s) What arithmetic properties of X can we read from ζ Xp (s)? ζ Xp (t) obeys a functional equation and satisfies the Riemann hypothesis! What about ζ X (s)? 3 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

8 Elliptic curves E an elliptic curve over Q ζ E (s) := p ζ Ep (p s ) and ζ Ep (t) = L p (t) (1 t)(1 pt) 1 a p t + pt 2, good reduction, a p = p + 1 #E p (F p ) L p (t) = 1 ± t, non-split/split multiplicative reduction; 1 additive reduction; 4 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

9 Elliptic curves E an elliptic curve over Q ζ E (s) := p ζ Ep (p s ) and ζ Ep (t) = L p (t) (1 t)(1 pt) 1 a p t + pt 2, good reduction, a p = p + 1 #E p (F p ) L p (t) = 1 ± t, non-split/split multiplicative reduction; 1 additive reduction; ζ E (s) = p L p (p s ) ζ(s)ζ(s 1) (1 p s )(1 p s+1 = ) L E (s) 4 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

10 Elliptic curves E an elliptic curve over Q ζ E (s) := p ζ Ep (p s ) and ζ Ep (t) = L p (t) (1 t)(1 pt) 1 a p t + pt 2, good reduction, a p = p + 1 #E p (F p ) L p (t) = 1 ± t, non-split/split multiplicative reduction; 1 additive reduction; ζ E (s) = p L p (p s ) ζ(s)ζ(s 1) (1 p s )(1 p s+1 = ) L E (s) a p arithmetic information about E p E. 4 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

11 Elliptic curves E an elliptic curve over Q ζ E (s) := p ζ Ep (p s ) and ζ Ep (t) = L p (t) (1 t)(1 pt) 1 a p t + pt 2, good reduction, a p = p + 1 #E p (F p ) L p (t) = 1 ± t, non-split/split multiplicative reduction; 1 additive reduction; ζ E (s) = p L p (p s ) ζ(s)ζ(s 1) (1 p s )(1 p s+1 = ) L E (s) a p arithmetic information about E p E. Modularity theorem = L E satisfies a functional equation Birch Swinnerton-Dyer conjecture predicts ord s=1 L E (s) = rk(e). 4 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

12 ζ(s) vs ζ X (s) We always expect ζ X (s) to satisfy a functional equation. zero-dimensional varieties (number fields) elliptic curves over Q genus 2 curves? 5 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

13 ζ(s) vs ζ X (s) We always expect ζ X (s) to satisfy a functional equation. zero-dimensional varieties (number fields) elliptic curves over Q genus 2 curves? numerically 5 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

14 ζ(s) vs ζ X (s) We always expect ζ X (s) to satisfy a functional equation. zero-dimensional varieties (number fields) elliptic curves over Q genus 2 curves? numerically surfaces? 5 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

15 ζ(s) vs ζ X (s) We always expect ζ X (s) to satisfy a functional equation. zero-dimensional varieties (number fields) elliptic curves over Q genus 2 curves? numerically surfaces? Major difference easy to explicitly write down ζ(s) extremely difficult to calculate ζ Xp (t) for an arbitrary X 5 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

16 ζ(s) vs ζ X (s) We always expect ζ X (s) to satisfy a functional equation. zero-dimensional varieties (number fields) elliptic curves over Q genus 2 curves? numerically surfaces? Major difference easy to explicitly write down ζ(s) extremely difficult to calculate ζ Xp (t) for an arbitrary X Problem Given an explicit description of X, compute ζ Xp (t) := exp i 0 #X p (F p i) ti i Q(t) 5 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

17 The zeta function problem Let X be a smooth variety over a finite field F q of characteristic p, consider ζ X (t) := exp i 1 #X(F q i) ti i Problem Compute ζ X from an explicit description of X. 6 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

18 The zeta function problem Let X be a smooth variety over a finite field F q of characteristic p, consider ζ X (t) := exp i 1 #X(F q i) ti i Problem Compute ζ X from an explicit description of X. Theoretically this is trivial. The degree of ζ X is bounded by the geometry of X, and we can then enumerate X(F q i) for enough i to pinpoint ζ X. 6 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

19 The zeta function problem Let X be a smooth variety over a finite field F q of characteristic p, consider ζ X (t) := exp i 1 #X(F q i) ti i Problem Compute ζ X from an explicit description of X. Theoretically this is trivial. The degree of ζ X is bounded by the geometry of X, and we can then enumerate X(F q i) for enough i to pinpoint ζ X. This approach is only practical for very few classes of varieties, e.g., low genus curves and p small. 6 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

20 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

21 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) Isomorphism/Isogeny testing 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

22 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) Isomorphism/Isogeny testing Determining endomorphisms of an abelian variety 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

23 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) Isomorphism/Isogeny testing Determining endomorphisms of an abelian variety rank of the Picard lattice (and the order of the Brauer group) 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

24 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) Isomorphism/Isogeny testing Determining endomorphisms of an abelian variety rank of the Picard lattice (and the order of the Brauer group) Computing L-functions and their special values, e.g.: Birch Swinnerton-Dyer searching for Langlands correspondences 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

25 Real life applications Cryptography/Coding Theory, we are often interested in #X(F q ) Isomorphism/Isogeny testing Determining endomorphisms of an abelian variety rank of the Picard lattice (and the order of the Brauer group) Computing L-functions and their special values, e.g.: Birch Swinnerton-Dyer searching for Langlands correspondences Arithmetic statistics Sato Tate Lang Trotter 7 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

26 Common Approaches Very generic algorithms derived from Dwork s p-adic analytic proof that ζ X (t) Q(t) 8 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

27 Common Approaches Very generic algorithms derived from Dwork s p-adic analytic proof that ζ X (t) Q(t) l-adic: by computing the action of Frobenius on mod-l étale cohomology for many l. 8 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

28 Common Approaches Very generic algorithms derived from Dwork s p-adic analytic proof that ζ X (t) Q(t) l-adic: by computing the action of Frobenius on mod-l étale cohomology for many l. We need to have an effective description of the cohomology. E.g.: for abelian varieties we have Schoof-Pila s method However, only practical if g 2 or some extra structure is available. 8 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

29 Common Approaches Very generic algorithms derived from Dwork s p-adic analytic proof that ζ X (t) Q(t) l-adic: by computing the action of Frobenius on mod-l étale cohomology for many l. We need to have an effective description of the cohomology. E.g.: for abelian varieties we have Schoof-Pila s method However, only practical if g 2 or some extra structure is available. p-adic: based on Monsky Washnitzer cohomology Today New p-adic method to compute ζ X (t) that achieves a striking balance between practicality and generality. 8 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

30 Outline Toric hypersurfaces p-adic Cohomology Some examples 9 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

31 Toric hypersurfaces

32 Toy example, the Projective space There are many ways to define the P n 10 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

33 Toy example, the Projective space There are many ways to define the P n For example, let P d := homogeneous polynomials in n + 1 variables of degree d and consider the graded ring P := P d. d 0 Then we have P n := Proj P 10 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

34 Toy example, the Projective space There are many ways to define the P n For example, let P d := homogeneous polynomials in n + 1 variables of degree d and consider the graded ring 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. 10 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

35 Toric hypersurfaces f = α Z n c α x α R[x ± 1,..., x± n ] a Laurent polynomial f defines an hypersurface in the torus T n := Spec(R[x ± 1,..., x± n ]) 11 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

36 Toric hypersurfaces f = α Z n c α x α R[x ± 1,..., x± n ] a Laurent polynomial f defines an hypersurface in the torus T n := Spec(R[x ± 1,..., x± n ]) := Newton polytope of f = convex hull of the support of f in R n 11 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

37 Toric hypersurfaces f = α Z n c α x α R[x ± 1,..., x± n ] a Laurent polynomial f defines an hypersurface in the torus T n := Spec(R[x ± 1,..., x± n ]) := Newton polytope of f = convex hull of the support of f in R n To we can associate a graded ring and a projective variety. 11 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

38 Toric hypersurfaces f = α Z n c α x α R[x ± 1,..., x± n ] a Laurent polynomial f defines an hypersurface in the torus T n := Spec(R[x ± 1,..., x± n ]) := Newton polytope of f = convex hull of the support of f in R n To we can associate a graded ring and a projective variety. P := d 0 P d, P d := R[d Z n ] P := Proj P X f := Proj P /(f) P X f is an hypersurface in the toric variety P 11 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

39 Toric hypersurfaces f = α Z n c α x α R[x ± 1,..., x± n ] a Laurent polynomial f defines an hypersurface in the torus T n := Spec(R[x ± 1,..., x± n ]) := Newton polytope of f = convex hull of the support of f in R n To we can associate a graded ring and a projective variety. P := d 0 P d, P d := R[d Z n ] P := Proj P X f := Proj P /(f) P X f is an hypersurface in the toric variety P Examples X Conv(0, e 1,..., e n ) P n Conv(0, e 1, le 2,..., le n ) P n (l, 1,..., 1) Conv(0, e 1, e 2, e 1 + e 2 ) = [0, 1] 2 P 1 P 1 11 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

40 Toric hypersurfaces are everywhere Some familiar examples: Vertices of Resulting hypersurface ( ) d 1 0, de 1, de 2 Smooth plane curve of genus 2 0, (2g + 1)e 1, 2e 2 Odd hyperelliptic curve of genus g 0, ae 1, be 2 C a,b -curve 0, 4e 1, 4e 2, 4e 3 Quartic K3 surface 0, 2e 1, 6e 2, 6e 3 Degree 2 K3 surface 12 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

41 Toric hypersurfaces are everywhere Some familiar examples: Vertices of Resulting hypersurface ( ) d 1 0, de 1, de 2 Smooth plane curve of genus 2 0, (2g + 1)e 1, 2e 2 Odd hyperelliptic curve of genus g 0, ae 1, be 2 C a,b -curve 0, 4e 1, 4e 2, 4e 3 Quartic K3 surface 0, 2e 1, 6e 2, 6e 3 Degree 2 K3 surface (All the examples above are hypersurfaces in a weighted projective spaces.) 12 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

42 Toric hypersurfaces are everywhere Some familiar examples: Vertices of Resulting hypersurface ( ) d 1 0, de 1, de 2 Smooth plane curve of genus 2 0, (2g + 1)e 1, 2e 2 Odd hyperelliptic curve of genus g 0, ae 1, be 2 C a,b -curve 0, 4e 1, 4e 2, 4e 3 Quartic K3 surface 0, 2e 1, 6e 2, 6e 3 Degree 2 K3 surface (All the examples above are hypersurfaces in a weighted projective spaces.) K3 surfaces can arise as hypersurfaces: in P 3, as a quartic surface; 12 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

43 Toric hypersurfaces are everywhere Some familiar examples: Vertices of Resulting hypersurface ( ) d 1 0, de 1, de 2 Smooth plane curve of genus 2 0, (2g + 1)e 1, 2e 2 Odd hyperelliptic curve of genus g 0, ae 1, be 2 C a,b -curve 0, 4e 1, 4e 2, 4e 3 Quartic K3 surface 0, 2e 1, 6e 2, 6e 3 Degree 2 K3 surface (All the examples above are hypersurfaces in a weighted projective spaces.) K3 surfaces can arise as hypersurfaces: in P 3, as a quartic surface; in 95 weighed projective spaces; 12 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

44 Toric hypersurfaces are everywhere Some familiar examples: Vertices of Resulting hypersurface ( ) d 1 0, de 1, de 2 Smooth plane curve of genus 2 0, (2g + 1)e 1, 2e 2 Odd hyperelliptic curve of genus g 0, ae 1, be 2 C a,b -curve 0, 4e 1, 4e 2, 4e 3 Quartic K3 surface 0, 2e 1, 6e 2, 6e 3 Degree 2 K3 surface (All the examples above are hypersurfaces in a weighted projective spaces.) K3 surfaces can arise as hypersurfaces: in P 3, as a quartic surface; in 95 weighed projective spaces; in 4319 toric varieties. 12 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

45 Keeping our eyes on the prize Given f = α Z n c α x α F q [x ± 1,..., x± n ] efficiently compute ζ X (t) := exp i 1 #X(F q i) ti i = i det ( 1 t Frob H i et(x Fq, Q l ) ) ( 1) i+1 Q(t), where X := Proj P /(f) P 13 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

46 Keeping our eyes on the prize Given f = α Z n c α x α F q [x ± 1,..., x± n ] efficiently compute ζ X (t) := exp i 1 #X(F q i) ti i = i det ( 1 t Frob H i et(x Fq, Q l ) ) ( 1) i+1 Q(t), where X := Proj P /(f) P But under what assumptions on X? Is smoothness enough? 13 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

47 Keeping our eyes on the prize Given f = α Z n c α x α F q [x ± 1,..., x± n ] efficiently compute ζ X (t) := exp i 1 #X(F q i) ti i = i det ( 1 t Frob H i et(x Fq, Q l ) ) ( 1) i+1 Q(t), where X := Proj P /(f) P But under what assumptions on X? Is smoothness enough? We will need a bit more, we will nondegeneracy. 13 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

48 Nondegenerate toric hypersurfaces Geometric definition An hypersurface is nondegenerate if the cross-section by any bounding hyperplane (in any dimension) are all smooth in their respective tori. Equivalently, if for every face σ, f restricted to the torus associated to σ is nonsingular of codimension / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

49 Nondegenerate toric hypersurfaces Geometric definition An hypersurface is nondegenerate if the cross-section by any bounding hyperplane (in any dimension) are all smooth in their respective tori. Equivalently, if for every face σ, f restricted to the torus associated to σ is nonsingular of codimension 1. Example Let C be a plane curve in P 2, then C is nondegenerate if: C does not pass through the points (1, 0, 0), (0, 1, 0), (0, 0, 1); C intersects the coordinate axes x = 0, y = 0, z = 0 transversally; C is smooth on the complement of the coordinate axes. 14 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

50 Nondegenerate toric hypersurfaces Geometric definition An hypersurface is nondegenerate if the cross-section by any bounding hyperplane (in any dimension) are all smooth in their respective tori. Equivalently, if for every face σ, f restricted to the torus associated to σ is nonsingular of codimension 1. Example Let C be a plane curve in P 2, then C is nondegenerate if: C does not pass through the points (1, 0, 0), (0, 1, 0), (0, 0, 1); C intersects the coordinate axes x = 0, y = 0, z = 0 transversally; C is smooth on the complement of the coordinate axes. In terms of ideals, rad x x f, y y f, z z f, f = x, y, z 14 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

51 p-adic Cohomology

52 Goal Setup f = α Z n c α x α F q [x ± 1,..., x± n ] X := Proj P /(f) P a nondegenerate hypersurface Goal Compute ζ X (t) := exp #X(F q i)t i /i Q(t) i 1 = i det ( 1 t Frob H i et(x Fq, Q l ) ) ( 1) i+1 n 1 ( = Q(t) ( 1)n 1 ) bi, 1 q i t where Q(t) 1 + Z[t] and the b i are determined by. i=0 15 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

53 How can we do this? Naively computing #X(F q a) to deduce Q(t) is not practical! 16 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

54 How can we do this? Naively computing #X(F q a) to deduce Q(t) is not practical! In general, to recover Q(t) we need #X(F q a) for a = 1,..., deg Q/2. Some data points: Elliptic curve, Q(t) = 1 a p t + pt 2 Smooth plane curve, Q(t) = 1 + a p,1 t + a p,2 t 2 + pa p,1 t 3 + p 2 t 4 K3 surface, deg Q 21 Calabi-Yau 3fold, deg Q / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

55 How can we do this? Naively computing #X(F q a) to deduce Q(t) is not practical! In general, to recover Q(t) we need #X(F q a) for a = 1,..., deg Q/2. Some data points: Elliptic curve, Q(t) = 1 a p t + pt 2 Smooth plane curve, Q(t) = 1 + a p,1 t + a p,2 t 2 + pa p,1 t 3 + p 2 t 4 K3 surface, deg Q 21 Calabi-Yau 3fold, deg Q 204 Can we rewrite #X(F q ) = {x F q : f(x) = 0}? 16 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

56 How can we do this? Naively computing #X(F q a) to deduce Q(t) is not practical! In general, to recover Q(t) we need #X(F q a) for a = 1,..., deg Q/2. Some data points: Elliptic curve, Q(t) = 1 a p t + pt 2 Smooth plane curve, Q(t) = 1 + a p,1 t + a p,2 t 2 + pa p,1 t 3 + p 2 t 4 K3 surface, deg Q 21 Calabi-Yau 3fold, deg Q 204 Can we rewrite #X(F q ) = {x F q : f(x) = 0}? = {x X ( ) F q : FrobFq (x) = x} 16 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

57 Lefschetz fixed point theorem Given Z be a nice space; H be a nice cohomology theory; F : X X be a nice map. L(F n ) = #{x X : F n (x) = x} we have ( ) exp L(F n )t n /n = n=1 i det(1 tf H i (X)) ( 1)i+1 17 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

58 Lefschetz fixed point theorem Given Z be a nice space; H be a nice cohomology theory; F : X X be a nice map. L(F n ) = #{x X : F n (x) = x} we have ( ) exp L(F n )t n /n = n=1 i det(1 tf H i (X)) ( 1)i+1 Applying this to the Frobenius endomorphism, we get ζ XFq (t) = i det(1 t Frob H i (X Fq )) ( 1)i+1 17 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

59 Lefschetz fixed point theorem Given Z be a nice space; H be a nice cohomology theory; F : X X be a nice map. L(F n ) = #{x X : F n (x) = x} we have ( ) exp L(F n )t n /n = n=1 i det(1 tf H i (X)) ( 1)i+1 Applying this to the Frobenius endomorphism, we get ζ XFq (t) = i det(1 t Frob H i (X Fq )) ( 1)i+1 = det(1 q 1 t Frob H n( n 1 ) ( ) bi U Fq ) ( 1) n 1, 1 q i t where U := P \X 17 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces i=0

60 Master plan Setup f = c α x α F q [x ± 1,..., x± n ] α Z n X := Proj P /(f) P a nondegenerate hypersurface U := P \X the complement of X σ := p-th power Frobenius Goal Compute the matrix representing the action of σ in H n (U) with enough of p-adic precision to deduce Q(t) := det(1 q 1 t Frob H n( U ) ) 1 + Z[t]. 18 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

61 Master plan Setup f = c α x α F q [x ± 1,..., x± n ] α Z n X := Proj P /(f) P a nondegenerate hypersurface U := P \X the complement of X σ := p-th power Frobenius Goal Compute the matrix representing the action of σ in H n (U) with enough of p-adic precision to deduce Q(t) := det(1 q 1 t Frob H n( U ) ) 1 + Z[t]. We will work with H n = H,n, the Monsky Washnitzer cohomology. 18 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

62 Overall picture Goal Compute the matrix representing the action of σ in H,n (U) with enough p-adic precision. 19 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

63 Overall picture Goal Compute the matrix representing the action of σ in H,n (U) with enough p-adic precision. H n dr (U Q q ) id σ H,n (U) 19 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

64 Overall picture Goal Compute the matrix representing the action of σ in H,n (U) with enough p-adic precision. H n dr (U Q p ) explicit description over C [Dwork Griffiths, Batyrev Cox] id σ H,n (U) 19 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

65 Overall picture Goal Compute the matrix representing the action of σ in H,n (U) with enough p-adic precision. H n dr (U Q p ) explicit description over C [Dwork Griffiths, Batyrev Cox] id σ H,n (U) de Rham cohomology with overconvergent power series 19 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

66 Overall picture Goal Compute the matrix representing the action of σ in H,n (U) with enough p-adic precision. H n dr (U Q p ) explicit description over C [Dwork Griffiths, Batyrev Cox] id σ H,n (U) de Rham cohomology with overconvergent power series cohomology relations + commutative algebra = basis for H n dr (U Q q ) = { x β ω/f i} β + reduction algorithm 19 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

67 Generic algorithm Abbott Kedlaya Roe type H n dr (U Q q ) id σ H,n (U) { } x β 1. Compute f m ω a monomial basis for H n dr (U Q q ) β with ω := dx1 x 1 dxn x n Ω n 2. In H,n compute a series approximation for ( ) x β σ f m ω = p n xpβ f pm ω i 0 ( ) ( m σ(f) f p 3. Write the approximation in terms of basis elements, i.e., apply the de Rham relations Note: Originally for smooth hypersurfaces in the projective space. 20 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces i f p ) i

68 Schematically Abbott Kedlaya Roe vs C. Harvey Kedlaya K 1 ( m ) ( ) σ(f) f p i K 1 ( m )( m+k 1 ) i=0 i f p i=0 i K i 1 σ(f) i f p(m+i) (pdk) n+o(1) terms (dk) n+o(1) terms 21 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

69 Schematically Abbott Kedlaya Roe vs C. Harvey Kedlaya K 1 ( m ) ( ) σ(f) f p i K 1 ( m )( m+k 1 ) i=0 i f p i=0 i K i 1 σ(f) i f p(m+i) (pdk) n+o(1) terms (dk) n+o(1) terms 21 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

70 Schematically Abbott Kedlaya Roe vs C. Harvey Kedlaya K 1 ( m ) ( ) σ(f) f p i K 1 ( m )( m+k 1 ) i=0 i f p i=0 i K i 1 σ(f) i f p(m+i) (pdk) n+o(1) terms (dk) n+o(1) terms ρ : P l+1 P l l gω ρ(g)ω f l+1 f l 21 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

71 Schematically Abbott Kedlaya Roe vs C. Harvey Kedlaya K 1 ( m ) ( ) σ(f) f p i K 1 ( m )( m+k 1 ) i=0 i f p i=0 i K i 1 σ(f) i f p(m+i) (pdk) n+o(1) terms (dk) n+o(1) terms ρ : P l+1 P l l gω ρ(g)ω f l+1 f l lx π : P n P n α+β gω π(g)ω xβ f l+1 f l, x α P 1 21 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

72 Schematically Abbott Kedlaya Roe vs C. Harvey Kedlaya K 1 ( m ) ( ) σ(f) f p i K 1 ( m )( m+k 1 ) i=0 i f p i=0 i K i 1 σ(f) i f p(m+i) (pdk) n+o(1) terms (dk) n+o(1) terms ρ : P l+1 P l l gω ρ(g)ω f l+1 f l slice slice lx π : P n P n α+β gω π(g)ω xβ f l+1 f l, x α P 1 dot dot 21 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

73 Generic algorithm C. Harvey Kedlaya H n dr (U) id σ H,n (U Fp ) { } x β 1. Compute f m ω a monomial basis for H n dr (U Q q ) β 2. In H,n compute a sparse approximation for σ ( ) x β f m ω p n xpβ f pm N 1 ( m i i=0 )( m + N 1 N i 1 ) σ(f) i f p(m+i) 3. Apply sparse reduction algorithm to reduce expansion to basis elements. Involves multiplying together O(p) matrices of size #(n L) n n vol 22 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

74 Generic algorithm C. Harvey Kedlaya H n dr (U) id σ H,n (U Fp ) { } x β 1. Compute f m ω a monomial basis for H n dr (U Q q ) β 2. In H,n compute a sparse approximation for σ ( ) x β f m ω p n xpβ f pm N 1 ( m i i=0 )( m + N 1 N i 1 ) σ(f) i f p(m+i) 3. Apply sparse reduction algorithm to reduce expansion to basis elements. Involves multiplying together O(p) matrices of size #(n L) n n vol In a more convoluted process, we can reduce the matrix size to n! vol, saving a factor of e n n n /n! (e.g ) 22 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

75 Generic algorithm C. Harvey Kedlaya H n dr (U) id σ H,n (U Fp ) { } x β 1. Compute f m ω a monomial basis for H n dr (U Q q ) β 2. In H,n compute a sparse approximation for σ ( ) x β f m ω p n xpβ f pm N 1 ( m i i=0 )( m + N 1 N i 1 ) σ(f) i f p(m+i) 3. Apply sparse reduction algorithm to reduce expansion to basis elements. Involves multiplying together O(p) matrices of size #(n L) n n vol In a more convoluted process, we can reduce the matrix size to n! vol, saving a factor of e n n n /n! (e.g ) For large p, all the work is in step 3 22 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

76 Some Remarks Complexity First version of our new algorithm has complexity roughly p 1+o(1) vol( ) O(n) 23 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

77 Some Remarks Complexity First version of our new algorithm has complexity roughly and space complexity is only p 1+o(1) vol( ) O(n) log p vol( ) O(n). 23 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

78 Some Remarks Complexity First version of our new algorithm has complexity roughly and space complexity is only p 1+o(1) vol( ) O(n) log p vol( ) O(n). This allows us to handle examples with much larger p than any found in the literature. 23 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

79 Some Remarks Complexity First version of our new algorithm has complexity roughly and space complexity is only p 1+o(1) vol( ) O(n) log p vol( ) O(n). This allows us to handle examples with much larger p than any found in the literature. Implementation Projective hypersurfaces ( 2014): C++ with NTL and Flint Soon available in Sage 23 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

80 Some Remarks Complexity First version of our new algorithm has complexity roughly and space complexity is only p 1+o(1) vol( ) O(n) log p vol( ) O(n). This allows us to handle examples with much larger p than any found in the literature. Implementation Projective hypersurfaces ( 2014): C++ with NTL and Flint Soon available in Sage Toric hypersurfaces: beta version in C++ with NTL 23 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

81 Some examples

82 Example: random dense K3 surface X P 3 F p given by 9x 4 10x 3 y 9x 2 y 2 + 2xy 3 7y 4 + 6x 3 z + 9x 2 yz 2xy 2 z + 3y 3 z + 8x 2 z 2 + 6y 2 z 2 + 2xz 3 + 7yz 3 + 9z 4 + 8x 3 w + x 2 yw 8xy 2 w 7y 3 w + 9x 2 zw 9xyzw + 3y 2 zw xz 2 w 3yz 2 w + z 3 w x 2 w 2 4xyw 2 3xzw 2 + 8yzw 2 6z 2 w 2 + 4xw 3 + 3yw 3 + 4zw 3 5w 4 = 0 For p = 49999, in 1h5m5s, we obtain ζ X (t) = ((1 t)(1 pt)(1 p 2 t)q(t)) 1 where pq(t/p) =(1 t)(p t t t t t t t t t t t t pt 20 ) 24 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

83 Example: a quartic surface in the Dwork pencil Consider the surface X in P 3 F p for p = = given by x x x x x 0 x 1 x 2 x 3 = 0. Using the old projective code in 100h30m we compute that ζ X (t) = where the interesting factor 1 (1 t)(1 pt)(1 p 2 t)r 1 (pt) 3 R 2 (pt) 6 S(t) S(t) = (1 + pt)( t + p 2 t 2 ). The polynomials R 1 and R 2 arise from the action of Frobenius on the Picard lattice; by a p-adic formula of de la Ossa Kadir. 25 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

84 Example: a quartic surface in the Dwork pencil Consider the surface X in P 3 F p for p = = given by x x x x x 0 x 1 x 2 x 3 = 0. Using the toric old projective code in 6m32s 100h30m we compute ζ X (t) = where the interesting factor 1 (1 t)(1 pt)(1 p 2 t)r 1 (pt) 3 R 2 (pt) 6 S(t) S(t) = (1 + pt)( t + p 2 t 2 ). The polytope of X is much smaller than the full simplex (32/ vs 2/3 0.6), as the monomials defining X generate a sublattice of index 4 2 in Z / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

85 Example: a quintic threefold in the Dwork pencil Consider the threefold X in P 4 F p for p = given by x x x 0 x 1 x 2 x 3 x 5 = 0. In 667s, we compute that ζ X (t) = where the interesting factor is R 1 (pt) 20 R 2 (pt) 30 S(t) (1 t)(1 pt)(1 p 2 t)(1 p 3 t) S(t) = T pT p 3 T 3 + p 6 T 4. and R 1 and R 2 are the numerators of the zeta functions of certain curves (given by a formula of Candelas de la Ossa Rodriguez Villegas). Using the old projective code, it would have taken around 37h. 27 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

86 Example: another family of K3 surfaces Consider now the surface X in the weighted projective space P(8, 5, 4, 3) Fp given by taking the closure of the affine surface yz 5 + xz 4 + y 4 + z 4 + x = 0. For p = 49999, in 120s we compute that where ζ X (t) = 1 (1 t)(1 pt)(1 p 2 t)r(pt)s(t) ps(p 1 t) = p 14662t 31559t t t t 5 + pt 6. This example is from Miles Reid s list of 95 families of nondegenerate toric surfaces which are K3 surfaces. There is no other method that can handle dense surfaces in P(8, 5, 4, 3) in this p range. 28 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

87 Example: a hypergeometric motive (also a K3 surface) Consider the appropriate completion of the toric surface over F p with p = given by x 3 y + y 4 + z 4 12xyz + 1 = 0. In 61s, we compute that the interesting factor of ζ X (t) is t 42243pt p 2 t p 3 t p 4 t 5 + p 6 t 6. In P 3 this surface is degenerate, and would have taken us one hour to do the same computation with a dense model. 29 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

88 Example: a hypergeometric motive (also a K3 surface) Consider the appropriate completion of the toric surface over F p with p = given by x 3 y + y 4 + z 4 12xyz + 1 = 0. In 61s, we compute that the interesting factor of ζ X (t) is t 42243pt p 2 t p 3 t p 4 t 5 + p 6 t 6. In P 3 this surface is degenerate, and would have taken us one hour to do the same computation with a dense model. We can confirm the linear term with Magma: C2F2 := HypergeometricData([6,12], [1,1,1,2,3]); EulerFactor(C2F2, 2ˆ10 * 3ˆ6, 49999: Degree:=1); *$.1 + O($.1ˆ2) 29 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

89 Example: another Calabi Yau threefold Let X be the closure in the weighted projective space P(10, 11, 16, 19, 21) Fp for p = of the affine threefold y 7 + x 2 zw + zyzw + y 2 zw + z 3 w + w 3 + xz + yz = 0. In 401s, we we compute that the interesting factor of ζ X is t pt p 2 t p 4 t p 6 T 5 + p 9 T 6 By analogy with the Reid list, one can classify Calabi Yau threefolds arising as hypersurfaces in weighted projective spaces; there are 7555 such families. See 30 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

90 Other possible versions Space-time tradeoff We can reduce the time dependence on p to p 0.5+o(1) vol( ) O(n) 31 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

91 Other possible versions Space-time tradeoff We can reduce the time dependence on p to p 0.5+o(1) vol( ) O(n) Average polynomial time Given an hypersurface defined over Q, we may compute the zeta functions of its reductions modulo various primes at once. The average time complexity for each prime p < N is log(n) 4+o(1) vol( ) O(n) 31 / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces

92 Other possible versions Space-time tradeoff We can reduce the time dependence on p to p 0.5+o(1) vol( ) O(n) Average polynomial time Given an hypersurface defined over Q, we may compute the zeta functions of its reductions modulo various primes at once. The average time complexity for each prime p < N is log(n) 4+o(1) vol( ) O(n) These have not yet been implemented and we still need to write the paper / 31 Edgar Costa (Dartmouth College) Computing zeta functions of nondegenerate toric hypersurfaces