Mirror symmetry and K3 surface zeta functions Ursula Whitcher - - PowerPoint PPT Presentation

Mirror symmetry and K3 surface zeta functions

Mirror symmetry and K3 surface zeta functions

Ursula Whitcher whitchua@uwec.edu

University of Wisconsin–Eau Claire

October 2015

Mirror symmetry and K3 surface zeta functions

Collaborators

Tyler Kelly, Charles Doran, Steven Sperber, Ursula Whitcher, John Voight, Adriana Salerno

Mirror symmetry and K3 surface zeta functions

Outline

Experimental Evidence Classical Mirror Symmetry Arithmetic Mirror Symmetry Alternate Mirrors Berglund-H¨ ubsch-Krawitz Duality Results

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Five Interesting Quartics in P3

Family Equation F4 (Fermat/Dwork) x4

0 + x4 1 + x4 2 + x4 3

F2L2 x4

0 + x4 1 + x3 2x3 + x3 3x2

F1L3 (Klein-Mukai) x4

0 + x3 1x2 + x3 2x3 + x3 3x1

L2L2 x3

0x1 + x3 1x0 + x3 2x3 + x3 3x2

L4 x3

0x1 + x3 1x2 + x3 2x3 + x3 3x0

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Five Interesting Quartics in P3

Family Equation F4 (Fermat/Dwork) x4

0 + x4 1 + x4 2 + x4 3

F2L2 x4

0 + x4 1 + x3 2x3 + x3 3x2

F1L3 (Klein-Mukai) x4

0 + x3 1x2 + x3 2x3 + x3 3x1

L2L2 x3

0x1 + x3 1x0 + x3 2x3 + x3 3x2

L4 x3

0x1 + x3 1x2 + x3 2x3 + x3 3x0

Warnings

◮ These quartics are not isomorphic.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Five Interesting Quartics in P3

Family Equation F4 (Fermat/Dwork) x4

0 + x4 1 + x4 2 + x4 3

F2L2 x4

0 + x4 1 + x3 2x3 + x3 3x2

F1L3 (Klein-Mukai) x4

0 + x3 1x2 + x3 2x3 + x3 3x1

L2L2 x3

0x1 + x3 1x0 + x3 2x3 + x3 3x2

L4 x3

0x1 + x3 1x2 + x3 2x3 + x3 3x0

Warnings

◮ These quartics are not isomorphic. ◮ These quartics are not Fourier-Mukai partners. ◮ These quartics are not derived equivalent.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Counting Points

Prime F4 F2L2 F1L3 L2L2 L4 5 20 30 80 40 7 64 50 64 64 78 11 144 122 144 144 254 13 128 180 206 336 232 17 600 328 294 600 328 19 400 362 400 400 438 23 576 530 576 576 622 29 768 884 1116 1232 1000 31 1024 962 1024 1024 1334 37 1152 1300 1374 1744 1448

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Counting Points

Prime F4 F2L2 F1L3 L2L2 L4 5 20 30 80 40 7 64 50 64 64 78 11 144 122 144 144 254 13 128 180 206 336 232 17 600 328 294 600 328 19 400 362 400 400 438 23 576 530 576 576 622 29 768 884 1116 1232 1000 31 1024 962 1024 1024 1334 37 1152 1300 1374 1744 1448 Equality holds (mod p) for all p in this table.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Counting Points on Pencils

◮ We can add the deforming monomial −4ψxyzw to each of our

quartics to obtain pencils of quartics X⋄,ψ.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Counting Points on Pencils

◮ We can add the deforming monomial −4ψxyzw to each of our

quartics to obtain pencils of quartics X⋄,ψ.

◮ We can count the number of points on X⋄,ψ over Fp for

0 ≤ ψ < p.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Counting Points on Pencils

◮ We can add the deforming monomial −4ψxyzw to each of our

quartics to obtain pencils of quartics X⋄,ψ.

◮ We can count the number of points on X⋄,ψ over Fp for

0 ≤ ψ < p.

◮ For each ψ, the point counts on X⋄,ψ agree (mod p).

Mirror symmetry and K3 surface zeta functions Experimental Evidence

What about the zeta functions?

The zeta function for any smooth quartic will have the form Z(X/Fp, T) = 1 (1 − T)(1 − pT)(1 − p2T)P(T).

Mirror symmetry and K3 surface zeta functions Experimental Evidence

What about the zeta functions?

The zeta function for any smooth quartic will have the form Z(X/Fp, T) = 1 (1 − T)(1 − pT)(1 − p2T)P(T). Let’s use Edgar Costa’s code to look at P(T) for X⋄,ψ when p = 41.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

F4

ψ F4 (1 − 41T)19(1 − 18T + 412T 2) 1, 9, 32, 40 not smooth 2, 18, 23, 39 (1 − 41T)3(1 + 41T)16(1 − 50T + 412T 2) 3, 14, 27, 38 (1 − 41T)19(1 + 78T + 412T 2) 4, 5, 36, 37 (1 − 41T)13(1 + 41T)6(1 + 66T + 412T 2) 6, 13, 28, 35 (1 − 41T)3(1 + 41T)16(1 − 50T + 412T 2) 7, 19, 22, 34 (1 − 41T)3(1 + 41T)16(1 + 46T + 412T 2) 8, 10, 31, 33 (1 − 41T)13(1 + 41T)6(1 − 62T + 412T 2) 11, 17, 24, 30 (1 − 41T)5(1 + 41T)16 12, 15, 26, 29 (1 − 41T)3(1 + 41T)16(1 − 50T + 412T 2) 16, 20, 21, 25 (1 − 41T)3(1 + 41T)16(1 − 50T + 412T 2)

Mirror symmetry and K3 surface zeta functions Experimental Evidence

F2L2

ψ F2L2 (1 − 41T)11(1 + 41T)8(1 − 18T + 412T 2) 1, 9, 32, 40 not smooth 2, 18, 23, 39 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 3, 14, 27, 38 (1 − 41T)19(1 + 78T + 412T 2) 4, 5, 36, 37 (1 − 41T)9(1 + 41T)2(1 + 412T 2)4(1 + 66T + 412T 2) 6, 13, 28, 35 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 7, 19, 22, 34 (1 − 41T)11(1 + 41T)8(1 + 46T + 412T 2) 8, 10, 31, 33 (1 − 41T)9(1 + 41T)2(1 + 412T 2)4(1 − 62T + 412T 2) 11, 17, 24, 30 (1 − 41T)13(1 + 41T)8 12, 15, 26, 29 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 16, 20, 21, 25 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2)

Mirror symmetry and K3 surface zeta functions Experimental Evidence

L2L2

ψ L2L2 (1 − 41T)19(1 − 18T + 412T 2) 1, 9, 32, 40 not smooth 2, 18, 23, 39 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 3, 14, 27, 38 (1 − 41T)15(1 + 41T)4(1 + 78T + 412T 2) 4, 5, 36, 37 (1 − 41T)13(1 + 41T)6(1 + 66T + 412T 2) 6, 13, 28, 35 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 7, 19, 22, 34 (1 − 41T)15(1 + 41T)4(1 + 46T + 412T 2) 8, 10, 31, 33 (1 − 41T)13(1 + 41T)6(1 − 62T + 412T 2) 11, 17, 24, 30 (1 − 41T)17(1 + 41T)4 12, 15, 26, 29 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2) 16, 20, 21, 25 (1 − 41T)11(1 + 41T)8(1 − 50T + 412T 2)

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Experimental evidence foreshadows . . .

◮ We see interesting shared quadratic factors in the zeta

functions (also for other values of ⋄).

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Experimental evidence foreshadows . . .

◮ We see interesting shared quadratic factors in the zeta

functions (also for other values of ⋄).

◮ In fact, these are part of shared cubic factors.

Mirror symmetry and K3 surface zeta functions Experimental Evidence

Experimental evidence foreshadows . . .

◮ We see interesting shared quadratic factors in the zeta

functions (also for other values of ⋄).

◮ In fact, these are part of shared cubic factors. ◮ We can also make predictions about the structure of the other

factors.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Why?

The arithmetic patterns we observe are a consequence of mirror symmetry.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Are Strings the Answer?

String Theory proposes that “fundamental” particles are strings.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Extra Dimensions

For string theory to work as a consistent theory of quantum mechanics, it must allow the strings to vibrate in extra, compact dimensions.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Building a Model

Locally, space-time should look like M3,1 × V .

◮ M3,1 is four-dimensional space-time ◮ V is a d-dimensional complex manifold ◮ Physicists require d = 3 (6 real dimensions) ◮ V is a Calabi-Yau manifold

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Calabi-Yau Manifolds

We can define an n-dimensional Calabi-Yau manifold as a simply connected, smooth . . .

◮ Variety with trivial canonical bundle ◮ Ricci-flat K¨

ahler-Einstein manifold

◮ Complex manifold with a unique (up to scaling) holomorphic

n-form

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Calabi-Yau Manifolds

We can define an n-dimensional Calabi-Yau manifold as a simply connected, smooth . . .

◮ Variety with trivial canonical bundle ◮ Ricci-flat K¨

ahler-Einstein manifold

◮ Complex manifold with a unique (up to scaling) holomorphic

n-form Calabi-Yau 2-folds are also known as K3 surfaces.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

A-Model or B-Model?

The worldsheet of a string is a Riemann surface. We consider maps from our string worldsheet to our space-time.

Choosing Variables

◮ z = a + ib, w = c + id ◮ z = a + ib, w = c − id

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Mirror Symmetry

Physicists say . . .

◮ Calabi-Yau manifolds appear in pairs (V , V ◦). ◮ The universes described by M3,1 × V and M3,1 × V ◦ have the

same observable physics.

Mathematicians say . . .

◮ Calabi-Yau manifolds appear in paired families (Vα, V ◦ α). ◮ Mirror symmetry interchanges complex and K¨

ahler structures.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Greene-Plesser Mirror Symmetry

◮ We want to know the mirror of smooth quintics in P4

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Greene-Plesser Mirror Symmetry

◮ We want to know the mirror of smooth quintics in P4 ◮ Consider the Fermat quintic pencil Xψ given by

x5

0 + x5 1 + x5 2 + x5 3 + x5 4 − 5ψx0x1x2x3x4 = 0

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Greene-Plesser Mirror Symmetry

◮ We want to know the mirror of smooth quintics in P4 ◮ Consider the Fermat quintic pencil Xψ given by

x5

0 + x5 1 + x5 2 + x5 3 + x5 4 − 5ψx0x1x2x3x4 = 0 ◮ The pencil admits a group action by (Z/5Z)3

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Greene-Plesser Mirror Symmetry

◮ We want to know the mirror of smooth quintics in P4 ◮ Consider the Fermat quintic pencil Xψ given by

x5

0 + x5 1 + x5 2 + x5 3 + x5 4 − 5ψx0x1x2x3x4 = 0 ◮ The pencil admits a group action by (Z/5Z)3 ◮ Taking the quotient by the group action and resolving

singularities yields the mirror family Yψ

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Counting Deformation Parameters

◮ Smooth quintics in P4 have many complex deformation

parameters

◮ The mirror family Yψ is a one-parameter family

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

The Hodge Diamond

Calabi-Yau Threefolds

1 h1,1(V ) 1 h2,1(V ) h2,1(V ) 1 h1,1(V ) 1

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Hodge diamond for the quintic and its mirror

Smooth quintics 1 1 1 101 101 1 1 1 Yψ 1 101 1 1 1 1 101 1

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Arithmetic Mirror Symmetry?

Figure: Philip Candelas Figure: Xenia de la Ossa Figure: Fernando Rodriguez Villegas

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Dwork and the Weil Conjectures

◮ Z(X/Fq, T) is rational ◮ We can factor Z(X/Fq, T) using polynomials with integer

coefficients: Z(X/Fp, T) := n

j=1 P2j−1(T)

n

j=0 P2j(T) , ◮ dimX = n ◮ P0(t) = 1 − T and P2n(T) = 1 − pnT ◮ For 1 ≤ j ≤ 2n − 1, degPj(T) = bj, where bj = dim Hj dR(X).

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Arithmetic Mirror Symmetry for Threefolds

If X and Y are mirror Calabi-Yau threefolds, we can expect a relationship between Z(X/Fq, T) and Z(Y /Fq, T) due to the interchange of Hodge numbers.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Arithmetic Mirror Symmetry for Pencils in Weighted Projective Space

Let Xψ be a generalized Fermat pencil of Calabi-Yau varieties in a weighted projective space and let Yψ be the mirror family.

◮ Physicists claim: Z(Xψ/Fq, T) and Z(Yψ/Fq, T) share a

common factor.

◮ Daqinq Wan:

#Xψ(Fpd) ≡ #Yψ(Fpd) (mod pd).

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Greene-Plesser Mirror for Quartics in P3

◮ Start with all smooth quartics in P3 ◮ Consider the Fermat pencil Xψ : x4 + y4 + z4 + w4 − 4ψxyzw ◮ The pencil admits an action of G = (Z/(4))2 (multiply

coordinates by 4th roots of unity)

◮ Resolve singularities in the quotient Xψ/G to obtain Yψ ◮ Yψ is the mirror family to smooth quartics in P3 ◮ Smooth quartics in P3 have many complex deformation

parameters; Yψ has 1

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

The Fermat quartic pencil

Let Xψ be the Fermat quartic pencil. Xenia de la Ossa and Shabnam Kadir (building on results of Dwork) showed: Z(Xψ/Fp, T) = 1 (1 − T)(1 − pT)(1 − p2T)P(T) P(T) = R(T)Q3(T)S12(T)

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

The Fermat quartic pencil

Let Xψ be the Fermat quartic pencil. Xenia de la Ossa and Shabnam Kadir (building on results of Dwork) showed: Z(Xψ/Fp, T) = 1 (1 − T)(1 − pT)(1 − p2T)P(T) P(T) = R(T)Q3(T)S12(T) where (with choices of ± depending on p and ψ)

◮ R(T) = (1 ± pT)(1 − aψT + p2T) ◮ Q(T) = (1 ± pT)(1 ± pT) ◮ S(T) =

• [(1 − pT)(1 + pT)]1/2

when p ≡ 3 mod 4 (1 ± pT)

• therwise

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Mirror Quartics

Let Yψ be the mirror family to quartics in P3 (constructed using Greene-Plesser and the Fermat pencil). Then de la Ossa and Kadir showed: Z(Yψ/Fp, T) = 1 (1 − T)(1 − pT)19(1 − p2T)R(T).

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

Mirror Quartics

Let Yψ be the mirror family to quartics in P3 (constructed using Greene-Plesser and the Fermat pencil). Then de la Ossa and Kadir showed: Z(Yψ/Fp, T) = 1 (1 − T)(1 − pT)19(1 − p2T)R(T). The factor R(T) corresponds to periods of the holomorphic form and its derivatives, and is invariant under mirror symmetry.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry

A Modular Curve

◮ The mirror quartics are naturally associated (via a

Shioda-Inose structure) to products of elliptic curves with a 2-isogeny (see work of Dolgachev, notes of Elkies and Sch¨ utt).

◮ The pairs of elliptic curves are parametrized by the modular

curve X0(2)/w.

Mirror symmetry and K3 surface zeta functions Alternate Mirrors

Alternate Mirrors

The Fermat pencil is not the only highly symmetric pencil one can use to construct the mirror to smooth quartics in P3. Family Equation Symmetries F4 x4

0 + x4 1 + x4 2 + x4 3 − 4ψx0x1x2x3

(Z/4Z)2 F2L2 x4

0 + x4 1 + x3 2x3 + x3 3x2 − 4ψx0x1x2x3

Z/8Z F1L3 x4

0 + x3 1x2 + x3 2x3 + x3 3x1 − 4ψx0x1x2x3

Z/7Z L2L2 x3

0x1 + x3 1x0 + x3 2x3 + x3 3x2 − 4ψx0x1x2x3

Z/4Z × Z/2Z L4 x3

0x1 + x3 1x2 + x3 2x3 + x3 3x0 − 4ψx0x1x2x3

Z/5Z

Mirror symmetry and K3 surface zeta functions Alternate Mirrors

Guiding Intuition

◮ These distinct pencils of quartics are each mirror to a

• ne-parameter family representing mirror quartics

◮ The pencils should share properties corresponding to this

common mirror.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Why These Pencils?

◮ One can use the Berglund-H¨

ubsch-Krawitz (BHK) mirror construction to characterize the pencils we use to construct alternate mirrors.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

A matrix polynomial

Consider a polynomial FA that is the sum of n + 1 monomials in n + 1 variables FA :=

n

• i=0

n

• j=0

xaij

j .

We view FA as determined by an integer matrix A = (aij) (with rows corresponding to monomials).

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Invertible polynomials

We say FA is invertible if:

◮ The matrix A is invertible ◮ There exist positive integers called weights qj so that

d := n

j=0 qjaij is the same constant for all i ◮ The polynomial FA has exactly one critical point, namely at

the origin.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Calabi-Yau Condition

Calabi-Yau Condition

We say an invertible polynomial FA satisfies the Calabi-Yau condition if d = n

j=0 qj.

Consequences

If the Calabi-Yau condition is satisfied:

◮ The weights determine a weighted projective space

WPn(q0, . . . , qn)

◮ FA determines a Calabi-Yau hypersurface XA in this weighted

projective space.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Classifying Invertible Polynomials

Kreuzer and Skarke proved that any invertible polynomial FA can be written as a sum of invertible potentials, each of which must be

• f one of the three atomic types:

WFermat := xa, Wloop := xa1

1 x2 + xa2 2 x3 + . . . + xam−1 m−1 xm + xam m x1, and

Wchain := xa1

1 x2 + xa2 2 x3 + . . . xam−1 m−1 xm + xam m .

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Specifying an Orbifold

◮ The torus (C∗)n+1 acts coordinate-wise on the weighted

projective space W Pn(q0, . . . , qn).

◮ Let H be the subgroup of the torus that acts nontrivially and

symplectically on XA (fixes the holomorphic n − 1-form).

◮ H is a finite abelian group, and the coordinates of each

element of H are roots of unity.

◮ For any subgroup G of H, we may define the

Berglund-H¨ ubsch-Krawitz mirror of the orbifold XA/G.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

The BHK Mirror

We start with an orbifold XA/G, corresponding to a matrix A.

◮ Take the transpose matrix AT. ◮ Consider the polynomial FAT . ◮ We obtain a dual orbifold XAT /G T.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

The BHK Mirror

We start with an orbifold XA/G, corresponding to a matrix A.

◮ Take the transpose matrix AT. ◮ Consider the polynomial FAT . ◮ We obtain a dual orbifold XAT /G T.

BHK duality is a true duality: the mirror of the mirror yields the

• riginal orbifold.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Pencils

BHK duality for a polynomial FA extends naturally to the pencil of hypersurfaces described by FA − (n + 1)ψx0 . . . xn.

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Pencils of Interest

Invertible quartic pencils in P3 with BHK mirrors that are orbifolds

• f quartics in P3:

Family Equation Symmetries F4 x4

0 + x4 1 + x4 2 + x4 3 − 4ψx0x1x2x3

(Z/4Z)2 F2L2 x4

0 + x4 1 + x3 2x3 + x3 3x2 − 4ψx0x1x2x3

Z/8Z F1L3 x4

0 + x3 1x2 + x3 2x3 + x3 3x1 − 4ψx0x1x2x3

Z/7Z L2L2 x3

0x1 + x3 1x0 + x3 2x3 + x3 3x2 − 4ψx0x1x2x3

Z/4Z × Z/2Z L4 x3

0x1 + x3 1x2 + x3 2x3 + x3 3x0 − 4ψx0x1x2x3

Z/5Z

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Other Pencils

Other invertible quartic pencils in P3: Quartic Family Symmetries x3

0x1 + x4 1 + x4 2 + x4 3 − 4ψx0x1x2x3

Z/4Z x3

0x1 + x3 1x2 + x4 2 + x4 3 − 4ψx0x1x2x3

none x3

0x1 + x3 1x2 + x3 2x3 + x4 3 − 4ψx0x1x2x3

none x3

0x1 + x4 1 + x3 2x3 + x4 3 − 4ψx0x1x2x3

Z/6Z x3

0x1 + x4 1 + x3 2x3 + x3 3x2 − 4ψx0x1x2x3

Z/8Z

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Other Mirrors

BHK Mirror Pencil

{y3

0 +y0y4 1 +y4 2 +y4 3 −4ψy0y1y2y3=0}⊆WP(4,2,3,3)

Z4:[0,0,3,1]

{y3

0 + y0y3 1 + y1y4 2 + y4 3 − 4ψy0y1y2y3 = 0}

⊆ WP(12, 8, 7, 9) {y3

0 + y1y3 2 + y2y3 3 + y3y4 4 = 0} ⊆ WP(9, 6, 7, 5) {y3

0 +y1y4 2 +y3 2 +y3y4 4 −4ψy0y1y2y3}⊆WP(2,1,2,1)

Z/2Z:[0,1,0,1] {y3

0 +y0y4 1 +y3 2 y3+y2y3 3 −4ψy0y1y2y3=0}⊆WP(4,2,3,3)

Z/8Z:[0,2,5,1]

Mirror symmetry and K3 surface zeta functions Berglund-H¨ ubsch-Krawitz Duality

Experimental evidence

We detect relationships between point counts on the Fermat pencil and point counts on the pencils of interest, but not the other quartic pencils.

Mirror symmetry and K3 surface zeta functions Results

A common factor

Theorem (DKSSVW)

Let ⋄ ∈ {F4, F2L2, F1L3, L2L2, L4} signify one of the five K3

• families. Let X⋄,ψ be a smooth fiber of the family ⋄.

Let P⋄,ψ(T) be the degree 21 factor of Z(X⋄,ψ/Fq, T). Then the following statements hold.

Mirror symmetry and K3 surface zeta functions Results

A common factor

Theorem (DKSSVW)

Let ⋄ ∈ {F4, F2L2, F1L3, L2L2, L4} signify one of the five K3

• families. Let X⋄,ψ be a smooth fiber of the family ⋄.

Let P⋄,ψ(T) be the degree 21 factor of Z(X⋄,ψ/Fq, T). Then the following statements hold.

◮ We have a factorization

P⋄,ψ(T) = Q⋄,ψ(T)Rψ(T) in Z[T] with deg Q⋄,ψ = 18 and deg Rψ = 3.

Mirror symmetry and K3 surface zeta functions Results

A common factor

Theorem (DKSSVW)

Let ⋄ ∈ {F4, F2L2, F1L3, L2L2, L4} signify one of the five K3

• families. Let X⋄,ψ be a smooth fiber of the family ⋄.

Let P⋄,ψ(T) be the degree 21 factor of Z(X⋄,ψ/Fq, T). Then the following statements hold.

◮ We have a factorization

P⋄,ψ(T) = Q⋄,ψ(T)Rψ(T) in Z[T] with deg Q⋄,ψ = 18 and deg Rψ = 3.

◮ The reciprocal zeros of Q⋄,ψ(T) are of the form q times a

root of unity.

Mirror symmetry and K3 surface zeta functions Results

A common factor

Theorem (DKSSVW)

Let ⋄ ∈ {F4, F2L2, F1L3, L2L2, L4} signify one of the five K3

• families. Let X⋄,ψ be a smooth fiber of the family ⋄.

Let P⋄,ψ(T) be the degree 21 factor of Z(X⋄,ψ/Fq, T). Then the following statements hold.

◮ We have a factorization

P⋄,ψ(T) = Q⋄,ψ(T)Rψ(T) in Z[T] with deg Q⋄,ψ = 18 and deg Rψ = 3.

◮ The reciprocal zeros of Q⋄,ψ(T) are of the form q times a

root of unity.

◮ The polynomial Rψ(T) is independent of ⋄.

Mirror symmetry and K3 surface zeta functions Results

Changing fields

For Fq containing sufficiently many roots of unity, we have that Z(X⋄,ψ/Fq, T) = 1 (1 − T)(1 − qT)19(1 − q2T)Rψ(T). We may say our zeta functions are potentially equal.

Mirror symmetry and K3 surface zeta functions Results

Comparing to the mirror

Let Yψ be the family of mirror quartics we constructed earlier. Then Z(X⋄,ψ) and Z(Yψ) are potentially equal for any ⋄.

Mirror symmetry and K3 surface zeta functions Results

A note on Picard ranks

◮ By Tate’s conjecture, smooth members of our families ⋄ have

Picard rank at least 20 over fields of odd characteristic.

◮ We can look for further factorization of Rψ or use the

Shioda-Inose structure on Yψ to identify supersingular family members.

Mirror symmetry and K3 surface zeta functions Results

A note on Picard ranks

◮ By Tate’s conjecture, smooth members of our families ⋄ have

Picard rank at least 20 over fields of odd characteristic.

◮ We can look for further factorization of Rψ or use the

Shioda-Inose structure on Yψ to identify supersingular family members.

◮ Over C, the Picard rank of a general member of the families ⋄

is 19.

◮ Can one give an elementary description of the Picard lattice

for all of our families?

Mirror symmetry and K3 surface zeta functions Results

Coming up

◮ The factor Rψ is closely associated to periods of the

holomorphic form.

◮ We can describe the structure of Q⋄,ψ(T) using periods of

• ther forms.