# Effective Computations in Arithmetic Mirror Symmetry (13frg165)

Arriving in Banff, Alberta Sunday, March 17 and departing Sunday March 24, 2013

## Organizers

Charles Doran (University of Alberta, Canada)

Adriana Salerno (Bates College)

Ursula Whitcher (American Mathematical Society)

## Objectives

Suppose we have an algebraic variety $X$ over a finite field $mathbb{F}_p$ (e.g, the set of common zeros of a finite set of polynomials in $n$ variables with coefficients in $mathbb{F}_p$). The emph{congruent zeta function} (also known as the Hasse-Weil zeta function) is a generating function for the number of $mathbb{F}_{p^d}$-valued points on $X$, that is, the number of $n$-tuples in $mathbb{F}_{p^d}^n$ on the variety. Formally, the congruent zeta function is defined as:

[mathrm{Zeta}(X/mathbb{F}_p,t):=expleft(sum_{dgeq1}frac{ Card( X(mathbb{F}_{p^d}))t^d}{d}right).]

This zeta function is actually rational cite{dwork}, and can be factored in terms of polynomials with integer coefficients:

[ mathrm{Zeta}(X/mathbb{F}_p,t):=frac{prod_{j=1}^nP_{2j-1}(t)}{prod_{j=0}^nP_{2j}(t)},]

noindent Here $n$ is the dimension of the variety $X$. Furthermore, $P_0(t)=1-t$, $P_{2n}(t)=1-p^nt$, and for each $1leq jleq 2n-1$, the polynomials $P_j(t)$ have degree $b_j$, the Betti numbers of the variety, that is $b_j=dim H_{dR}^j(X)$.

Mirror symmetry for Calabi-Yau threefold mirror pairs predicts that the Hodge numbers $h^{1,1}$ and $h^{2,1}$ are interchanged. The possible implications of this exchange for the arithmetic structure of the varieties were first explored in the physics literature in 2000 by Candelas, de la Ossa, and Rodriguez Villegas (see cite{CORV}). In particular, because the Hodge numbers control the Betti numbers, it follows that mirror symmetry will be reflected in the congruent zeta functions of mirror pairs. Shabnam Kadir studied this phenomenon for specific mirror pairs of Calabi-Yau threefolds in cite{kadir}, using a two-parameter deformation of a diagonal hypersurface in a weighted projective space. Candelas and de la Ossa described techniques for computing the congruent zeta function of the pencil of Fermat threefolds in cite{candelas}, and Kloosterman developed techniques for computing the zeta function of arbitrary deformations of a Fermat hypersurface in cite{kloost}. Recently, Goto, Kloosterman, and Yui have studied the congruent zeta function for a class of diagonal hypersurfaces in weighted projective spaces (see cite{GKY}). The higher regulator maps of Beilinson-Bloch have image a lattice of maximal rank whose volume is related to the value of the Hasse-Weil zeta function cite{Bl}. These higher regulator maps are investigated for anticanonical hypersurfaces in toric Fano varieties in connection with local mirror symmetry in the recent work of Doran and Kerr cite{DK}.

New research by computational number theorists offers a framework for understanding the congruent zeta function for varieties other than deformations of diagonal hypersurfaces. Much of the number theory literature focuses on properties of curves, but forthcoming work by Sperber and Voight describes an algorithm for computing the zeta function for hypersurfaces described by sparse polynomials in toric varieties (see cite{SV}). These computational advances offer an unprecedented opportunity to explore the arithmetic relationships between varieties that have been predicted by mirror symmetry.

We will investigate arithmetic mirror symmetry for a broad class of hypersurfaces, including non-diagonal hypersurfaces in projective space and specific families of hypersurfaces in toric varieties. The open-source computer algebra software Sage offers a convenient framework for investigations blending number theory and toric geometry. The time is ripe for a modern, coherent approach to mathematical experimentation.

The first mirror symmetry construction, due to Greene and Plesser, used a one-parameter family of Calabi-Yau threefolds with a $(mathbb{Z}/5mathbb{Z})^3$ action to construct the mirror family to all quintic hypersurfaces in $mathbb{P}^4$. This construction gives an expansion of the mirror map near the Fermat quintic. In cite{GPR}, Greene, Plesser, and Roan used one-parameter families of quintic hypersurfaces with other abelian group actions to describe the mirror correspondence near other quintic hypersurfaces. This construction offers a way to study the relationship between the family of all quintic hypersurfaces and its mirror family at specific points in moduli space. More recently, cite{DGJ} computed the Picard-Fuchs equations of the one-parameter families of Calabi-Yau hypersurfaces used in the cite{GPR} mirror construction; they showed that while the Picard-Fuchs equation of the holomorphic period is always the same (and identical to the Picard-Fuchs equation for the mirror family, see also cite{BvGK}), the Picard-Fuchs equations of non-holomorphic differential forms are sensitive to the particular discrete group action. We conjecture that the congruent zeta functions will encode this information about the common structure of these quintic pencils. In this way, mirror symmetry provides a powerful tool for making predictions about the arithmetic properties of Calabi-Yau varieties.

Because all K3 surfaces have the same Hodge diamond, mirror symmetry constructions for K3 surfaces are more subtle: the constructions depend on choosing a emph{lattice polarization}, which specifies a sublattice of the Picard group of algebraic curves on the K3 surface. We will investigate the Picard-Fuchs equations and congruent zeta functions for families of (ADE singular/lattice-polarized) quartic hypersurfaces defined in analogy with the quintic families studied in cite{GPR}, and study the relationship between the congruent zeta function and the Picard group structure.

Recent work by McOrist, Morrison, and Sethi builds on work of Clingher-Doran and Clingher-Doran-Lewis-Whitcher in order to investigate the properties of the string theory construction known as emph{F-theory} described by certain families of K3 surfaces with high Picard rank (see cite{MMS, CD, CD2, CDLW}). These families may be explicitly realized in two ways: as resolutions of singular hypersurfaces in projective space, and as hypersurfaces in toric varieties. The properties of the congruent zeta function for singular varieties are not well understood. Developing computational methods for studying the congruent zeta function for hypersurfaces in both projective space and toric varieties will offer a means to investigate the properties of the zeta function for singular hypersurfaces as compared to related, nonsingular varieties.

BIRS has been a center for research activity in mathematical mirror symmetry for almost a decade. This Focussed Research Group builds on connections made at the ``Number Theory and Physics at the Crossroads'' event in May 2011 while incorporating new expertise from computational number theorists and mathematical physicists. The participants hail from geographically and culturally diverse institutions; the FRG offers an opportunity for sustained research interactions that would otherwise be difficult to accomplish.

begin{thebibliography}{9}

bibitem{BvGK} G. Bini, B. van Geemen, and T. Kelly, ``Mirror quintics, discrete symmetries, and Shioda maps,'' arXiv:0809.1791, 2008.

bibitem{Bl} S. Bloch, emph{Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves}, CRM Monograph Series, Vol. 11, 97 pages, 2000.

bibitem{candelas} Candelas, P. and de la Ossa, X. , emph{The Zeta-Function of a p-adic Manifold, {D}work Theory for Physicists}, arxiv:0705.2056v1, 2008.

bibitem{CORV}Candelas, P., de la Ossa, X., and Rodriguez Villegas, F. emph{Calabi-Yau manifolds over finite fields, {I}}, arXiv:hep-th/0012233v1, 2000.

bibitem{CD}

A. Clingher and C.F. Doran,``Modular Invariants for Lattice Polarized K3 Surfaces,''

{em Michigan Mathematical Journal}, Vol. 55, Issue 2, 2007.

bibitem{CD2}

A. Clingher and C.F. Doran, ``Lattice Polarized K3 Surfaces and Siegel Modular Forms,'' emph{Advances in Mathematics}, Vol. 231, Issue 1, 2012.

bibitem{CDLW}

A. Clingher, C.F. Doran, J.M. Lewis, and U. Whitcher, ``Normal forms, K3 surface moduli, and modular parametrizations,'' emph{Groups and Symmetries: Proceedings of the CRM Conference in Honor of John McKay}, edited by John Harnad and Pavel Winternitz, American Mathematical Society, 2009.

bibitem{DGJ}

C.F. Doran, B. Greene, and S. Judes, emph{Families of quintic {C}alabi-{Y}au 3-folds with discrete

symmetries}, Communications in Mathematical Physics, vol. 280, no. 8, 2008.

bibitem{DK}

C.F. Doran and M. Kerr, ``Algebraic K-Theory of Toric Hypersurfaces,'' emph{Communications in Number Theory and Physics}, Vol. 5, No. 2, p. 397--600, 2011.

bibitem{dwork} Dwork, B., emph{On the rationality of the zeta function of an algebraic variety}, American Journal of Mathematics, Vol. 82, No. 3, p. 631--648, 1960.

bibitem{GKY} Goto, Y., Kloosterman, R., and Yui, N. emph{Zeta-functions of certain K3-fibered Calabi-Yau threefolds,} Internat. J. Math. 22, no. 1, 67, 129, 2011.

bibitem{GPR} Greene, B., Plesser, M., and Roan, S. , emph{New constructions of mirror manifolds: Probing moduli space far from Fermat points}, Mirror Symmetry I, Somerville, MA, International Press, 1998.

bibitem{kadir} Kadir, S., emph{The Arithmetic of Calabi--Yau Manifolds and Mirror Symmetry}, Oxford DPhil Thesis (arXiv: hep-th/0409202), 2004.

bibitem{kloost} Kloosterman, R. emph{The zeta function of monomial deformations of {F}ermat hypersurfaces}, { Algebra & Number Theory}, 1:421--450, 2007.

bibitem{MMS} McOrist, J., Morrison, D., and Sethi, S. emph{Geometries, non-geometries, and fluxes}, Adv. Theor. Math. Phys. 14 (2010) 1515-1583.

bibitem{SV} Sperber, S., and Voight, J. emph{Computing zeta functions of nondegenerate hypersurfaces with few monomials} (arXiv:1112.4881v2 [math.AG]). To appear in the London Mathematical Society J. Comp. Math.

end{thebibliography}

[mathrm{Zeta}(X/mathbb{F}_p,t):=expleft(sum_{dgeq1}frac{ Card( X(mathbb{F}_{p^d}))t^d}{d}right).]

This zeta function is actually rational cite{dwork}, and can be factored in terms of polynomials with integer coefficients:

[ mathrm{Zeta}(X/mathbb{F}_p,t):=frac{prod_{j=1}^nP_{2j-1}(t)}{prod_{j=0}^nP_{2j}(t)},]

noindent Here $n$ is the dimension of the variety $X$. Furthermore, $P_0(t)=1-t$, $P_{2n}(t)=1-p^nt$, and for each $1leq jleq 2n-1$, the polynomials $P_j(t)$ have degree $b_j$, the Betti numbers of the variety, that is $b_j=dim H_{dR}^j(X)$.

Mirror symmetry for Calabi-Yau threefold mirror pairs predicts that the Hodge numbers $h^{1,1}$ and $h^{2,1}$ are interchanged. The possible implications of this exchange for the arithmetic structure of the varieties were first explored in the physics literature in 2000 by Candelas, de la Ossa, and Rodriguez Villegas (see cite{CORV}). In particular, because the Hodge numbers control the Betti numbers, it follows that mirror symmetry will be reflected in the congruent zeta functions of mirror pairs. Shabnam Kadir studied this phenomenon for specific mirror pairs of Calabi-Yau threefolds in cite{kadir}, using a two-parameter deformation of a diagonal hypersurface in a weighted projective space. Candelas and de la Ossa described techniques for computing the congruent zeta function of the pencil of Fermat threefolds in cite{candelas}, and Kloosterman developed techniques for computing the zeta function of arbitrary deformations of a Fermat hypersurface in cite{kloost}. Recently, Goto, Kloosterman, and Yui have studied the congruent zeta function for a class of diagonal hypersurfaces in weighted projective spaces (see cite{GKY}). The higher regulator maps of Beilinson-Bloch have image a lattice of maximal rank whose volume is related to the value of the Hasse-Weil zeta function cite{Bl}. These higher regulator maps are investigated for anticanonical hypersurfaces in toric Fano varieties in connection with local mirror symmetry in the recent work of Doran and Kerr cite{DK}.

New research by computational number theorists offers a framework for understanding the congruent zeta function for varieties other than deformations of diagonal hypersurfaces. Much of the number theory literature focuses on properties of curves, but forthcoming work by Sperber and Voight describes an algorithm for computing the zeta function for hypersurfaces described by sparse polynomials in toric varieties (see cite{SV}). These computational advances offer an unprecedented opportunity to explore the arithmetic relationships between varieties that have been predicted by mirror symmetry.

We will investigate arithmetic mirror symmetry for a broad class of hypersurfaces, including non-diagonal hypersurfaces in projective space and specific families of hypersurfaces in toric varieties. The open-source computer algebra software Sage offers a convenient framework for investigations blending number theory and toric geometry. The time is ripe for a modern, coherent approach to mathematical experimentation.

The first mirror symmetry construction, due to Greene and Plesser, used a one-parameter family of Calabi-Yau threefolds with a $(mathbb{Z}/5mathbb{Z})^3$ action to construct the mirror family to all quintic hypersurfaces in $mathbb{P}^4$. This construction gives an expansion of the mirror map near the Fermat quintic. In cite{GPR}, Greene, Plesser, and Roan used one-parameter families of quintic hypersurfaces with other abelian group actions to describe the mirror correspondence near other quintic hypersurfaces. This construction offers a way to study the relationship between the family of all quintic hypersurfaces and its mirror family at specific points in moduli space. More recently, cite{DGJ} computed the Picard-Fuchs equations of the one-parameter families of Calabi-Yau hypersurfaces used in the cite{GPR} mirror construction; they showed that while the Picard-Fuchs equation of the holomorphic period is always the same (and identical to the Picard-Fuchs equation for the mirror family, see also cite{BvGK}), the Picard-Fuchs equations of non-holomorphic differential forms are sensitive to the particular discrete group action. We conjecture that the congruent zeta functions will encode this information about the common structure of these quintic pencils. In this way, mirror symmetry provides a powerful tool for making predictions about the arithmetic properties of Calabi-Yau varieties.

Because all K3 surfaces have the same Hodge diamond, mirror symmetry constructions for K3 surfaces are more subtle: the constructions depend on choosing a emph{lattice polarization}, which specifies a sublattice of the Picard group of algebraic curves on the K3 surface. We will investigate the Picard-Fuchs equations and congruent zeta functions for families of (ADE singular/lattice-polarized) quartic hypersurfaces defined in analogy with the quintic families studied in cite{GPR}, and study the relationship between the congruent zeta function and the Picard group structure.

Recent work by McOrist, Morrison, and Sethi builds on work of Clingher-Doran and Clingher-Doran-Lewis-Whitcher in order to investigate the properties of the string theory construction known as emph{F-theory} described by certain families of K3 surfaces with high Picard rank (see cite{MMS, CD, CD2, CDLW}). These families may be explicitly realized in two ways: as resolutions of singular hypersurfaces in projective space, and as hypersurfaces in toric varieties. The properties of the congruent zeta function for singular varieties are not well understood. Developing computational methods for studying the congruent zeta function for hypersurfaces in both projective space and toric varieties will offer a means to investigate the properties of the zeta function for singular hypersurfaces as compared to related, nonsingular varieties.

BIRS has been a center for research activity in mathematical mirror symmetry for almost a decade. This Focussed Research Group builds on connections made at the ``Number Theory and Physics at the Crossroads'' event in May 2011 while incorporating new expertise from computational number theorists and mathematical physicists. The participants hail from geographically and culturally diverse institutions; the FRG offers an opportunity for sustained research interactions that would otherwise be difficult to accomplish.

begin{thebibliography}{9}

bibitem{BvGK} G. Bini, B. van Geemen, and T. Kelly, ``Mirror quintics, discrete symmetries, and Shioda maps,'' arXiv:0809.1791, 2008.

bibitem{Bl} S. Bloch, emph{Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves}, CRM Monograph Series, Vol. 11, 97 pages, 2000.

bibitem{candelas} Candelas, P. and de la Ossa, X. , emph{The Zeta-Function of a p-adic Manifold, {D}work Theory for Physicists}, arxiv:0705.2056v1, 2008.

bibitem{CORV}Candelas, P., de la Ossa, X., and Rodriguez Villegas, F. emph{Calabi-Yau manifolds over finite fields, {I}}, arXiv:hep-th/0012233v1, 2000.

bibitem{CD}

A. Clingher and C.F. Doran,``Modular Invariants for Lattice Polarized K3 Surfaces,''

{em Michigan Mathematical Journal}, Vol. 55, Issue 2, 2007.

bibitem{CD2}

A. Clingher and C.F. Doran, ``Lattice Polarized K3 Surfaces and Siegel Modular Forms,'' emph{Advances in Mathematics}, Vol. 231, Issue 1, 2012.

bibitem{CDLW}

A. Clingher, C.F. Doran, J.M. Lewis, and U. Whitcher, ``Normal forms, K3 surface moduli, and modular parametrizations,'' emph{Groups and Symmetries: Proceedings of the CRM Conference in Honor of John McKay}, edited by John Harnad and Pavel Winternitz, American Mathematical Society, 2009.

bibitem{DGJ}

C.F. Doran, B. Greene, and S. Judes, emph{Families of quintic {C}alabi-{Y}au 3-folds with discrete

symmetries}, Communications in Mathematical Physics, vol. 280, no. 8, 2008.

bibitem{DK}

C.F. Doran and M. Kerr, ``Algebraic K-Theory of Toric Hypersurfaces,'' emph{Communications in Number Theory and Physics}, Vol. 5, No. 2, p. 397--600, 2011.

bibitem{dwork} Dwork, B., emph{On the rationality of the zeta function of an algebraic variety}, American Journal of Mathematics, Vol. 82, No. 3, p. 631--648, 1960.

bibitem{GKY} Goto, Y., Kloosterman, R., and Yui, N. emph{Zeta-functions of certain K3-fibered Calabi-Yau threefolds,} Internat. J. Math. 22, no. 1, 67, 129, 2011.

bibitem{GPR} Greene, B., Plesser, M., and Roan, S. , emph{New constructions of mirror manifolds: Probing moduli space far from Fermat points}, Mirror Symmetry I, Somerville, MA, International Press, 1998.

bibitem{kadir} Kadir, S., emph{The Arithmetic of Calabi--Yau Manifolds and Mirror Symmetry}, Oxford DPhil Thesis (arXiv: hep-th/0409202), 2004.

bibitem{kloost} Kloosterman, R. emph{The zeta function of monomial deformations of {F}ermat hypersurfaces}, { Algebra & Number Theory}, 1:421--450, 2007.

bibitem{MMS} McOrist, J., Morrison, D., and Sethi, S. emph{Geometries, non-geometries, and fluxes}, Adv. Theor. Math. Phys. 14 (2010) 1515-1583.

bibitem{SV} Sperber, S., and Voight, J. emph{Computing zeta functions of nondegenerate hypersurfaces with few monomials} (arXiv:1112.4881v2 [math.AG]). To appear in the London Mathematical Society J. Comp. Math.

end{thebibliography}