We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of $p$ and from Weil's general bounds $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore, the critical $L$-values of the modular form are predicted to be $\mathbb Q$-proportional to the values of a related basis of solutions to the hypergeometric differential equation.
This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard--Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generat…
We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice A_3. Our proof is based on geometric analysis.
We prove that (not necessarily rigid) Calabi-Yau threefolds defined over the rationals which contain sufficiently many elliptic ruled surfcaes are modular (under mild restrictions on the primes of bad reduction). Our proof uses the results of Dieulefait and Manoharmayum who proved modularity of rigid Calabi-Yau threefolds.
In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.
We prove the modularity of Maschke's octic and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group, as conjectured by Bini and van Geemen. The proofs rely on automorphisms of the varieties and isogenies of K3 surfaces.
The purpose of this paper is to provide answers to some questions raised in a paper by Kaneko and Koike about the modularity of the solutions of a differential equations of hypergeometric type. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in other cases. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.
In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which "numerical evidence of modularity" was found in the second Author's thesis.…
We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.
We prove modularity for a huge class of rigid Calabi-Yau threefolds over $\Q$. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in e…
We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w \}$ such that $2 w < \ell$ (and $\ell > 3$) and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p \}) = \Q $. This result applies in particular to any motivic compatible family of odd two-dimensional Galois representations of $\Gal(\bar{\Q}/\Q)$ if the motive has rational coefficients, good reduction at 3, and the "ordinarity at 3" conditi…
Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.
In a recent preprint of F. Gouvea and N. Yui (see arXiv:0902.1466) a detailed account is given of a patching argument due to Serre that proves that the modularity of all rigid Calabi-Yau threefolds defined over the rationals follows from Serre's modularity conjecture. In this note (a letter to N. Yui) we give an alternative proof of this implication. The main difference with Serre's argument is that instead of using as main input residual modularity in infinitely many characteristics we just require residual modularity in a suitable characteristic. This is combined with effective Cebotarev.
Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over $\mathbb{Q}$, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in $\mathbb{P}^{4}(1,1,2,2,2)$.
Let $X$ be a Calabi--Yau threefold fibred over ${\mathbb P}^1$ by non-constant semi-stable K3 surfaces and reaching the Arakelov--Yau bound. In [STZ], X. Sun, Sh.-L. Tan, and K. Zuo proved that $X$ is modular in a certain sense. In particular, the base curve is a modular curve. In their result they distinguish the rigid and the non-rigid cases. In [SY] and [V] rigid examples were constructed. In this paper we construct explicit examples in non-rigid cases. Moreover, we prove for our threefolds that the ``interesting'' part of their $L$-series is attached to an automorphic form, and hence that …
In their paper Livné and Yui (math.AG/0304497) discuss several examples of non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the modularity of the L-function of these examples. The purpose of this note is to point out that the examples which were listed in their paper, but which do not lead to semi-stable fibrations, are still modular in the sense that their L-function is associated to modular forms. We treat the case associated to the group Gamma_1(7) in detail, but our technique also …
It is believed that any p-adic Galois representation which is potentially semistable arises from a modular form. The main theorem of Wiles establishes this modularity when the representation in question satisfies various technical restrictions, together with the key hypothesis that its reduction modulo p arises itself from a modular form. This article explains how a strong version of Wiles' "lifting theorem" implies the modularity of all hypergeometric abelian varieties - so-called because their periods are expressed in terms of values of classical hypergeometric functions. While this strong v…
We consider the fourteen families $W$ of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with four periods of the holomorphic $(3,0)$-form over a symplectic basis of $H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under $Γ_0(25)$, whose Hecke eigenvalues are determined by the Hasse-…
We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.
In recent work, we conjectured that Calabi-Yau threefolds defined over $\mathbb{Q}$ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results?…
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\leq n\leq 4$. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric Calabi-Yau equations are the fu…
The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Γ_1(6). By Tate's conjecture, this should imply that Y, …
We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livne method to induced four-dimensional Galois representations over QQ. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.
In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.
In a previous article (a joint work with J. Manoharmayum) the modularity of a large class of rigid Calabi-Yau threefolds was established. To make that result more explicit, we recall (and re-prove) a result of Serre giving a bound for the conductor of an "integral" 2-dimensional compatible family of Galois representations and apply this result to give an algorithm that determines the level of a modular rigid Calabi-Yau threefold. We apply the algorithm to three examples.
By the modularity theorem every rigid Calabi-Yau threefold $X$ has associated modular form $f$ such that the equality of $L$-functions $L(X,s)=L(f,s)$ holds. In this case period integrals of $X$ are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of $X$. It is given in terms of certain variants of a Mellin transform of $f$. We provide numerical evidence towards this interpretation based on a case of double octics.
Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and their Fourier coefficients encode distinguished Maass cusp forms. The cuspidality of these Maass waveforms is proven by making use of the Habiro ring representations of the associated quantum modular forms. Thus, we provide an example of using the $q$-hypergeometric structure of associated series to establish modularity properties which are otherwise non-obvio…
We prove that the ancestor Gromov-Witten correlation functions of one-dimensional compact Calabi-Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov-Ruan's B-model technique. We first show that genus zero modularity is obtained from the phenomenon that the system of WDVV equations is essentially equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for a certain modular group associated to the orbifold curve. Higher genus modularity then follows by using tautological re…
We prove that if two Calabi-Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard--Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard--Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.
The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi-Yau threefold. We show that its Mahler measure is a rational linear combination of a special L-value of the normalized newform in S_4(Gamma_0(8)) and a Riemann zeta value. This is equivalent to a new formula for a 6F5-hypergeometric series evaluated at 1.
By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in ${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function ${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic continuation of the ratio $F(Λ)/F(Λ^p)$, where $F(Λ)$ is a solution of the $A$-hypergeome…
In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.
In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified 22 possible supercongruences. We provide a framework of congruences covering all 22 cases. Using this framework we prove one of the outstanding supercongruence conjectures between a special value of a truncated ordinary hypergeometric series and the $p$-th Fourier coefficient of a modular form. In the course of this work we also establish two new binomial coefficient-harmon…
Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/π^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new examples of series of non-hypergeometric type. To motivate our theory we begin with the simpler case of Ramanujan-Sato series for $1/π$.
In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over Q this mo…
