Caterina Consani, Jasper Scholten
The paper investigates some aspects of the geometry and the arithmetic of a non-rigid Calabi-Yau threefold. Particular emphasis is given to the study of its L-function L(H^3,s) and the Galois representation.
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We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds.
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold which is modular, in the sense that the L-function of its middle-dimensional cohomology is associated to a classical modular form of weight 4 and level 55.
We prove that both classical Chevalley-Warning-Ax and Tsen theorems hold for the blowing up of a quintic 3-fold along a line of multiplicity 3. Both proofs, which are of the same spirit than the original ones, involve the description of this blowing-up as a subvariety of a toric variety.
Construction of diagonal quintic threefolds with infinitely many rational points
本研究构造了一族定义在有理数域上、各含无穷多有理点的对角五次三维簇，并证明存在无穷多组互素整数四元组，使得对应五次方程对合适的依赖参数有无穷多正整数解。
This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in IP^3 with maximum Picard number 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.
We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.
We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.
In a recent paper, Rosen and Silverman showed that Tate's conjecture on the order of vanishing of L(E,s) implies Nagao's formula, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. The aim of this article is to extend their result to the case of elliptic threefolds, and deduce, from Tate's conjecture, a Nagao-type formula for the rank of an elliptic threefold E. This will require a two-pronged approach: on the one hand, we need some cohomological results in order to derive a Shioda-Tate-like formula for elliptic threefolds; on the other…
We study Calabi-Yau threefolds fibered by abelian surfaces, in particular, their arithmetic properties, e.g., Néron models and Zariski density.
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
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-…
In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb{C}$…
In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realizati…
We construct Calabi-Yau threefolds defined over $\mathbb{Q}$ via quotients of abelian threefolds, and re-verify the rigid Calabi-Yau threefolds in this construction are modular by computing their L-series, without \cite{Dieulefait} or \cite{GouveaYui}. We compute the intermediate Jacobians of the rigid Calabi-Yau threefolds as complex tori, then compute a $\mathbb{Q}$-model for the 1-torus given a $\mathbb{Q}$-structure on the rigid Calabi-Yau threefolds, and find infinitely many examples and counterexamples for a conjecture of Yui about the relation between the $L$-series of the rigid Calabi-…
We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.
We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions $F_g^{alg}, g\geq 1$, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-Länge in an algebraic conte…
We study various examples of Calabi-Yau threefolds over finite fields. In particular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p-adic cohomologies of some Calabi-Yau threefolds constructed by Cynk-van Straten which have remarkable arithmetic properties, as well as those of the Hirokado threefold. These examples and computations answer some outstanding questions of B. Bhatt, T. Ekedahl, van der Geer-Katsura and Patakfalvi-Zdanowicz, and shed new light on the Beauville-Bogomolov decomposition in positive …
Hirzebruch-Zagier classes and rational elliptic curves over quintic fields
在Hilbert模曲面étale上同调猜想等条件下，本文通过细化twisted triple product p-adic L-函数并构造Hirzebruch-Zagier圈，利用显式互反律建立秩0等变BSD猜想新实例，应用于五次数域有理椭圆曲线算术。
The Kodaira dimension of Hilbert modular threefolds
本文通过Thomas-Vasquez和Grundman的方法，证明了许多算术亏点为0和1的希尔伯特模三维为一般类型，部分具有非负柯西-亚罗维茨维数。
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-function…
In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the `type' of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke/Matsumoto and Hernández-Mada. We also study `maximally unipotent' degenerations of Calabi--Yau threefolds, following Kollár/Xu, showing in this case that the d…
Local Zeta Functions of Multiparameter Calabi-Yau Threefolds from the Picard-Fuchs Equations
本文提出了一种改进的框架，用于计算多参数卡拉比-叶三重流形的局部ζ函数，并通过实例验证了方法的实用性。
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