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All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In our previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers, which is related to the moduli space of unordered 12 points on $\mathbb P^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza., Comment: References updated, minor improvements in exposition

We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubics surfaces., Comment: Final version, to appear in Moduli

The paper describes of the Ryze Tello drone to move autonomously using a basic vision system. The drone's position is determined by identifying AprilTags' position relative to the drone's built-in camera. The accuracy of the drone's position readings and distance calculations was tested under controlled conditions, and errors were analysed. The study showed a decrease in absolute error with decreasing drone distance from the marker, a little change in the relative error for large distances, and a sharp decrease in the relative error for small distances. The method is satisfactory for determining the drone's position relative to a marker., Comment: Submitted to MMAR 2023 - 27th International Conference on Methods and Models in Automation and Robotics

Moduli spaces of (polarized) Enriques surfaces can be described as open subsets of modular varieties of orthogonal type. It was shown by Gritsenko and Hulek that there are, up to isomorphism, only finitely many different moduli spaces of polarized Enriques surfaces. Here we investigate the possible arithmetic groups and show that there are exactly $87$ such groups up to conjugacy. We also show that all moduli spaces are dominated by a moduli space of polarized Enriques surfaces of degree $1240$. Ciliberto, Dedieu, Galati, and Knutsen have also investigated moduli spaces of polarized Enriques surfaces in detail. We discuss how our enumeration relates to theirs. We further compute the Tits building of the groups in question. Our computation is based on groups and indefinite quadratic forms and the algorithms used are explained., Comment: 39 pages, 1 figure, 6 Tables

The moduli space of $8$ points on $\mathbb{P}^1$, a so-called ancestral Deligne-Mostow space, is, by work of Kond\={o}, also a moduli space of K3 surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal compactification of the corresponding ball quotient. Moreover, we show that these spaces are not $K$-equivalent, even though they are natural blow-ups at the unique cusps and have the same cohomology. This is analogous to the work of Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is the Fulton-MacPherson compactification of the configuration space of points on $\mathbb{P}^1$, play an important role in the proof. We further relate our computations to new developments in the minimal model program and recent work of Odaka. We briefly discuss other cases of moduli space of points on $\mathbb{P}^1$ where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon., Comment: v2: We describe the relationship with the minimal model program and prove that the Kirwan blow-up in the unordered case is not a semi-toric compactification in Subsection 4.3. We also refer to the relationship with the work of Kudla-Rapoport in the introduction

The moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel compactification of a ball quotient, and as a compactified $K$-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup, while from the ball quotient point of view it is natural to consider the toroidal compactification. Both these spaces have the same cohomology and and it is therefore natural to ask whether they are isomorphic. Here we show that this is in fact not the case. Indeed, we show the more refined statement that both spaces are equivalent in the Grothendieck ring, but not $K$-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients., Comment: v2: Relation to work of Kudla-Rapoport and Zheng provided in the introduction; new Remark 4.7 gives a short independent proof of the result of Gallardo, Kerr, Scheffler on the isomorphism between Naruki's cross ratio variety and the toroidal compactification of the ball quotient. v3: final version. Nagoya Math J, to appear

In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarised K3 surfaces. The paper contains an appendix by M. Kirschmer providing computational results on the 1-dimensional ambi-typical strata., Comment: 55 pages, appendix by Markus Kirschmer, to appear in Algebraic Geometry

We investigate a toroidal compactification of the moduli space of K3 surfaces of degree $2$ originating from the program formulated by Gross-Hacking-Keel-Siebert. This construction uses Dolgachev's formulation of mirror symmetry and the birational geometry of the mirror family. Our main result in an analysis of the toric fan. For this we use the methods developed by two of us in a previous paper., Comment: 50 pages, comments welcome!

We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $\overline{\mathcal A}_3$ of the moduli space ${\mathcal A}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $g\ge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone toroidal compactification of ${\mathcal A}_g$ for any $g\ge 3$., Comment: Final version, to appear in Moscow Mathematical Journal

In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in degree $2$ as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans., Comment: Substantially revised final version

We generalize the classical semi-continuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism from X to an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes, and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed by Gulbrandsen-Halle-Hulek., Comment: 27 pages, major revision, final version, to appear in Michigan Math. J

We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ball quotient model, due to Allcock-Carlson-Toledo. Our starting point is Kirwan's method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli of cubic surfaces is discussed in an appendix., Comment: 101 pages, AMS LaTeX, minor revisions, to appear in Mem. Amer. Math. Soc

Using the Laza-Sacc\`a-Voisin construction, we give a simple proof for the fact that the Euler characteristic of a hyper-K\"ahler manifold of OG10 type is 176,904, a result previously established by Mozgovoy., Comment: 10 pages; To appear in the Proceedings of the ICM 2018 Satellite conference "Moduli Spaces in Algebraic Geometry and Applications", Campinas

Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of $K3$ surfaces we show that the stack ${\mathcal I}^n_{X/C} \to C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai., Comment: 53 pages. To appear in J. Algebraic Geom

We survey old and new results about the cohomology of the moduli space $A_g$ of principally polarized abelian varieties of genus $g$ and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur's endoscopic classification of automorphic representations for symplectic groups., Comment: 57 pages. Appendix by Olivier Ta\"ibi. To appear: Proceedings of the Abel Symposium 2017. v3: numbering of theorems changed to agree with the printed version

In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic., Comment: accepted for publication in Jahresbericht der Deutschen Mathematiker-Vereinigung, obituary, 17 pages, 2 figures, 1 photo

We study the possible Picard numbers of abelian varieties of given dimension $g$. If $R_g$ denotes the set of realizable Picard numbers, then $R_g$ is bounded by $g^2$. We show that, for $g$ at least $3$, the set $R_g$ always has gaps and we analyze the nature of these gaps. We further prove that the Picard numbers are asymptotically complete in $[1,g^2]$ as $g$ goes to infinity. Finally we show that every Picard number which can be realized over the complex numbers can already be realized by an abelian variety defined over a number field., Comment: Final version, to appear in Ann. Sc. Norm. Super. Pisa

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree., Comment: 29 pages. Appendix by Mathieu Dutour Sikiric

We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of degenerations of degree $n$ Hilbert schemes of a simple degeneration with two components., Comment: 50 pages; to appear in Documenta Mathematica

We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved., Comment: Final version, to appear in Math. Annalen

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2., Comment: 56 pages; v2: multiple updates and clarification in response to detailed referee's comments

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure., Comment: Main result slightly strengthened, minor corrections, improved presentation

We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism X -> Spec A to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points., Comment: v4: minor corrections

The main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to different toroidal compactifications of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactification, but we also apply this to other toroidal compactifications, in particular the perfect cone compactification, for which we obtain a combinatorial characterization of the indeterminacy locus, as well as a geometric description up to codimension six, and an explicit toroidal resolution of the Prym map up to codimension four., Comment: 53 pages, AMS LaTeX, Appendix by Mathieu Dutour Sikiric, minor revisions, to appear in JEMS

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to 8, we describe explicitly the generators of the cohomology. We also discuss various approaches to computing all of the stable cohomology in arbitrary degree., Comment: Final version, to appear in Crelle journal

We study the cones in the first Voronoi or perfect cone decomposition of quadratic forms with respect to the question which of these cones are basic or simplicial. As a consequence we deduce that the singular locus of the moduli stack ${\mathcal A_g^{\mathop{Perf}}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g \geq 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${\mathcal A_{g}^{\mathop{Vor}}}$ has singularities in codimension $3$ for $g\geq 5$., Comment: 16 pages; to appear in Algebraic Geometry

The purpose of this note is twofold. We first review the theory of Fourier-Mukai partners together with the relevant part of Nikulin's theory of lattice embeddings via discriminants. Then we consider Fourier-Mukai partners of K3 surfaces in the presence of polarisations, in which case we prove a counting formula for the number of partners., Comment: 35 pages

We survey the geometry of the theta divisor and discuss various loci of principally polarized abelian varieties (ppav) defined by imposing conditions on its singularities. The loci defined in this way include the (generalized) Andreotti-Mayer loci, but also other geometrically interesting cycles such as the locus of intermediate Jacobians of cubic threefolds. We shall discuss questions concerning the dimension of these cycles as well as the computation of their class in the Chow or the cohomology ring. In addition we consider the class of their closure in suitable toroidal compactifications and describe degeneration techniques which have proven useful. For this we include a discussion of the construction of the universal family of ppav with a level structure and its possible extensions to toroidal compactifications. The paper contains numerous open questions and conjectures., Comment: Final version; remark 6.5 on the non-basic codimension 5 stratum missed in our previous work added

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that the moduli space of Kummer surfaces associated to (1,21)-polarised abelian surfaces is uniruled., Comment: 14 pages

In this paper we compute the pullback of divisor classes under the Prym map (extended to the boundary), and apply this result to get a lower bound on the slope of effective divisors on the perfect cone compactification of the moduli space of principally polarized abelian fivefolds. In the appendix by Klaus Hulek, the notion of slope for arbitrary toroidal compactifications is discussed, and the slope bound is shown to hold in general.

In this paper we compute the cohomology groups of the second Voronoi compactification of the moduli space of abelian fourfolds in all degrees with the exception of the middle degree 10. We also compute the cohomology groups of the perfect cone compactification in degree < 10. The main tool is the investigation of the strata of the compactification corresponding to semi-abelic varieties with constant torus rank., Comment: v2: 41 pages, mostly expository changes

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces., Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Math

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the locus of intermediate jacobians of cubic threefolds in the Chow ring of the perfect cone compactification of the moduli space of principally polarized abelian fivefolds.

The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil's programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. We apply our geometric-automorphic method to study moduli spaces of both polarised K3 surfaces and irreducible symplectic varieties., Comment: 71 pages. Fairly minor revisions. To appear in "Handbook of Moduli"

We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively.

We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field d. Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study on Neron-Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces., Comment: 32 pages; v2: Section 2 expanded, minor additions and edits

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction., Comment: 35 pages, 6 figures; v2: minor additions and corrections

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces.

We prove that the cohomology groups with rational coefficients of the Voronoi compactification of the moduli space of abelian threefolds coincide with the Chow groups of that space, as determined by Van der Geer., Comment: 15 pages; exposition improved, first sections expanded, last section slightly shortened

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case., Comment: Revised paper; final section is shortened; correction of typos

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12., Comment: Exposition improved, Reference to work of Debarre and Voisin added, Corollary 1.6 removed

We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space $A_g$ of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of $A_{g-3}$ in the Satake compactification.

For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2, 8m+2) is of general type if m is at least 5., Comment: 19 pages

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60., Comment: 47 pages

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., Comment: Further references added

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface $S$, defined over a number field K such that S^[2](K) is not empty, then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example., Comment: Modified some proofs

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces S_k(\Gamma) of cusp forms of weight k, as k goes to infinity. We compute this in a number of examples, which are important for geometric applications., Comment: Minor corrections, bibliography updated