Tony Shaska
In this paper we study genus 2 function fields K with degree 3 elliptic subfields. We show that the number of Aut(K)-classes of such subfields of K is 0,1,2, or 4. Also we compute an equation for the locus of such K in the moduli space of genus 2 curves.
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We study genus 2 function fields with elliptic subfields of degree 2. The locus $Ł_2$ of these fields is a 2-dimensional subvariety of the moduli space $\mathcal M_2$ of genus 2 fields. An equation for $Ł_2$ is already in the work of Clebsch and Bolza. We use a birational parameterization of $Ł_2$ by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parameterized …
We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the modul…
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $\mathfrak{p}$ contains three irreducible components of genus $1$.
The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where $q$ is a prime power and $n \ge 1$ an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of $K_n$. In case $\gcd(q+1,n)=1$ and either $q$ is even or $q \equiv 1 \pmod{4},$ we find a complete list of Galois subfields of $K_n.$ Our construction adds several previously unknown genera to the genus spectrum of maximal curves.
We discuss the possibility of Mathieu group M24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M24. In this paper we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.
We prove a genus formula for modular curves of $D$-elliptic sheaves. We use this formula to show that the reductions of modular curves of $D$-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of $D$ tends to infinity.
Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.
We define the singular elliptic genus for arbitrary normal surfaces, prove that it is a birational invariant, and show that it generalizes the singular elliptic genus of Borisov and Libgober and the stringy $χ_y$ genus of Batyrev and Veys.
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
We present explicit models for non-elliptic genus one Shimura curves X_0(D, N) with Gamma_0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan's work in his Ph. D Thesis on points with complex multiplication on Shimura curves.
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier.
We discuss an algebro-geometric description of Witten's phases of N=2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met. For Landau-Ginzburg phase one recovers elliptic genus of LG models proposed in physics literature in early 90s. For certain transitions between phases we derive invariance of elliptic genus from an equivariant form of McKay correspondence for elliptic genus. As special cases one obtains Landau-Giznburg/Calabi-Yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybr…
We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.
A maximal function field of genus $17$ over $\mathbb{F}_{11^2}$
本文通过人工智能发现了一个新的在$\mathbb{F}_{11^2}$上的亏格17的极大函数域$G$，证明其不能作为Hermitian函数域的子域，并计算了其自同构群。
Borisov and Libgober recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties to deduce a formula for the elliptic genus of a generalised Kummer variety.
Hodge-Elliptic genera, K3 surfaces and Enumerative Geometry
本文将K3曲面的Elliptic亏格推广为Hodge-Elliptic亏格，并据此给出K3 × E精细化BPS不变量的显式乃至闭式计算方法。
We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which…
Curves of genus two with maps of every degree to a fixed elliptic curve
本文分类了所有亏格2曲线C和椭圆曲线E对，使得对每个整数n>1都存在从C到E的n次映射，并证明每条亏格2曲线都存在一个不超过59的整数n使得没有最小n次映射到椭圆曲线。
The (complex) Hodge-elliptic genus and its conformal field theoretic counterpart were recently introduced by Kachru and Tripathy, refining the traditional complex elliptic genus. We construct a different, so-called chiral Hodge-elliptic genus, which is expected to agree with the generic conformal field theoretic Hodge-elliptic genus, in contrast to the complex Hodge-elliptic genus as originally defined. For K3 surfaces X, the chiral Hodge-elliptic genus is shown to be independent of all moduli. Moreover, employing Kapustin's results on infinite volume limits it is shown that it agrees with the…
Instantons and multibananas: Relating elliptic genus and cohomological Donaldson-Thomas theory
本文利用Descombes双曲局部化公式计算局部多香蕉三维流形的上同调Donaldson-Thomas理论，证明其结果由带标架瞬子层模空间的椭圆亏格精确给出，并推导了无穷楔空间迹公式。
E. Kani has shown that the Hurwitz functor, which parametrizes the (normalized) genus 2 covers of degree 3 of an elliptic curve, is representable. In this paper the corresponding moduli scheme and the universal family are explicitly calculated and described by short equations.
In the paper we study two types of relations: a one is between the elliptic genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac-Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus.
We explain the relationship between the sigma orientation and Witten genus on the one hand and the two-variable elliptic genus on the other. We show that if E is an elliptic spectrum, then the Theorem of the Cube implies the existence of canonical SU-orientation of the associated spectrum of Jacobi forms. In the case of the elliptic spectrum associated to the Tate curve, this gives the two-variable elliptic genus. We also show that the two-variable genus arises as an instance of the circle-equivariant sigma orientation.
We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic different from 2 is birational to a product of two elliptic curves over the projective line. We formulate and prove a generalization of this theorem for the relative si…
In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.
Superspecial genus-$4$ double covers of elliptic curves
本文研究椭圆曲线二重覆盖得到的亏格4曲线，给出显式方程与非奇异性判据，证明相关长多项式在特征不等于2、3时不可约，并在小特征下枚举超特殊亏格4二重覆盖。
We develop the quantum Kodaira-Spencer theory on the elliptic curve and establish the corresponding higher genus B-model. We show that the partition functions of the higher genus B-model on the elliptic curve are almost holomorphic modular forms, which can be identified with partition functions of descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry is established at all genera.
In this survey we study the genus 2 curves with $(n, n)$-split Jacobian for even $n$.
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric …
A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $ψ: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split. The locus of $C$, denoted by $Ł_n$, is an algebraic subvariety of the moduli space $\M_2$. The space $Ł_2$ was studied in Shaska/Völklein and Gaudry/Schost. The space $Ł_3$ was studied in Shaska (2004) were an algebraic description was given as sublocus of $\M_2$. In this survey we give a br…
We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.
We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at gi…
Invariance of elliptic genus under wall-crossing
该研究提出一个拓扑判据，证明利用主空间得到的各类椭圆亏格墙交叉公式具有平凡性，并将其应用于GIT商、层模空间、Donaldson-Thomas及Vafa-Witten理论等场景。
We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ its elliptic genus satisfies a modular differential equation of degree $5$. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko--Zag…
We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the…
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical…
Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus $\sum Ell_{orb}(X^n,Σ_n)p^n$ for symmetric groups $Σ_n$ acting on $n$-fold products coincides with the one proposed by Dijkgraaf,…
Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}. When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the sublocus of $\mathcal M_2$ of curves having a degree 4 elliptic subcover.
Four paths from birational geometry to the elliptic genus
本文通过四条原因说明椭圆亏格是可推广至奇异空间的最一般特征类，证明其特征类（加额外因子）在特定修改下唯一不变，证实并扩展Totaro的结论，方法基于等变上同调局部演算。
Extending the method of Part I (alg-geom/9704004), we give recursive formulae for: the genus-1 Severi degree (formula first found by Getzler), the degree of the variety of 1-cuspidal curves of genus 0 or 1, and the linear (sectional) geometric genus of the genus-0 and genus-1 Severi varieties. The proof is short and elementary.
We match the elliptic genus of a Berglund-Hübsch model with the supertrace of $y^{J[0]}q^{L[0]}$ on a vertex algebra $V_{{\bf 1}, {\bf 1}}$. We show that it is a weak Jacobi form and the elliptic genus of one theory is equal to (up to a sign) the elliptic genus of its mirror.
We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small $h^{1, 1}$ the fraction of polytopes in the KS database that do not have a genu…
We define and compute the dressed elliptic genus of N = 2 heterotic compactifications with torsion that are principal two-torus bundles over a K3 surface. We consider the most general gauge bundle compatible with supersymmetry, a stable holomorphic vector bundle over the base together with an Abelian bundle over the total space, generalizing the computation previously done by the authors in the absence of the latter. Starting from a (0,2) gauged linear sigma-model with torsion we use supersymmetric localization to obtain the result. We provide also a mathematical definition of the dressed elli…
Examples of infinitely many $(-1)$-curves with higher genus on elliptic surfaces
本文构造了第一个具有无穷多条亏格 $g>1$ 的光滑 $(-1)$-曲线的椭圆曲面，解决了 Bauer 等人提出的开放问题。
In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes ${\cal H}(1,X(d),n)$ of morphisms of degree $n$ from elliptic curves to the modular curve $X(d)$, $d\geq 3$. Ultimately, we show that the moduli spaces in…
Multi-fan is an analogous notion of fan. As a fan is associated to a toric variety a multi-fan is associated to a torus orbifold. Orbifold elliptic class and orbifold elliptic genus are defined for a triple of a multi-fan, a set of generating integral vectors of one dimensional cones and a $\mathbb{Q}$ divisor. They are shown to behave functorially with respect to birational morphisms between these triples. The result may be considered as a combinatorial or topological analogue of the main result of Borisov and Libgober, McKay correspondence for elliptic genera, Ann. of Math., 161 (2005).
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by taking an orthogonal linear section of the spinor tenfold. This is an inverse of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a curve of genus 7 to a Fano threefold of genus 7. An explicit geometric construction of both isomorphisms and a similar result f…
The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…
We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $Ł_n$ of the moduli space $\M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $Ł_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $Ł_5$ as subvarieties …
We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also show that, if we want such a curve to be just geometrically hyperelliptic, so having a degree two map to a conic, then there are some with genus 3.
Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties
本文研究Calabi-Yau簇椭圆亏格的模微分方程，证明严格六维情形最小阶为四，并为六维超Kähler簇构造7阶方程，刻画非一般解的结构。
We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic…
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then character…
The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and $G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively $\ma…
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
We perform a refined count of BPS states in the compactification of M-theory on $K3 \times T^2$, keeping track of the information provided by both the $SU(2)_L$ and $SU(2)_R$ angular momenta in the $SO(4)$ little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $K3 \times T^2$, simultaneously refining Katz, Klemm, and Pandharipande's motivic Donaldson-Thomas counts on $K3$ and Oberdieck-Pandharipande's Gromov-Witten counts on $K3 \times T^2$. This provides the first full answer for motivic curve counts of a compact Calabi-Y…
The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hy…
Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.
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