Ikuo Satake
In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.
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The recently discovered fourth class of Frobenius manifolds by Combe--Manin in opened and highlighted new geometric domains to explore. The guiding mantra of this article is to show the existence of hidden geometric aspects of the fourth Frobenius manifold, which turns out to be related to so-called causality conditions. Firstly, it is proved that the fourth class of Frobenius manifolds is a Pseudo-Elliptic one. Secondly, this manifold turns out to be a sub-manifold of a non-orientable Lorentzian projective manifold. Thirdly, Maurer--Cartan structures for this manifold and hidden geometrical p…
Good Basic Invariants for Elliptic Weyl Groups and Frobenius Structures
本文为椭圆根系定义椭圆Weyl群的优良基本不变量，证明其给出Saito平坦不变量，且Taylor系数决定Frobenius结构乘法常数。
We construct Frobenius structures on the $\mathbb{C}^{\times}$-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a trigonometric version of Frobenius algebras in terms of root systems and a new class of Frobenius manifolds. We also determine their potential functions.
On the number of Frobenius periodic vector bundles on elliptic curves
研究特征p>0代数闭域上椭圆曲线上弗罗贝尼乌斯周期向量丛数量，通过转化为平展基本群的连续表示，得出显式生成函数和精确周期公式，分普通和超奇异情形处理。
Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether-Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.
We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an arbitrary field $\mathbb{K} \subset \mathbb{C}$ via the theory of modular forms. By an arithmetic property of an elliptic curve $\mathbb{E}_τ$ defined over $\mathbb K$ associated to such a Frobenius manifold, it is proved that there are only two such Frobenius manifolds defined over $\mathbb C$ satisfying a certain symmetry assumption and thirteen Frobenius man…
Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks
该研究通过根栈Gromov-Witten理论，建立了Gross-Siebert与Tseng-You两类对数Calabi-Yau镜像代数的结构常数关联，给出对数镜像代数结合性新证，完整证明弱Frobenius结构定理，并引入相关扭化不变量研究分歧基变换下的行为。
Frobenius manifold for the nodal quiver
本文从 Weierstrass 椭圆函数出发，结合导出范畴与同调镜像对称性构造并同构了两种 Frobenius 结构，并研究了模群下 Lyashko--Looijenga 映射度与辫群作用下全容许例外范畴数量及 Gamma 积分结构的关系。
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
Frobenius on the cohomology of thickenings
本文研究正特征域上射影空间中光滑簇加厚的Frobenius映射单射性，对光滑完全交和椭圆曲线射影嵌入给出与特征无关的一致界，并在超曲面和椭圆曲线情形证明界的精确性。
In this paper, we give a new formulation of invariant theory for elliptic Weyl group using the group O(2,n). As an elliptic Weyl group quotient, we define a suitable $\C^*$-bundle. We show that it has a conformal Frobenius structure which we define in this paper. Then its good section could be identified with a Frobenius manifold which we constructed in arXiv:math/0611553.
Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $ϕ:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=ϕ^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $κ_P$ of $P$ are of the form $q_P^{1/2}e^{iθ_P},q_{P}e^{-iθ_P}$, where $q_P=q^{°(P)}$ and $0\leθ_P\leπ$. The goal of this note is to determine given an integer $B\ge 1$, $α,β\in[0,π]$ the number of $P\in C$ where the reduction of $E$ is goo…
In this paper, we introduce a family of hyper-elliptic curves. For this family we compute the matrix of the divided Frobenius and we obtain general formulas. We use a recent results of Huyghe-Wach : the divided Frobenius coincides with the morphism built by Deligne-Illusie in 1987.
In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms. Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities. Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities. The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain La…
We describe a variation of Dworks unit-root method to determine the degree four Frobenius polynomial for members of a 1-modulus Calabi-Yau family over $\mathbb{P}^1$ in terms of the holomorphic period near a point of maximal unipotent monodromy. The method is illustrated on a couple of examples.
Let C be a smooth irreducible projective curve defined over a finite field $\mathbb{F}_{q}$ of q elements of characteristic p>3 and $K=\mathbb{F}_{q}(C)$ its function field and $ϕ_{\mathcal{E}}:\mathcal{E}\to C$ the minimal regular model of $\mathbf{E}/K$. For each $P\in C$ denote $\mathcal{E}_P=ϕ^{-1}_{\mathcal{E}}(P)$. The elliptic curve $E/K$ has good reduction at $P\in C$ if and only if $\mathcal{E}_P$ is an elliptic curve defined over the residue field $κ_P$ of $P$. This field is a finite extension of $\mathbb{F}_q$ of degree $°(P)$. Let $t(\mathcal{E}_P)=q^{°(P)}+1-#\mathcal{E}_P(κ_P)$ b…
The explicit description of the Frobenius structure for the elliptic root system of type $D_4^{(1,1)}$ in terms of the characters of an affine Lie algebra of type $D_4^{(1)}$ is given.
When written in terms of $\vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) of the elliptic trilogarithm function introduced by Beilinson and Levin. A differential identity satisfied by this function is also derived. These generalized Frobenius-Stickelberger identities play a fundamental role in the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde…
An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties G/P are Frobenius split. The aim of this article is to give in this case a complete and self contained representation theoretic approach to this method. The geometric Frobenius method in (char k=p>0) will here be replaced by Lusztig's Frobenius maps for quantum groups at roots of unity (which exist not only for primes but any odd integer \ell >1).
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-polarise…
Local root numbers of elliptic curves over dyadic fields
本文研究了二元域上具有加性且潜在良好约化的椭圆曲线，通过分析其三挠点生成的有限Galois扩张，给出了计算该椭圆曲线局部根数的公式。
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root…
Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteri…
We show the density of rational points on non-isotrivial elliptic surfaces by studying the variation of the root numbers among the fibers of these surfaces, conditionally to two analytic number theory conjectures (the squarefree conjecture and Chowla's conjecture). This is a weaker statement than the one found in a preprint of Helfgott which proves (under the same assumptions) that the average root number is $0$ when the surface admits a place of multiplicative reduction. However, we use a different technique. The conjectures involved impose a restriction on the degree of the irreducible facto…
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