E. Izadi, H. Lange, V. Strehl
We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tjurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.
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In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.
Quadratic unilateral polynomials over split quaternions
本文推导了分裂四元数代数中$a$不可逆时计算二次单边多项式$ax^{2}+bx+c=0$根的显式公式，并借鉴Opfer等人的方法在伴随多项式不恒为零时验证了结果。
A local-global correspondence for perfectoid purity
本文引入了完美oid分裂概念，建立项目性方案的完美oid分裂与其戈林斯坦截面环的完美oid纯度之间的对应关系，并构造了新的完美oid纯环例子。
Addition-deletion for conic-line arrangements with split Chern polynomial
本研究提出复射影平面曲线对数向量场丛陈多项式整分解诱导的组合/几何障碍，推广了射影直线排列与自由曲线的相关结果，并给出秩2向量丛基于光滑圆锥曲线限制的分裂准则。
A generic categorical local Langlands correspondence for quasi-split reductive groups
针对一大类拟分裂p进约化群，构造了从自守侧一般Bernstein块的稳定∞-范畴到L-参数模栈上ind-凝聚层的稳定∞-范畴的完全忠实函子，推广了GL_n情形的工作，并建立了经典对应提升为∞-范畴对应的框架。
How to split two-dimensional Jacobians: a geometric construction
针对代数曲线分歧覆盖的雅可比簇等分解问题，研究人员证明了当亏格g(Y)=2、g(X)=1时，该分解由满足伽罗瓦图的曲线代数对应诱导，并给出互补曲线W的几何构造等结果。
According to McKay (1980) the irreducible characters of finite subgroups of SU(2) are in a natural 1-1 correspondence with the extended Coxeter-Dynkin graphs of type ADE. We show that the character values themselves can be given by an uniform formula, as special values of polynomials which arise naturally as numerators of Poincare series associated to finite subgroups of SU(2) acting on polynomials in two variables. These polynomials have been the subject of a number of investigations, but their interpretation as characters has apparently not been noticed.
Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${\mathbb A}^{2n}$ by the symmetric group $S_n$. A short proof of a similar derived …
We classify absolutely split vector bundles on proper $k$-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer--Severi varieties.
In this paper, we introduce modular polynomials for the congruence subgroup $Γ_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respec…
Smooth correspondences between quiver varieties
本文提出了一种新的光滑对应于淬火品种，用于构建奇点解析并证明其不可约性和Cohen-Macaulay性质，扩展了之前关于希尔伯特方案的研究。
Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic
本文研究正特征下 incidence 对应上线丛的上同调，提出偶进位多项式解决 Liu-Gao-Raicu 的递推关系，得到 n=3 时的显式公式，并推测一般情况下的上同调结构。
In this thesis we study when a homogeneous polynomial $f$ decomposes or "splits" additively. Up to base change this means that it is possible to write $f = g + h$ where $g$ and $h$ are polynomials in independent sets of variables. This simple idea leads us to define a set $M_f$ of matrices associated to $f$. Surprisingly, $M_f$ turns out to be a commutative matrix algebra when $deg f \ge 3$. We show how all (regular) splittings $f = g_1 + ... + g_n$ can be computed from $M_f$. Next we show how to find the minimal free resolution of the graded Artinian Gorenstein quotient $R/\ann f$, assuming t…
The Matsuo-Cherednik correspondence is an isomorphism from solutions of Knizhnik-Zamolodchikov equations to eigenfunctions of generalized Calogero-Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials, recently introduced in math-ph/0105014. We compute the Poincare' polynomials for the space H_m and of its isotypical components corresponding to each irreducible representation of the group G.…
Cycles on splitting models of Shimura varieties
本文通过分裂模型构造了 Shimura 变种的 exotic Hecke 对应关系，并验证了 Tate 猜想的通用实例。
Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we prove that there exists an effectively computable constant c(m,n) depending only on m and n such that the following holds. Let f_1,...,f_n be polynomials with coefficients in K, which are disintegrated polynomials of degree at most m and let F be the induced coordinate-wise self-map of the n-th dimens…
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