Teruyoshi Yoshida
We relate a part of the abelian etale fundamental group of curves over local fields to the component group of the Neron model of the jacobian. We apply the result to the modular curve X_0(p)/Q_p to show that the unramified abelian covering X_1(p) over X_0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X_0(p).
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We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.
For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite etale coverings: we show that this class is different from the one used by de Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.
This paper contains a loose collection of remarks on F1-schemes. Etale morphisms and universal coverings are introduced. The relation to toric varieties, at least for integral schemes, is clarified.In this paper it is shown that integral F1-schemes of finite type are essentially the same as toric varieties. A description of the F1-zeta function in terms of toric geometry is given. Etale morphisms and universal coverings are introduced.
The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic étale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $κ$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.
On $ψ$-lattices in modular $(φ,Γ)$-modules
本文推广Colmez的$\psi$-格概念至多变量环上的étale $(\varphi,\Gamma)$-模，并给出定义、性质与实例。
We give an effective proof of Faltings' theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields. We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of $\mathrm{GL}_2$-type over an odd-degree totally real field. We deduce for example an effective height bound for $K$-points on the curves $C_a : x^6 + 4y^3 = a^2$ ($a\in K^\times$) when $K$ is odd-degree totally real. (Over $\overline{\mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an étale cover dominating $C_1$.)
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
We use the adelic language to show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof is based on a study of the actions of $\mathrm{GL}_2$ and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet's twisting operators on modular abelian varieties.
We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are étale local on X. If X is connected, its geometric coverings form a tame infinite Galois category, and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs," making it similar to geometric coverings of schemes studied by Bhatt and Scholze as well as semicoverings of topological spaces introduced by Brazas.
The geometric fundamental group of the affine line over a finite field
该研究以有限域上仿射直线与穿孔仿射直线为基准，通过Tannaka对偶重构构造泛仿射射影代数群，推导其商群与仿射子簇平展Galois覆叠的关系，并给出两类仿射簇对应泛仿射群的显式描述。
Consider three normalised cuspidal eigenforms of weight $2$ and prime level $p$. Under the assumption that the global root number of the associated triple product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel-Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associat…
These are the notes from a survey talk given at Arbeitstagung 2001 covering the author's work with Lev Borisov and Sorin Popescu on toric varieties, modular forms, and equations of modular curves.
We extend our previous theory of etale parallel transport to a larger class of slope zero vector bundles on p-adic curves. The new class is stable under pullback by ramified coverings. We also construct p-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.
The moduli space of representations of the modular group into $G_2$
本文构造了模群到 $G_2$ 的四维表示族，分析了该模空间的分量，并给出了特化满射到 $G_2(\mathbf{F}_p)$ 的代数条件，提供了 $\phi$-同余子群的新实例。
We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sysπ_{1}(M_{I})\geq \frac{4}{3\sqrt{n}}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $M_{I}$.
The classical Chevalley-Weil theorem asserts that for an étale covering of projective varieties over a number field K, the discriminant of the field of definition of the fiber over a K-rational point is uniformly bounded. We obtain a fully explicit version of this theorem in dimension 1.
Locally analytic representations in the étale coverings of the Lubin-Tate moduli space
本文将Lubin-Tate模空间上自同构群诱导局部解析表示的结果推广至$\boldsymbol{\mathbb{Q}_p}$有限扩张赋值环上的形式模形变及带Drinfeld level-$m$结构的形变空间情形，并确定了对应截面空间的局部有限向量空间的结构。
We compute the stable reduction of the Lubin-Tate space $X(π^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(π^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(π^2)$. Our method is very explicit and elementary.
We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the cont…
Hypergeometric Distributions and Joint Families of Elliptic Curves
本文将椭圆曲线族超几何函数极限分布的研究推广到${_4F_3}$超几何函数，通过推广Michel的Sato-Tate律工作并结合平展上同调等理论，确定了椭圆曲线对的弗罗贝尼乌斯迹和的极限分布。
We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by $\mathbf{K}_2$ -- the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over $\mathbb R$. An appendix surveys torsors and gerbes on the étale site, as they are used in the construction of…
We incorporate nonlinear covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. This L-group is an extension of the absolute Galois group of a local or global field $F$ by a complex reductive group. The L-group depends on an extension of a quasisplit reductive $F$-group by $\mathbf{K}_2$, a positive integer $n$ (the degree of the cover), an injective character $ε\colon μ_n \rightarrow {\mathbb C}^\times$, and a separable closure of $F$. Our L-group is consistent with previous work on covering groups, and its construction is contravaria…
The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf_0(3) consistent with properties…
Global $\mathbb{A}^1$ degrees of covering maps between modular curves
本文定义了光滑曲线有限态射的两种整体 $\mathbb{A}^1$ 度数，并证明在特定条件下模曲线间覆盖映射的两种度数均等于 Grothendieck-Witt 环中的双曲元素之和。
This paper studies a covering map phi from the upper half plane to the complex plane with a triangular lattice excised. This map is interesting as it factorises Klein's J invariant. Its derivative has properties which are a slight generalisation of modular functions, and (phi')^6 is a modular function of weight 12. There is a homomorphism from the modular group Gamma to the affine transformations of the complex plane which preserve the excised lattice. With respect to this action phi is a map of Gamma-sets. Identification of the excised lattice with the root lattice of sl_3(C) allows functions…
Formal Siegel modular forms for arithmetic subgroups
本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。
We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental group approximates the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statement about vector bundles which are true in characteristic zero are proved. Some of these are positive characteristic avatar of notions which appear in the second aut…
Let f: S' --> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps α_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and β_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that α_{r}oβ_{r}=β_{r}oα_…
Factorisation de la cohomologie de systèmes locaux $p$-adiques sur le demi-plan de Drinfeld
本研究计算Drinfeld p进半平面覆叠塔上泛平展p进局部系统对称代数的第一上同调群，得到基于Kisin环上族p进Langlands对应的分解形式，推广了平凡系数情形的已有结果。
Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $ρ$ of $π_1(U)$ corresponds to an étale $F$-isocrystal. By work of Tsuzuki and Crew an $F$-isocrystal is overconvergent precisely when $ρ$ has finite monodromy at each $x \in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth $F$-isocrystals …
As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of $l$-adic sheaves on a variety over a finite field, we show the finiteness of étale coverings of such a variety with given degree whose ramification bounded along an effective Cartier divisor. This can be thought of a higher dimensional analogue of the classical Hermite-Minkowski theorem.
Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one to one correspondence between the set of finite abelian étale coverings and the set of finite index open subgroups of the idèle class group as in the classical class field theory under some assumptions.
Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute…
We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…
This thesis develops some of the basic model theory of covers of algebraic curves. In particular, an equivalence between the good model-theoretic behaviour of the modular j-function, and the openness of certain Galois representations in the Tate-modules of abelian varieties is described.
A genus one curve C of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We prove a result characterising the covariants for these models in terms of their restrictions to the family of curves parametrised by the modular curve X(5). We then construct covariants describing the covering map of degree 25 from C to its Jacobian and give a practical algorithm for evaluating them.
Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves
该研究构造了PSL(2, F_p)表示的几何实现，建立了模曲线定义理想、PSL(2, F_p)表示与伽罗瓦表示间的对应，是多个数论几何核心理论的阿贝尔几何对应物。
The Heisenberg covering of the Fermat curve
本文给出第N个海森堡曲线作为费马曲线覆盖的模型，证明N=3时Manin-Drinfeld原理成立、N=5时不成立，解释了费马曲线尖点子群描述，并研究了雅可比簇态射像及X'_N同调。
Through the lens of the Eisenstein ideal
本文通过Eisenstein理想建立模曲线几何与分圆域算术的联系，定义两个猜想互逆的映射，并给出其等变构造及Euler系统的应用。
p-adic equidistribution of modular geodesics and of Heegner points on Shimura curves
本文提出 Duke 定理的 p-adic 版本，研究模曲线上闭测地线及 Shimura 曲线上 Heegner 点的 p-adic 等分布，方法涉及分裂二次域和 p-adic 覆盖。
In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modu…
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