Werner Bley, Cristian D. Popescu
We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by Greither and Popescu, in their proof of the classical Equivariant Main Conjecture along the (arithmetic) cyclotomic Iwasawa tower.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient condition for the conjecture to hold. An advantage of our method is that it is effective, in the sense that we know how to construct the eventual solution.
Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-Mézard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state a conjectural generalisation to n-dimensional representations of the absolute Galois group of an arbitrary finite extension of Qp, and give a conditional proof of this conjecture, subject to a certain R = T-type theorem together with a strong version of the weight part of Serre's conjecture for rank n unitary groups. We deduce an unconditional result in the c…
We give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for separated schemes of finite type over a finite field, as stated and proved by Burns and the author. We will also comment briefly on versions of the main conjecture for function fields.
The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations
本文在二维模p伽罗瓦表示的模叠上，建立了潜在Barsotti-Tate表示的Breuil-Mézard猜想及Serre猜想权重部分的几何化。
In this paper we will explore the solutions to the diophantine equation in the Erdős-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$ We will separate the types of solutions into two cases. In particular we will argue that the most common relationship found is $$ x = \lfloor \frac{py}{4y-p} \rfloor + 1.$$ Finally, we will make a few conjectures to motivate further research in this area.
Zaremba's Conjecture for Geometric Sequences: An Algorithm
本文提出了一种改进的算法，利用连续分数的折叠引理，扩展并改进了原有算法，从而发现新的满足齐纳尔巴猜想强版本的序列示例。
We prove the geometric Bogomolov conjecture over a function field of characteristic zero.
Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a "geometric" variant of the "algebraic" Serre weight conj…
We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from an incompatibility of geometric invariants.
The Geometric Unitary Kudla Conjecture
本文基于几何单位元Kudla猜想，证明对称形式傅里叶-雅科比级数收敛并等于Hermitian Hilbert模形式的展开，验证了猜想并在任意余维数上去除Li-Liu的算术内积公式的模性假设。
This is a report of the author's talk at Kinosaki Algebraic Geometry Symposium 2018. We discuss some recent progress on the geometry of thin exceptional sets in Manin's Conjecture.
On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the …
This paper proposes a conjecture on special values of L-functions of geometric motives over Z. This includes L-functions of mixed motives over Q and Hasse-Weil zeta-functions of schemes over Z. We conjecture the following: the order of L(M, s) at s=0 is given by the negative Euler characteristic of motivic cohomology of $M^\vee(-1)$. Up to a nonzero rational factor, the L-value at s=0 is given by the determinant of the pairing of Arakelov motivic cohomology of M with the motivic homology of M. Under standard assumptions concerning mixed motives over Q, $F_p$, and Z, this conjecture is essentia…
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.
Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.
We construct a quartic threefold with L-rational singularities which has torsion in its middle homology group. This answers a question of Brown and Schnetz for all fields of characteristic zero.
We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive $p$-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
On the geometric Serre weight conjecture for Hilbert modular forms
本文证明了 Diamond 和 Sasaki 提出的几何 Serre 权重猜想，在无奇偶性假设下适用于 $p \geq 5$ 惰性的实二次域和 $p \geq \min\{5, [F:\mathbb{Q}]\}$ 完全分裂的全实域。
Parshin's method and the geometric Bombieri-Lang conjecture
本文综述 Parshin 对几何 Bombieri–Lang 猜想的证明，并用其为 Xie–Yuan 关于空特殊轨迹且可有限映射至无迹阿贝尔簇的射影簇结果提供替代证明。
Geometric Bogomolov conjecture for semiabelian varieties
本文证明了函数域上半阿贝尔簇的几何Bogomolov猜想，刻画了闭子簇包含Zariski稠密小点集的充要条件，并揭示了特殊子簇可能缺乏高度为0点的Zariski稠密集这一新现象。
We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang's reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang's to extend to modular abelian varieties over totally real fields.
Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture
本文提出利用部分高度与 Brody 引理从无穷有理点序列构造全纯曲线的新方法，并在部分高度非退化猜想下证明了特征 0 函数域上具有到阿贝尔簇有限态射的双曲射影簇的几何 Bombieri-Lang 猜想。
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually periodic continued fractions have partial quotients which have exponential growth when iteratively multiplied by $n$, for $n$ any fixed, natural number.
We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\ell$-adic Mumford-Tate conjecture.
The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties
该研究针对特征 0 函数域上映射到平凡迹 Abel 簇的射影簇，借助整曲线构造并以 Lie 代数显式刻画整曲线，证明了几何 Bombieri–Lang 猜想。
The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvarieties. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function field, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler's result. The key in establishing the previous results on the Bogomolov conjecture is the …
A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality
该研究通过高亏分数的圆法推广，证明了对于一般低次超曲面，不存在增加Fujita不变量的累积映射，从而在特定范围内建立了这些模空间的庞加莱对偶性。
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
On the Serre conjecture for Artin characters in the geometric case
本文研究几何情形下的Serre猜想，针对有限群作用的正则局部环，在等特征条件下证明了Serre定义的类Artin特征函数对应群的ℚ_ℓ-有理表示的猜想。
Functional Transcendence of Periods and the Geometric André--Grothendieck Period Conjecture
本文通过证明代数簇族中代数形式积分的函数超越性，给出混合 Hodge 结构变分的 Ax–Schanuel 结果，并据此建立几何形式的 André–Grothendieck 周期猜想。
Recent progress on the geometric Bombieri--Lang conjecture
综述特征零函数域上几何Bombieri-Lang猜想的最新进展，重点介绍Xie-Yuan和高国庆的工作，证明了有限态射到阿贝尔簇的簇的猜想。
Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections
本文研究了Chabauty-Kim猜想与局部几何截面猜想，证明了在满足Kim猜想的条件下，X/Q满足截面猜想，并推广到超几何曲线的S-整点。
The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example
针对Batyrev和Tschinkel提出的Fano 5-维Fermat三次曲面丛X，在特征0域上显式构造有限薄映射，证明其生成的薄集与Lehmann-Sengupta-Tanimoto猜想的Manin猜想例外集一致。
Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.
A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties
本文通过新的计数论证，证明了有理函数域上阿贝尔簇的分支覆盖满足几何Bombieri-Lang猜想，扩展了Xie-Yuan的结果。
The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we furthermore show that the torsion is bounded in terms of the $\mathit{gonality}$ of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the…
A higher genus circle method and an application to geometric Manin's conjecture
该研究从几何角度重新解释Hardy-Littlewood圆法并推广至高亏格光滑射影曲线，证明Davenport收缩引理的高亏格变体，还应用结果证明低次光滑超曲面真子簇的Fujita不变量小于1。
Mordell--Lang plus Geometric Bogomolov
本文证明了特征零半阿贝尔簇上 Poonen 定理的几何版本，推广了 Mordell--Lang 与几何 Bogomolov 猜想。
In 1980 Zelevinsky introduced commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss--Leclerc--Schröer condition that occurs in the conjectural characterization of $\square$-irreducible representations. We verify some special cases of the new conjectu…
On some algebraic and geometric extensions of Goldbach's conjecture
本文研究域上仿射代数簇正则函数环的哥德巴赫猜想，定义牛顿多面体的哥德巴赫条件，构造性证明任意至少两个变量的多项式可表示为至多2r个绝对不可约多项式之和，并验证强迫代数及局部化等情形。
Geometric Langlands duality for periods
本文研究Ben-Zvi--Sakellaridis--Venkatesh关于几何朗兰兹对偶中周期与L-函数关系的范畴化猜想，并通过导出傅里叶分析和手征代数在低秩例子中提供证据。
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ngô-Yun, and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries we obtain combinatorial identities for counts of rational curve…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。