首页/论文/math/0206001

arXiv math/02060012002-06-25math.AGmath.NT

Independence of l in Lafforgue's theorem

CheeWhye Chin

备注：: 19 pages, AMSTeX; revised version 2 to appear in Advances in Mathematics
发表信息：: Adv. Math. 180 (2003), no. 1, 64--86
DOI：: 10.1016/S0001-8708(02)00082-8

英文摘要：

Let X be a smooth curve over a finite field of characteristic p, let l be a prime number different from p, and let L be an irreducible lisse l-adic sheaf on X whose determinant is of finite order. By a theorem of Lafforgue, for each prime number l' different from p, there exists an irreducible lisse l'-adic sheaf L' on X which is compatible with L, in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for L and L' are equal. We prove an "independence of l" assertion on the fields of definition of these irreducible l'-adic sheaves L' : namely, that there exists a number field F such that for any prime number l' different from p, the l'-adic sheaf L' above is defined over the completion of F at one of its l'-adic places.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

The work of Laurent Lafforgue

Laurent Lafforgue has been awarded the Fields Medal for his proof of the Langlands correspondence for the full linear groups $\mathop{\rm GL}\nolimits_{r}$ ($r\geq 1$) over function fields. This article is a brief introduction to the Langlands correspondence and to Lafforgue's theorem.

2002-12-01math.AGmath.NT

关于 Lafforgue 的一个定理

On a theorem of Lafforgue

本文给出了 Laurent Lafforgue 关于刚性拟阵在任何给定域上仅有有限多个表示射影等价类这一民间定理的新证明及若干推广。

2023-09-11math.AGmath.CO

A rank inequality for the Tate Conjecture over global function fields

Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be wri…

2015-08-08math.AGmath.NT

Independence of ell of Monodromy Groups

Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and consider an E-compatible system of lisse sheaves on the curve X. For each place lambda of E not lying over p, the lambda-component of the system is a lisse E_lambda-sheaf on X, whose associated arithmetic monodromy group is an algebraic group over the local field E_lambda. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the E-compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these m…

2004-07-31math.AGmath.NT

A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)

Revised: just some typos, reorganized a bit the article. It will be published in the VIASM Annual meeting, Hanoi. We give a detailed account of Deligne's letter to Drinfeld dated June 18, 2011, in which he shows that there are finitely many irreducible lisse $\bar \Q_\ell$-sheaves with bounded ramification, up to isomorphism and up to twist, on a smooth variety defined over a finite field. The proof relies on Lafforgue's Langlands correspondence over curves. In addition, Deligne shows the existence of affine moduli of finite type over $\mathbb{Q}$. A corollary of Deligne's finiteness theorem i…

2012-11-29math.AGmath.NT

Independence of $\ell$ for the supports in the Decomposition Theorem

In this note, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the Decomposition Theorem, as well as for the family of local systems on each support. It generalizes Gabber's result on the independence of $\ell$ of intersection cohomology to the relative case.

2018-08-15math.AG

更多相关论文收起更多论文

Algebraic independence for values of integral curves

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series $E_2,E_4,E_6$. The main technical improvement in our a…

2018-05-17math.AGmath.NT

A Linear independence result for $p$-adic $L$-values

The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $χ$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,χω^{1-i}) K )\geq \frac{(1-ε)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.

2020-06-24math.AGmath.NT

On l-independence for the etale cohomology of rigid spaces over local fields

We investigate the action of the Weil group on the compactly supported l-adic etale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of l when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on l-independence for nearby cycle cohomology, which leads to our l-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of R. Huber, which requires the restriction on the c…

2006-02-13math.AGmath.NT

Tropical Independence I: Shapes of Divisors and a Proof of the Gieseker-Petri Theorem

We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.

2014-11-02math.AG

Skew-invariant curves and the algebraic independence of Mahler functions

For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ an…

2022-03-09math.AGmath.DSmath.NT

On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

Let $Π$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $λ$ of the field of algebraic numbers which is prime to p, consider the $λ$-adic pro-semisimple completion of $Π$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $λ$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its c…

2018-01-18math.AGmath.NTmath.RT