刚性几何函数L（Ⅰ）：F-模的可转换性或超可转换性及Dwork猜想
Fonctions L en géométrie rigide I: F-modules convergents ou surconvergents et conjecture de Dwork
摘要：
本文是专门讨论L-函数的三篇系列文章中的第一篇。本文利用Teichmüller提升定义了收敛或过收敛F-模的L-函数，并建立了闭单位圆盘上收敛F-模的L-函数的亚纯性。Wan在一系列的三篇文章中建立了Dwork猜想，由于Katz的等距定理，他的证明被简化为普通的情形：这里我们在与椭圆曲线族有关的两个显式例子上证明了普通过收敛F-模上的斜率滤子不提升为过收敛滤子.作为一个副产品，我们证明了单位根的次F-同晶的de Rham上同调的勒让德家庭的正常椭圆曲线是不是过收敛的Berthelot的意义。在第二篇文章中，我们将用上同调的方法给出F-（iso）晶体的L-函数的定义，并说明它与本文给出的定义是如何匹配的：它回复了Katz或Etesse在晶体上同调中所使用的定义，或回复了Etesse-Le Stum在刚性上同调中所使用的定义，或回复了Wan所使用的定义;目的是利用刚性上同调证明Katz猜想关于这些L-函数的p-adic单位根和极点.在第三篇文章中，我们给出了这些结果的一个显式形式。
英文摘要：
This article is the first one of a series of three articles devoted to L-functions. In this one we give a definition of the L-functions of convergent or overconvergent F-modules with the help of Teichmüller liftings and we establish the meromorphy of the L-functions of convergent F-modules in the closed unit disk. Wan has established Dwork conjecture in a series of three articles; owing to an isogeny theorem of Katz, his proof reduces to the ordinary case: here we prove, on two explicit examples related to families of elliptic curves, that the slope filtration on an ordinary overconvergent F-module does'nt lift to an overconvergent filtration. As a by-product we show that the unit-root sub-F-isocrystal of the de Rham cohomology of the Legendre family of ordinary elliptic curves is not overconvergent in Berthelot's sense. In the second article we'll give a definition of the L-functions of F-(iso)crystals by cohomological means and we'll show how it matches with the one given here: it gives back the one used in crystalline cohomology by Katz or Etesse, or the one used in rigid cohomology by Etesse-Le Stum, or the one used by Wan ; the aim is then to give a proof of Katz conjecture on p-adic unit roots and poles of these L-functions using rigid cohomology. In the third article we give an explicit form of these results for ordinary abelian schemes.
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