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Meromorphic continuation of $$L$$-functions of $$p$$-adic representations. (English) Zbl 0868.14011
Let $$p$$ be a prime number and let $$\mathbb{F}_q$$, $$q=p^n$$, be the field with $$q$$ elements. Let $$\mathbb{Q}_p$$ be the field of $$p$$-adic numbers with $$\overline {\mathbb{Q}}_p$$ an algebraic closure and $$\Omega_p$$ a completion of $$\overline {\mathbb{Q}}_p$$. Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$ contained in $$\Omega_p$$ and let $$R_p$$ be the ring of integers of $$K$$. Let $$X=(X_1, X_2,\dots, X_n)$$ be a system of indeterminates. Let $$R_p\{\{X\}\}$$ be the ring of power series in $$\{X_1,\dots, X_n\}$$ with coefficients in $$R_p$$ and which converge in the closed unit polydisc in $$\Omega^n_p$$. Let $$B(X)$$ be an $$r\times r$$ matrix with coefficients in $$R_p\{\{X\}\}$$ and let $$\mathbb{A}^n/\mathbb{F}_q$$ be the affine $$n$$-space. For each closed point $$\overline {x}$$ of $$\mathbb{A}^n$$, we let $$x$$ be a Teichmüller lifting to characteristic 0 (i.e., we lift some geometric point which lies above $$\overline {x}$$). Finally, let $$d(x)$$ be the degree of $$\overline {x}$$ over $$\mathbb{F}_q$$. One then defines the $$L$$-function of $$B(X)$$ on $$\mathbb{A}^n/\mathbb{F}_q$$ by $L(B/\mathbb{A}^n,u):= \prod_{\overline {x} \text{ closed}} \text{det}\bigl( 1-u^{d(x)} B(x^{q^{d(x)-1}})\cdots B(x^q)B(x) \bigr)^{-1}.$ Since both $$N\times M$$ and $$M\times N$$ have the same characteristic polynomials for arbitrary $$r\times r$$ matrices $$M$$, $$N$$, it follows directly that the above definition does not depend on the choice of the Teichmüller lifting of $$\overline{x}$$.
If the matrix $$B(X)$$ is invertible upon embedding $$R\{\{X\}\}$$ into $$R\{\{X\}\}\otimes \mathbb{Q}_p$$, then the matrix $$B$$ arises from an $$F$$-crystal and the $$L$$-function is the $$L$$-function of the $$F$$-crystal. In this case, N. M. Katz conjectured in the early 1970’s that the $$L$$-function is a $$p$$-adic meromorphic function.
In this important paper the author shows that the general form of this conjecture is false by exhibiting explicit counter-examples. It is also shown that if the coefficients of $$B(X)$$ are actually polynomials, then Katz’s conjecture is indeed true and, in fact, the $$L$$-series is a rational function in $$u$$. The author’s main tool is the Dwork trace formula. The interested reader would do well to at least read the introduction to this paper where the history of the conjecture and a summary of the author’s results are well presented.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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