# Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the variety, which is comparible to the (Hasse) zeta function of a proper, regular model the variety over $\mathcal{O}_K$. I will refer to the Hasse-Weil L-functions appearing in the factorization of the zeta function as L-factors of the zeta function. For example, when all the special fibres are smooth, the two zeta functions are equal. In general they disagree at finitely many (bad) primes. One is interested in basic analytic questions, such as meromorphic continuation and functional equation.

One way in which these questions can be addressed is by proving each Hasse-Weil L-factor is automorphic. According to Langlands in the short article "where stands functoriality today?", there are some deficiencies with this strategy (see, for example, his remarks in section $6$).

Bearing in mind the difficulty of these questions, I have found myself trying to study the zeta functions through methods that do not rely on automorphicity of the L-factors. One thing I have always taken for granted is that we do not expect the zeta function itself to be automorphic. But, if I am honest, I do not think I understand why not. In an attempt to improve this, I am lead to three several specific questions:

1) Are there explicit examples of varieties whose zeta function is (respectively is not) the L-function of an automorphic representation?

2) Is the first option ("is") ever expected to be possible?

2) What is wrong with the idea of trying to understand Hasse-Weil zeta functions as associated to "virtual" automorphic representations?

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I was completely confused when I first read this by your usage of "L-factor". The usual usage of "L-factor" is to mean the factor of an Euler product at some specific place of $K$. But you're using it, if I understand correctly, to mean the factors in the expression for the zeta function as an alternating product of $L$-functions associated to the cohomology groups of the variety in each degree $0 \le i \le 2 \operatorname{dim} X$. –  David Loeffler Oct 30 '13 at 14:45
Yes, I am sorry, that is what I mean. In the article I mentioned, Langlands also uses the term "factor", and I couldn't think of a better term. In my confusing terminology, the L-factors are what I denote $L(H^m,s)$ in this question: mathoverflow.net/questions/144285/… –  Tom163 Oct 30 '13 at 15:40