# 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?

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

• 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 $0 \le i \le 2 \operatorname{dim} X$. Oct 30, 2013 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/… Oct 30, 2013 at 15:40

The problem is with the premise that "we do not expect the zeta function itself to be automorphic".

The analytic properties of the Hasse-Weil zetas and L-functions are closely related, and both of them are definitely conjectured to be automorphic in the usual sense.

In particular, the Hasse-Weil zeta function of an elliptic curve over $\mathbb{Q}$ gives alredy a positive answer to questions (1) and (2).

The issue of for which varieties the Hasse-Weil L-function equals a single automorphic L-function or when the automorphic forms involved are cuspidal is much more delicate and conjectural.

To sum up, all Hasse-Weil zeta functions are expected to be a product of standard automorphic L-functions.

• Indeed. Perhaps many younger people do not appreciate that there is very little chance of proving meromorphic continuation of Dirichlet series with Euler products if they're not (in the most congenial, known, understood, ... class of) automorphic $L$-functions. That is, there's simply no "general method" to prove analytic continuation otherwise. From a defensible viewpoint, barring proof of the most grandiose form of "Functoriality", we simply have no devices to prove that interesting zeta/L -functions have meromorphic continuations... and, indeed, we must not be naive: Estermann phenomenon! Aug 9, 2016 at 23:01
• ... and one should definitely mention N. Kurokawa's mid-1980s work on natural boundaries... Aug 9, 2016 at 23:05
• Just to clarify, is "The issue of for which varieties the Hasse-Weil L-function equals a single automorphic L-function or when the automorphic forms involved are cuspidal is much more delicate and conjectural." the whole point of motives?
– Pig
Aug 10, 2016 at 2:29