Is the conductor of an L-function F the absolute value of the discriminant ofsome number field related to F?

Question

In the theory of automorphic forms, ramified primes of an L-function divide the so-called conductor thereof. On the other hand, one can define for a number field $ K $ an integral invariant $ \Delta_{K} $ equal to the square of the determinant of some square matrix defined through a basis of the ring of integers $ O_{K} $ and complex embeddings. It appears that the conductor of the Dedekind zeta function of a number field $ K $ is $ \vert \Delta_{K}\vert $.

Hence my question : is it possible for any L-function $ F $ to define a related number field whose absolute value of the discriminant is the conductor of $ F $? If yes, how is it defined ?

Answer 1

$\ $ There is a terribly ineffective procedure for Dedekind and Artin L-functions, but it is worth looking at it anyway:

For $f(x)=\sum_{n=0}^N a_n x^n \in \mathbb{Z}[x]$, let $v(f) = (N,\sum_{n=0}^{N} |a_n|, a_0,\ldots,a_N,0,\ldots) \in \mathbb{Z}^{\infty}$. Applying the lexical order to $v(f)$ we obtain an order on the integer polynomials satisfying the finite descending chain condition.

Then assume we are given the Euler product of a Dedekind zeta function $\zeta(K,s)=\prod_p\zeta_p(K,s)$ with $K$ an unknown number field.

Pick a primitive element $\alpha$ ($K = \mathbb{Q}(\alpha)$) and let $h \in \mathbb{Z}[x]$ be its minimal polynomial.

For $n=1,2,3,\ldots\ $ find the least irreducible polynomial $f_n \in \mathbb{Z}[x]$ satisfying $$\forall p \le n, \qquad \zeta_p(\mathbb{Q}[x]/(f_n),s)=\zeta_p(K,s)$$

The sequence $f_n$ is increasing in the order on monic polynomials, and it is bounded above by $h$.

Thus $f_n$ converges, to a polynomial $f$ satisfying $$\zeta(\mathbb{Q}[x]/(f),s)=\zeta(K,s)$$

Do you think $\mathbb{Q}[x]/(f)$ and $ K$ are isomorphic ? What if $K/\mathbb{Q}$ is Galois ?

For $L(s,\rho,K)$ an Artin L-function, the idea is the same, comparing the factors of $L(s,\rho,K)$ with that of each Artin L-function $L(s,\psi,\mathbb{Q}[x]/(f_n))$.

Restricting to $f_n$ irreducible is for simplicity. If $f_n$ isn't irreducible then $\mathbb{Q}[x]/(f_n(x))$ isn't a field and its ring of integers is ill-defined. But we can still look at the ring structure of $\mathbb{Z}[x]/(f_n(x),p)$ to create (for the $p$ where $f_n$ has $N$ distinct roots) an Euler factor and compare it with $\zeta_p(K,s)$, obtaining that $f_n$ converges to a polynomial $f$ satisfying $\zeta_p(K,s) = \zeta_p(\mathbb{Z}[x]/(f),s)$ for every $p$ unramified in $O_K$ and $\mathbb{Z}[x]/(f)$.

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2nd part : proving that some L-functions are not Artin L-functions. A conjecture predicts automorphic L-functions $= $ motivic L-functions.