# on the conductor of a scheme over $\mathbb{Z}$

Let $X$ be a regular, projective flat scheme over $Spec(\mathbb{Z})$, of relative dimension $d$, and look at the $L$-function $L(X, s)$, that is, the Hasse-Weil zeta function completed by the Gamma factors.

Conjecturally, it satisfies a functional equation $$L(X, s)=\varepsilon(X) A(X)^{-s} L(X, d+1-s)$$ where $\varepsilon(X)$ and $A(X)$ (the so called conductor) are real numbers which can be defined unconditionally.

1) How is the conductor defined?

2) It is a theorem or essentially part of the definition that $X$ has everywhere good reduction if and only if $|A(X)|=1$?

1) The conductor measures ramification in the Galois representations of $X$. It is the product (or alternating product?) of the Artin conductors of all these Galois representations.
2) No. Very much not so. $A(X)$ only measures bad reduction in the Galois representation of $X$, or it is a property only of the $L$-function. But different varieties can share the same Galois representation and $L$-function, and some of the varieties can have good reduction and some not. The simplest example is genus $0$ curves. These all have the same $L$-fucntion, and so the same conductor, $1$, but for every finite set of primes there is a genus $0$ curve with bad reduction at those priems.
• I think so. Also, the $L$-function is a motivic invariant, and the equation in the question enables you to derive the conductor from the $L$-function, so it had better be. Jan 25, 2014 at 0:52