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Trivial \(L\)-functions for the rational function field. (English) Zbl 1318.11147

From the text: In this paper, the author describes a number of interesting \(l\)-adic representations \(V\) of the Galois group of the rational function field with trivial \(L\)-function: \(L(V,s)=1\).
Let \(k\) be a global function field, over the finite field \(E\) with \(q\) elements. Let \(k^s\) be a separable closure of \(k\), and let \(E^s\) be the separable closure of \(E\) in \(k^s\).
The \(L\)-function \(L(V, s)\) of a semi-simple \(l\)-adic Galois representation \(\mathrm{Gal}(k^s/k) \rightarrow \mathrm{GL}(V)\) contains very little of the local information involved in its definition. Indeed, the cancellation of the local terms in the infinite Euler product defining \(L(V, s)\) ultimately results in a function which \(a\) is quotient of two polynomials in \(q^{-s}\). This led Weil to define the notion of a (formal) Dirichlet series belonging to \(k\), which keeps track of the local terms.
In this paper, the author studies an extreme case of cancellation over the rational function field \(k = E(T)\). If we assume that the geometric Galois group \(\mathrm{Gal}(k^s/kE^s)\) has no non-trivial invariants on \(V\) and that the degree \(f(V)\) of the Artin conductor of \(V\) is twice the dimension of \(V\), then \(L(V, s)\) is a polynomial of degree 0 in \(q^{-s}\) with constant coefficient 1. Hence \(L(V, s) = 1\) is a constant function! We call these trivial \(L\)-functions for the rational function field (although they arise in many non-trivial situations).
After a brief introduction to the cohomological theory of Weil, Grothendieck, and Deligne, he presents several examples of Galois representations of the rational function field \(k = E(T)\) with trivial \(L\)-functions. In all of his irreducible examples, the representation \(V\) remains geometrically irreducible and the set \(S\) of ramified places is contained in \(\{\infty, 0, 1\} \subset \mathbb P^1(E)\). More precisely, \(V\) corresponds either to an irreducible representation of \(\pi_1(\mathbb G_m) = \pi_1(\mathbb P^1 - \{\infty, 0\})\) which is tamely ramified at \(T=0\), or to an irreducible representation of \(\pi_1(\mathbb P^1 - \{\infty, 0, 1\})\) which is tamely ramified at all three places.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R58 Arithmetic theory of algebraic function fields
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