On root numbers connected with special values of \(L\)-functions over \(\mathbb{F}_ q(T)\)
In the paper ``Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)'' [J. Number Theory 60, 165-209 (1996; Zbl 0960.52407)], \textit{G. Anderson} introduced a number of very interesting objects of study in the arithmetic of the polynomial algebra \(\mathbb{F}_q[T]\). Among these are ``special points'', and certain ``root numbers'' which play a role not very different from classical root numbers. Now in classical number theory, root numbers are intimately related to Gauss sums and it is natural to wonder if there is a relationship between Anderson's root numbers and Thakur's Gauss sums for function fields. In this important paper such a connection is presented. Moreover, the answer is quite intriguing and leaves one wondering if something remarkable underlies it. More precisely, Thakur's Gauss sums are made up in the ``Carlitzian'' fashion from ``basic Gauss sums''. That is, for each \(j\) one has a basic Gauss sum \(g(j)\), which is a sum very similar to that of classical arithmetic. Then if \(i\) is an integer written \(q\)-adically as \(\sum^t_{j=0} c_jq^j\), \(0\leq c_j\leq q-1\) for all \(j\), one defines \(g(i):= \prod^t_{j=0} g(j)^{c_j}\). What is so remarkable about the author's results, as illustrated by his Example 2, is that the root numbers are expressed as products of the basic Gauss sums, but where the powers may be strictly greater than \(q-1\). This is most peculiar! Motivated by the present paper, Anderson went on to give another, equally intriguing, basis for the special points. We refer the reader to Section 10.6 of ``Basic structures of function field arithmetic'' [\textit{D. Goss}, Springer (1996; Zbl 0874.11004)]. One again wonders what underlying phenomenon explains these remarkable formulas.
scientific article; zbMATH DE number 992063
scientific article