Zeros of Eisenstein series, quadratic class numbers and supersingularity for rational function fields
Let \(\mathbb{F}_q\) be the finite field with \(q\)-elements and let \(A= \mathbb{F}_q[t]\) be the 1-variable polynomial ring in one indeterminate with quotient field \(k = \mathbb{F}_q(t)\). The completion of the field \(k\) at the place given by infinity is \(k_\infty = \mathbb{F}_q((1/t))\); one lets \(C\) denote the completion of a fixed algebraic closure of \(k_\infty\) equipped with its canonical valuation. The set \(C - k_\infty\) is a connected rigid analytic space with an action of \(Gl_2(A)\) on it in the standard fashion. The Eisenstein series \(E_k(z)\) are defined, in analogy with classical theory, by \(E_k(z) :=\sum_{(0,0)\neq (c,d)\in A^2}(cz + d)^{-k}\); they are non-trivial precisely when \(k\) is divisible by \(q -1\). In the paper being reviewed, the author establishes a number of interesting results about the zeros of such functions. Such results are not known in the classical setting, but the author does believe that much of what he has discovered will carry over.
scientific article; zbMATH DE number 1337030
scientific article