Supersingular reduction of Drinfel'd modules
Let \(q\) be an odd prime power, \(\mathbb{F}_q\) the field with \(q\) elements, \(R= \mathbb{F}_q [T]\) and \(F= \mathbb{F}_q (T)\) the ring of polynomials and the field of rational functions, respectively, in an indeterminate \(T\) over \(\mathbb{F}_q\). It is well known that Drinfeld \(R\)-modules \(\varphi\) of rank two behave in many respects like elliptic curves. E.g. there is a modular \(j\)-invariant that classifies \(\varphi\) over algebraically closed \(R\)-fields, a theory of Drinfeld modular forms, notions of singular (or ``complex multiplication'') and supersingular Drinfeld modules, etc., all of which are quite parallel to their classical counterparts. Let now \(\varphi\) be defined over \(F\), with invariant \(j_\varphi\in F\), and without complex multiplication. Following Elkies' (i.e., Euclid's) idea in the elliptic case, \textit{M. L. Brown} [Bull. Lond. Math. Soc. 20, No. 4, 293-296 (1988; Zbl 0654.14019)]\ proved: The number \(N_\varphi (x)= \# \{\)primes \(p\) of \(R\) of norm \(\leq x\) s.t. \(\varphi\) has good supersingular reduction at \(p\}\) behaves like \((*)\) \(N_\varphi (x) \gg \log \log \log x\), provided that a) \(\varphi\) is not exceptional ((loc. cit.) Definition 1.1.5) and b) \(q\) is odd. In the present article, the estimate \((*)\) is sharpened (under the same assumptions) to \(N_\varphi (x)\gg \log \log x\), where the main progress comes from a better bound (theorem 4.1, compared to (loc. cit.) 4.2.12) for the quantity \(N_d (\varphi)=\) numerator of \(P_d (j_\varphi)\). Here \(P_d (X)\) is the polynomial \(P_d (X)= \prod (X- j_\nu)\), where \(j_\nu\) runs through the singular invariants associated with the fundamental discriminant \(d\). Many intriguing questions remain: -- What is the precise asymptotic behavior of \(N_\varphi (x)\)? -- Can one give similar estimates if \(\varphi\) is exceptional or \(q\) is even? (The case of even \(q\) differs significantly from the treated case ``\(q\) odd'', which is modelled after the classical case.) -- Is there a simple rule that governs the distribution of Frobenius elements \(\varphi_p\) at places \(p\) of good reduction of \(\varphi\)? In other words, is there some kind of Tate-Sato conjecture for Drinfeld modules? -- What about generalizations: to higher-rank Drinfeld modules, or to Drinfeld R-modules, where \(R\) is a more complex \(\mathbb{F}_q\)-algebra than \(\mathbb{F}_q [T]\)?
scientific article; zbMATH DE number 815768