Deligne's congruence and supersingular reduction of Drinfeld modules
Let \(A\) be the polynomial ring \(\mathbb{F}_q [T]\) over the finite field \(\mathbb{F}_q\). Drinfeld \(A\)-modules \(\varphi\) of rank two over the quotient field \(\mathbb{F}_q(T)\) share many properties with elliptic curves over the field \(\mathbb{Q}\) of rationals. In particular, there is the notion of supersingular reduction of \(\varphi\) at a prime \({\mathfrak p}\) of \(A\). It is known that, for given \(\varphi\), the number of supersingular primes \({\mathfrak p}\) is infinite with density zero, if \(\varphi\) is neither of CM-type nor ``exceptional'' [\textit{M. L. Brown}, Invent. Math. 110, 419-439 (1992; Zbl 0767.11028); \textit{C. David}, Duke Math. J. 78, 399-412 (1995; Zbl 0842.11025); \textit{B. Poonen}, Math. Res. Notices 3, 151-159 (1998; Zbl 0898.11024)]. The author proves estimates (with explicit constants) for the number \(S(k,\varphi)\) of supersingular \({\mathfrak p}\) of degree less or equal to \(k\in \mathbb{N}\). These estimates are rather weak asymptotically, where they yield \(S(k,\varphi)= O(q^{k-2}/k)\), but supply meaningful information if \(k\) is small. The proof relies on the theory of Drinfeld modular forms, notably on ``Deligne's congruence'' between the Hasse invariant of \(\varphi\) reduced at \({\mathfrak p}\) and the Eisenstein series of weight \(q^{\deg {\mathfrak p}}-1\).
scientific article; zbMATH DE number 1309372