Frobenius distributions of Drinfeld modules of any rank
Let \(A =\mathbb{F}_q[T]\) be the polynomial ring in an indeterminate \(T\) over the finite field \(\mathbb{F}_q\), with quotient field \(F =\mathbb{F}_q(T)\). Drinfeld \(A\)-modules, and in particular Drinfeld modules of rank two, offer striking similarities with elliptic curves over the rationals. For example, if \(E/\mathbb{Q}\) is an elliptic curve without complex multiplication, \textit{S. Lang} and \textit{H. Trotter} [Lect. Notes Math. 504, Springer, Berlin (1976; Zbl 0329.12015)] formulated a precise conjecture about the growth of the function \(x\longmapsto \#\{p \leq x \mid \text{tr}(E,p) = t\}\), where \(t \in \mathbb{Z}\) is fixed and \(\text{tr}(E,p)\) is the Frobenius trace of the reduction of \(E\) at the prime \(p \in {\mathbb{N}},\) neglecting the finite number of \(p\) with bad reduction of \(E\). The Lang-Trotter conjecture, suitably adapted, may be translated to Drinfeld modules. The present paper is dedicated to a proof of a result toward the ``Lang-Trotter'' conjecture on Drinfeld modules. It is as follows. Let \(\Phi\) be a Drinfeld \(A\)-module of rank \(r \in {\mathbb{N}}\) over \(F\), \(t \in A\), and put NEWLINE\[NEWLINE\pi_t(k) := \# \{\text{primes }\mathfrak p \subset A~|\text{ deg } \mathfrak p = k\text{ and tr}(\Phi,\mathfrak p) = t\}.NEWLINE\]NEWLINE Then the author's result Theorem 1.1 states that NEWLINE\[NEWLINE\pi_t(k) \ll \frac{r}{k}q^{k\theta(r)},NEWLINE\]NEWLINE where \(\theta(r) = 1-\frac{1}{2(r^2+2r)}\) depends only on \(\Phi\), which is supposed to possess no ``complex multiplications'' other than the operators in \(A\). The result, which is the first in this direction, is much weaker than the actual conjecture, both since \(k\cdot \theta(r)\) is larger than the expected correct exponent \(k(1-\frac{1}{r})\) and since it doesn't provide lower estimates of \(\pi_t(k)\). The proof is modelled after \textit{J.-P. Serre}'s proof of a corresponding result for elliptic curves [see Publ. Math., Inst. Hautes Étud. Sci. 54, 123-202 (1981; Zbl 0496.12011)]. An important ingredient (the openness of the image of Galois in the adelic representation associated with \(E\)) is presently not available for Drinfeld modules. Instead, a weaker result of \textit{R. Pink} [Ann. Math. (2) 135, 483-525 (1992; Zbl 0796.14007)] about Galois representations attached to Drinfeld modules is used.
scientific article; zbMATH DE number 1684890
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