Singular moduli and supersingular moduli of Drinfeld modules
Let \({\mathcal C}\) be a smooth, projective curve over the finite field \(\mathbb{F}_ r\), \(r=p^ n\) where we assume that \(\mathbb{F}_ r\) is the full field of constants on \({\mathcal C}\). Let \(\infty\) be a fixed place of \({\mathcal C}\) and let \({\mathbf A}\) be the affine ring of functions regular on \({\mathcal C}-\infty\). So \({\mathbf A}\) is a Dedekind ring with finite class group and unit group \(\simeq\mathbb{F}_ r^*\); we denote its field of fractions by \({\mathbf k}\). By its very construction, the ring \({\mathbf A}\) is analogous to \(\mathbb{Z}\), or the integers \({\mathfrak O}\) in a complex quadratic field, in that it has a unique prime at \(\infty\). The prototypical example of such \({\mathbf A}\) is the polynomial algebra \(\mathbb{F}_ r[T]\). For \(\mathbb{Z}\) one has lattices \((\subset\mathbb{C})\) of rank 1 or 2; for a complex quadratic field one has \({\mathfrak O}\)-lattices (again \(\subset\mathbb{C})\) of \({\mathfrak O}\)-rank 1 (and thus of \(\mathbb{Z}\)-rank 2). It is therefore reasonable to look for ``\({\mathbf A}\)-lattices'' contained in the algebraic closure of \({\mathbf k}_ \infty\); such objects are easy to find and lead to algebraic entities called ``elliptic \({\mathbf A}\)-modules''. The theory of elliptic \({\mathbf A}\)-modules (or, more commonly, ``Drinfeld modules'') was inaugurated in 1973 in \textit{V. G. Drinfel'd}'s paper of the same name [Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]; however a very important specific rank 1 example was worked out by \textit{L. Carlitz} in the 1930's [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)]. This example is called, naturally, the ``Carlitz module'' and it acts as a very suitable analog of \(\mathbb{G}_ m\) for \(\mathbb{F}_ r[T]\). In general, a Drinfeld module \(\psi\) over a field equips the field with a new exotic action of \({\mathbf A}\); it thus makes sense to discuss division points of \(\psi\). If the fields of ``finite characteristic \({\mathfrak P}\)'' (i.e., the structure map from \({\mathbf A}\) to the field factors through \({\mathbf A}/{\mathfrak P})\), then one says that \(\psi\) is ``supersingular'' if the \({\mathbf A}\)-module of \({\mathfrak P}\)-division points over the algebraic closure is trivial; clearly this is the direct analog of the concept for elliptic curves. We now fix \({\mathbf A}\) to be \(\mathbb{F}_ r[T]\), as does the paper being reviewed, and we focus on rank 2 Drinfeld modules -- the rank of a Drinfeld module being the rank of the associated lattice. Let \(L\) be a finite extension of \({\mathbf k}\) and let \(\tau=\tau_ r\) be the \(r\)th power mapping \(\tau(x)=x^ r\). A Drinfeld module \(\varphi\) of rank 2 over \(L\) is described by the additive polynomial \[ \varphi_ T=T\tau^ 0+g\tau+\Delta \tau^ 2=Tx+gx^ r+\Delta x^{r^ 2}, \qquad \Delta\neq 0. \] The element \(g\) is a ``modular form'' (the definition being the analog of the classical definition involving lattices) of weight \(r-1\) and \(\Delta\) is a form of weight \(r^ 2-1\); they thus can be described as rigid analytic functions on the uniformizing (rigid) space \(\Omega^ 2\) with classical style invariance properties. Such forms were originally described, and the existence of \(q\)-expansions -- involving the reciprocal of the Carlitz exponential (these are called ``\(t\)- expansions'' by the author) -- established [in Compos. Math. 41, 3-38 (1980; Zbl 0388.10020)]. For a ``measure theoretic'' approach to such modular forms see \textit{J. Teitelbaum} [Rigid analytic modular forms: An integral transform approach, in ``The arithmetic of function fields'', de Gruyter, 189-207 (1992; Zbl 0802.11024)], and related papers in that volume. The forms \(g\) and \(\Delta\) maybe expressed in terms of Eisenstein series and so their \(q\)-expansions may be calculated. However, as in classical theory, the \(q\)-expansion for \(\Delta\) may also be given a very beautiful product expansion [see \textit{E.-U. Gekeler}, J. Number Theory 21, 135-140 (1985; Zbl 0572.10021)]. For a prime \(v\in\text{Spec}({\mathbf A})\), set \(n(v)=|{\mathbf A}/v|\), and let \(\varphi^ v\) be the reduction of \(\varphi\) at \(v\). Let \(z\) be an \(r\)th power; following the author we define \(N_ \varphi(z)=|\{v\in \text{Spec}({\mathbf A})|\) \(\varphi^ v\) is supersingular and \(n(v)\leq z\}|\). If \(\varphi\) has complex multiplication by an ``imaginary'' quadratic extension \(F\) of \({\mathbf k}\), then classical results imply \(N_ \varphi(z)\sim z/2\log_ rz\) as \(z\to\infty\) when \(F\neq\mathbb{F}_{r^ 2}(T)\); for \(F=\mathbb{F}_{r^ 2}(T)\) there is a similar result. Suppose now that \(\varphi\) does not have complex multiplication. Following the author we say that the \(j\)- invariant, \(j(\varphi)\in{\mathbf k}\), is exceptional if and only if (1) \(r\equiv 1\pmod 4\), (2) \(j(\varphi)\) is a square in the completion \({\mathbf k}_ \infty\), and (3) the prime factors of the numerator of \(j(\varphi)\) of even degree \(>0\) have even multiplicities. The author is then able to establish the following important result: Theorem: Suppose that \(j(\varphi)\) is not exceptional and that \(p\neq 2\). Then we have \(N_ \varphi(z)\gg\log\log\log z\) as \(z\to\infty\). While this is far from what the author mentions as expected, it does have the very important corollary that most Drinfeld modules of rank 2 as above have infinitely many places of supersingular reduction. The proof of the theorem is motivated by \textit{N. Elkies'} paper [Invent. Math. 89, 561-567 (1987; Zbl 0631.14024)], which, in turn, is a ``rank 2'' version of the classical Jensen theorem on the existence of infinitely many irregular primes. The proof of the author's theorem involves delicate estimates on modular forms, quadratic reciprocity, the Chebotarev density theorem, etc.
scientific article; zbMATH DE number 167403