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Regular singularity of Drinfeld modules. (English) Zbl 0806.11029

Let \(\mathbb{F}_ q\) be the finite field with \(q=p^ n\) elements. Let \(K\) be a field containing \(\mathbb{F}_ q\) and let \(\tau: K\to K\) be the map \(x\mapsto x^ q\). So \(\tau\) gives rise to an embedding of \(K\) into itself which is surjective if and only if \(K\) is perfect. Let \(\overline {K}\) be a fixed algebraic closure of \(K\) with the obvious extension of \(\tau\). Let \(P:= P(\tau)\) be a polynomial in \(\tau\) with coefficients over \(K\); we view \(P\) as an \(\mathbb{F}_ q\)-linear operator on \(\overline {K}\). The kernel of \(P\) is then an \(\mathbb{F}_ q\)-vector space of dimension at most the degree of \(P\); there is equality if and only if the constant term of \(P\) is nonvanishing.
In an analogous fashion, let \(D=d/dx\) and \(P(D)\) a complex differential operator. Then, under good conditions, the solutions to \(P(D)y=0\) form a finite dimensional space over \(\mathbb{C}\). This analogy, which goes back at least to O. Ore, can be pushed remarkably far. For instance, let \(\{x_ 1\dots, x_ n\}\) be \(n\) elements of \(K\). Then the determinant \(\Delta:= \text{det} (x_ i^{q^{j-1}})_{1\leq i,j\leq n}\) is nonzero if and only if \(\{x_ 1,\dots, x_ n\}\) is a linearly independent set over \(\mathbb{F}_ q\). This determinant, which plays the role of the Wronskian, is called “the Moore determinant”. There is a closed form expression for \(\Delta\), due to Moore, which is analogous to Abel’s formula for the Wronskian; in fact, it can be derived in a matter similar to that of Abel.
The constants of \(D\) are obviously obtained by solving the differential equation \(Dy=0\). However, the “constants” \(\mathbb{F}_ q\) of \(\tau\) are obtained by solving \(\tau y=y\); 0 obviously being the only solution to \(\tau y=0\). If one looks for an operator \(\delta\) with \(\mathbb{F}_ q= \{y\mid \delta y=0\}\), one is clearly led to the Artin-Schreier-like operator \(\delta y:= y^ q-y\). It turns out that this operator possesses many pleasing derivation-type formulae. For instance, one sees immediately that \(\delta(xy)= x\delta y+\tau(y) \delta x\), (so \(\delta\) is a “\(\tau\)-derivation”); if one sets for \(u\in K- \mathbb{F}_ q\) and \(y\in K\) \(\delta y/\delta u:= (\tau y-y)/ (\tau u-u)\), then one has the “chain rule” for \(x\in K- \mathbb{F}_ q\) \[ {{\delta y} \over {\delta x}} = {{\delta y} \over {\delta u}} {{\delta u} \over {\delta x}}. \] For more results on this “Galois calculus” (some of a much less elementary flavor) we refer the reader to Y. Hellegouarch [in ‘The arithmetic of function fields’, de Gruyter, 33-50 (1992; Zbl 0794.11025)]. In fact, Hellegouarch points out (using a cohomological calculation) that every \(\tau\)-derivation is of the form \(\lambda\delta\) for \(\lambda\in K\); thus, as might be expected, the space of \(\tau\)-derivations is one dimensional.
Using \(\delta\), and its composites, one can construct a “Wronskian” for elements \(\{x_ 1,\dots, x_ n\}\) in the obvious fashion; it is an elementary calculation that this determinant agrees with the Moore determinant. Moreover, for \(\lambda\in K\) let \(\Psi_ \lambda\) be the \(2\times 2\) matrix \({{\lambda\;\delta\lambda} \choose {0\;\tau\lambda}}\).
Then the fact that \(\delta\) is a \(\tau\)-derivation is equivalent to having \(\Psi\) be an \(\mathbb{F}_ q\)-algebra injection of \(K\) into \(M_ 2(K)\). This is similar to what occurs in the de Rham theory of Drinfeld modules [see, e.g., the reviewer, in Proc. Symp. Pure Math., Vol. 55, Pt. 2, 309-362 (1994)].
With the analogies between \(D\) and \(\tau\) in mind, it is natural to pursue other “differential structures” in the theory of additive polynomials and Drinfeld modules. One such structure would be that of an “additive polynomial regular at a place \(v\)” as analog of “regular singular point of an ordinary differential equation”. The author studies such structures for Drinfeld modules and “\(\varphi\)-modules” (roughly \(\varphi\)-modules stand in the same relationship to the operator \(\tau\) as connections on coherent modules do to \(D\)). A number of very nice characterizations of regularity are then presented.
Finally, it is a very curious thing that the arithmetic of function fields is awash with various types of differential formalism. We can think of at least three distinct types:
1. The usual differential formalism of \(D\), supplemented through the use of hyper-derivatives.
2. The Frobenius formalism of \(\delta\) as above, and the somewhat similar de Rham formalism of Drinfeld modules.
3. The commutator formalism on additive polynomials \((f\mapsto [g,f]\), for some additive \(g\)). These results are used, e.g., to define analogs of classical hypergeometric functions [see D. Thakur, Hypergeometric functions for function fields, Finite Fields Appl. (to appear)]. It would be very interesting to understand just how these operators interact.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11T99 Finite fields and commutative rings (number-theoretic aspects)

Citations:

Zbl 0794.11025
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