Zur Arithmetik von Drinfeld-Moduln
The theory of modular curves over global function fields has been developed by \textit{V. G. Drinfel'd} in his paper ''Elliptic modules'' [Math. USSR, Sb. 23 (1974), 561-592 (1976); translation from Mat. Sb. Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]. In the paper under review, Drinfeld's elliptic modules are referred as Drinfeld modules, and endomorphism rings of Drinfeld modules are extensively investigated. First, Drinfeld modules are defined: Algebraic as well as analytic descriptions are given. Let K be the global function field over a finite field \({\mathbb{F}}_ q\) of characteristic p, \(\infty\) a fixed place, A the ring of functions integral outside \(\infty\), \(K_{\infty}\) the completion of K with respect to the absolute value \(||_{\infty}=||\), \(\bar K_{\infty}\) the algebraic closure of \(K_{\infty}\) and \(C=\hat{\bar K}_{\infty}\) the completion of \(\bar K_{\infty}\). Let L be a field with A-algebra structure \(\gamma:A\to L.\) Let \({\mathbb{G}}_ a\) denote the additive group scheme, and let \(End_ L({\mathbb{G}}_ a)=:L\{\tau \}\) denote the ring of non- commutative polynomials over L in one variable \(\tau\) with the commutation relation \(f^ p\tau =\tau f\) (\(f\in L).\) Algebraic description: A ''Drinfeld module over L'' of rank \(r\in {\mathbb{N}}\) is a ring homomorphism \(\phi:A\to End_ L({\mathbb{G}}_ a)\) with the following properties: (a) for all \(a\in A\), \(\phi(a)=\phi_ a\) is an additive polynomial with linear coefficient \(\gamma(a)\), and (b) for all \(a\in A\), \(\| a\| =:\deg \phi_ a=| a|^ r.\) A ''homomorphism'' of Drinfeld modules \(\phi\to \psi\) is an element \(u\in End_ L({\mathbb{G}}_ a)\) such that for all \(a\in A\), \(u{\mathbb{O}}\phi_ a=\psi_ a{\mathbb{O}}u\); and u is called an ''isomorphism'' if \(u\in L^*\). \(\phi\) has the following properties: Let \(End(\phi)=End_ L(\phi)=\{u\in End_ L({\mathbb{G}}_ a)| \quad u{\mathbb{O}}\phi =\phi {\mathbb{O}}u\}.\) Then En\(d(\phi)\) is a projective A-module of rank \(\leq r^ 2\), and \(End(\phi)\otimes_ AK_{\infty}\) is a division ring, where En\(d(\phi)\) is discretely embedded. Moreover, if L has characteristic \(\infty\) (i.e. \(\gamma:A\to L\) is injective), then \(End(\phi)\) is commutative and has rank \(\leq r.\) Analytic description: An A-submodule \(G\subset C\) is called an A-lattice if for all balls B(s) in C of radius \(s>0\), \(G\cap B(s)\) is finite. Let \(e_ G:C\to C\) be the exponential function defined by the infinite product \(e_ G(t)=t\prod_{0\neq g\in G}(1-t/g).\) We have the following commutative diagram \[ \begin{tikzcd} 0 \ar[r] & G \ar[r]\ar[d,"a"] & C \ar[r,,"{e_ G}"]\ar[d,"a"] & C\ar[r]\ar[d,"\phi_ a^ G"] & 0 \\ 0 \ar[r] & G \ar[r] & C \ar[r,"{e_ G}"] & C \ar[r] & 0 \rlap{\,.}\end{tikzcd} \] This yields a ring homomorphism \(\phi^ G: A\to End_ C({\mathbb{G}}_ a)\) \((a\to \phi^ G_ a)\) with \(\phi_ a^{cG}{\mathbb{O}}c=c{\mathbb{O}}\phi^ G_ a.\) Hence \(\phi^ G\) is a Drinfeld module over C. - This establishes the following 1-1 correspondence: The map \(G\to \phi^ G\) induces a bijection of the sets \{Drinfeld modules over C of rank r\}/iso and \{r-dimensional A-lattices in C\}/isog. Furthermore, \(End(\phi^ G)\) is canonically isomorphic to the multiplier ring \(M(G)=\{c\in C| \quad cG\subset G\}.\) Tate parametrization and reductions of Drinfeld modules are discussed. Let \(\phi\) be a rank r Drinfeld module over L. Let \(\nu\) be a rank 1 valuation of L, with the valuation ring B and the residue field \(L(\nu)\). \(\phi\) is said to have ''stable reduction at \(\nu\) '' if there exists a Drinfeld module \(\phi'\) with values in \(B\{\tau\}\) and its reduction \(\overline{\phi '}\) is a Drinfeld module of rank \(r'\geq 1\) over \(L(\nu)\). If \(r'=r\), \(\phi\) is said to have ''good reduction at \(\nu\)''. Now assume that L is algebraically closed and complete. Let \(\phi\) be a Drinfeld module over L of rank \(r_ 1\) and let G be an A-lattice of rank \(r_ 2\) in \(\phi\). For \(a\in A\), let \(\psi_ a\in End_ L({\mathbb{G}}_ a)\) be defined by the following commutative diagram \[ \begin{tikzcd} 0 \ar[r] & G \ar[r]\ar[d,"\phi_a"] & C \ar[r,,"{e_ G}"]\ar[d,"\phi_a"] & C\ar[r]\ar[d,"\phi_ a^G"] & 0 \\ 0 \ar[r] & G \ar[r] & C \ar[r,"{e_ G}"] & C \ar[r] & 0 \rlap{\,.}\end{tikzcd} \] Then the association \(a\to \psi_ a\) defines a Drinfeld module \(\psi\) of rank \(r=r_ 1+r_ 2\) over L, with good reduction of rank \(r_ 1\). The pair \((\phi,G)\) is called the ''Tate parametrization'' of \(\psi\). then the map \((\phi,g)\to \psi\) is a bijection of the sets \{Drinfeld module \(\phi\) over L of rank \(r_ 1\) with good reduction, and lattice G of rank \(r_ 2\) in \(\phi\}\) and \{Drinfeld modules over L of rank \(r_ 1+r_ 2\) with reduction of rank \(r_ 1\}\). If \(\psi\) is a Drinfeld module over L with the Tate parametrization \((\phi,G)\), then \(End(\phi)\) is canonically isomorphic to the multiplier ring \(M(G)=\{c\in End(\phi)| \quad cG\subset G\}.\) Henceforth take \(K={\mathbb{F}}_ q(T)\) to be the field of rational functions over \({\mathbb{F}}_ q\) in T. Then \(A={\mathbb{F}}_ q[T]\). We confine ourselves to Drinfeld modules of rank 2 over L. Then a Drinfeld module is given by \[ \phi_ T\in End_ L({\mathbb{G}}_ a): \phi_ T(X)=\gamma(T)X+f_ 1X^ q+f_ 2X^{q^ 2},\quad f_ i\in L,\quad f_ 2\neq 0. \] A Drinfeld module \(\phi\) is denoted \(\phi =(f_ 1,f_ 2)\). the j-invariant of \(\phi =(f_ 1,f_ 2)\) is defined by \(J=j(\phi)=f_ 1^{q+1}/f_ 2.\) It is shown that if L is a field with rank 1 valuation \(\nu\) with the valuation ring B, then a Drinfeld module \(\phi\) has a ''potentially good reduction'' at \(\nu\) if and only if \(j(\phi)\) lies in B. Moreover, j defines an analytic parametrization \[ GL(2,A)\backslash \Omega \to^{\cong}C\quad(\Omega =C\backslash K_{\infty}). \] An A-lattice G (or the associated Drinfeld module \(\phi\), or \(j(\phi))\) is called ''singular'' if \(End_ C(\phi)\cong M(G)\neq A.\) It is proved that the singular invariant \(j(\phi)\) is an algebraic integer over A. Then the analogue of the classical theorems on complex multiplication are proved for Drinfeld modules. Let L/K be an imaginary quadratic extension with the ring of integers B. An order in L is a subring \(B'\subset B\) of finite index. Let \(I=I_ L\) denote the group of L-idèle without \(\infty\) components, and \(L^{ab}\) the maximal Abelian extension of L in which infinite places are completely split. For a Drinfeld module \(\phi =(f_ 1,f_ 2)\) with \(j(\phi)\neq 0\), let \(h_{\phi}:C\to C,\quad t\to f_ 1t^{q-1}\quad(t\to f_ 2t^{q^ 2-1}).\) Let \(L\subset C\) be an imaginary quadratic extension over K, \({\mathfrak a}\) a proper ideal of an order in L, \(\phi\) a Drinfeld module belonging to \({\mathfrak a}\) and \(u\in L/{\mathfrak a}\). Let \(W=\{s\in I| \quad s{\mathfrak a}={\mathfrak a}\text{ and }su=u\}.\) Then the subgroup \(WL^*\) of I coincides with the field \(L(j({\mathfrak a}),h_{\phi}(t))\) with \(t=e_{{\mathfrak a}}(u)\). Consequently, one has the following assertions. Let L/K be an imaginary quadratic extension, \({\mathfrak a}\) a proper ideal of an order \(M({\mathfrak a})\) in L. Then \(j({\mathfrak a})\in L^{ab}\), and for \(s\in I\), \([s,L](j({\mathfrak a}))=j(s^{-1}{\mathfrak a}),\) where [,L] is the Artin homomorphism. \(L^{ab}\) is generated over L by \(j({\mathfrak a})\) and \(h_{\phi}(t)\). The extension degree \(L(j):L\) and that of \(K(j):K\) are equal. If \(M({\mathfrak a})\) is the maximal order of L then L(j) is the maximal unramified subfield contained in \(L^{ab}.\) In the final section, supersingular Drinfeld modules of rank 2 are discussed in detail. Let \({\mathfrak p}\) be a place of A of degree d, \({\mathbb{F}}\) a finite extension of \({\mathbb{F}}_{{\mathfrak p}}=A/{\mathfrak p}\) of degree s. Let F denote the Frobenius automorphism of \({\mathbb{F}}\). For any Drinfeld module \(\phi\) over \({\mathbb{F}}\), F lies in \(End_{{\mathbb{F}}}(\phi)\). If \({\mathfrak q}\) (\(\neq {\mathfrak p})\) is a place of A, then there is a natural injective representation \(i_{{\mathfrak q}}:End(\phi)\otimes A_{{\mathfrak q}}\to End_{A_{{\mathfrak q}}}(T_{{\mathfrak q}}(\phi)).\)(Here \(t_{{\mathfrak q}}(\phi)\) denotes the Tate module of \(\phi\).) Then one can show that the characteristic polynomial of \(i_{{\mathfrak q}}(F)\) is of degree 2 with coefficients Tr(F), N(F) in A, and is independent of \({\mathfrak q}\). Moreover, \(N(F)=c.p^ s\) with \(c\in A^*\) and \(| Tr(F)| \leq | N(F)|^{\frac{1}{2}}.\) Recall that for any Drinfeld module \(\phi\) over \({\mathbb{F}}\), the element \(\phi_ p\in End_{{\mathbb{F}}}({\mathbb{G}}_ a)\) has the form \[ \phi_ p=g_ d\tau^ d+ g_{d+1}\tau^{d+1}+... +g_{2d}\tau^{2d}, g_ i\in {\mathbb{F}} (\tau =X^ q). \] The ''Hasse invariant'' \(H(\phi)\) of \(\phi\) is then defined to be the coefficient \(g_ d\) of \(\phi_ p\). Then the following conditions are equivalent for a Drinfeld module \(\phi\) over \({\mathbb{F}}: (a)\quad H(\phi)=0;\) (ii) \(\phi\) has no \({\mathfrak p}\)-torsion points; (iii) \(End_{{\bar {\mathbb{F}}}}(\phi)\) is non-commutative; and (iv) some power of F lies in A. In this case, \(End_{{\bar {\mathbb{F}}}}(\phi)\) is an order of the quaternion algebra over K in which only \({\mathfrak p}\) and \(\infty\) ramify, and \(j(\phi)\) lies in the quadratic extension of \({\mathbb{F}}_{{\mathfrak p}}\). When \(\phi\) satisfies the above equivalent conditions, \(\phi\) is called ''supersingular''. Furthermore, for a Drinfeld module over \({\mathbb{F}}\) of type \(\phi_ T=\gamma(T)X+\lambda X^ q+X^{q^ 2}\) with \(j(\phi)=\lambda^{q+1},\) singular values of \(\lambda\) are determined (as the analogue of the results on supersingular elliptic curves). The results are formulated as follows. Let \({\mathfrak p}\) be a place of A of degree d. If d is even then there exist exactly \((q^ d-1)/(^ 2-1)\) singular j-invariants in characteristic \({\mathfrak p}\). Here \(j=0\) is not singular. If d is odd, then there exist exactly \(q(q^{d-1}-1)/(q^ 2-1)+1\) singular j-invariants in characteristic \({\mathfrak p}\). In this case, \(j=0\) is singular. Finally there is a mass formula: \[ \sum \quad \frac{1}{\#Aut(\phi)}=\frac{\#({\mathbb{F}}_{{\mathfrak p}})-1}{(q^ 2-1)(q-1)} \] where the sum runs over all isomorphism classes of supersingular Drinfeld modules of rank 2 in characteristic \({\mathfrak p}\).
scientific article; zbMATH DE number 3851273