Bounds on the number of non-rational subfields of a function field
Let C be a smooth, geometrically connected, projective algebraic curve defined over a field \(K\). Denote by \(J_ C\) the Jacobian and by \(F=K(C)\) the function field of \(C\) over \(K\), and let \(g=g_ F\) be the genus of F. To every separable, nonconstant morphism \(f: C\to C'\) of \(C\) to an algebraic curve \(C'\) over \(K\) with function field \(F'=K(C')\) of genus g'\(\geq 2\), one can associate an invariant \(\gamma^*_ E\) in the K- endomorphism ring \(End_ K(J_ C)\) of \(J_ C\) by taking \[ \gamma^*_ E:=\gamma^*_ f:=f^*\circ f_*: J_ C\to^{f_*}J_{C'}\to^{f^*}J_ C\quad, \] where \(f_*\) (resp. \(f^*)\) is the direct (resp. inverse) image map acting on the respective Jacobian varieties and \(E\) is the image of \(F'\), \(E:=f^* F'\leq F\), a subfield of \(F\). The classical theorem of de Franchis states that a) the number of invariants \(\gamma^*_ E\in End_ K(J_ C)\) is finite, and b) for any fixed invariant \(\gamma^*_ E\in End_ K(J_ C)\), there are at most finitely many subfields \(E/K\) of \(F/K\). By modifying the classical proof of de Franchis in the setting of \textit{P. Samuel} [Lectures on old and new results on algebraic curves, Tata Inst. Fundam. Res. (Bombay 1966; Zbl 0165.241)], \textit{A. Howard} and \textit{A. J. Sommese} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 429-436 (1983; Zbl 0534.14016)], in the case of \(K={\mathbb{C}}\), gave an explicit upper bound for the number \(N_ F\) of non-rational subfields of \(F/K\) in terms of the genus g of \(F/K\). (An error in their result was corrected by the author.) In the present very interesting and important paper, an ''Angle theorem'' is proved and combined with a ''packing argument'' due to Mumford to obtain a new and effective proof of de Franchis' theorem. In fact the author establishes the following refined explicit bound. Let r be the rank of \(End_ K(J_ C)\). It is known that \(r\leq r(g)\) for \(r(g)=2g^ 2\) or \(=4g^ 2\), according as the characteristic of K is \(=0\) or \(\neq 0\), respectively. Then, if \(g>g'\geq 2\), the number \(N_ F(g')\) of separable subfields of \(F/K\) of genus \(g'\) satisfies the inequality \[ N_ F(g')\leq (C+1)^{r-1} - (C-1)^{r-1},\text{ where } C:=2 \sqrt{\frac{(g-g')g'}{(g'-1)g}}\quad. \] It follows from this estimate that the function field F/K has at most \[ (g-1) 2^{r(g)-2} (2^{r(g)-1} -1) \] separable subfields of genus \(\geq 2.\) Slightly weaker versions of the main theorems of this paper had been announced earlier [Tagungsberichte, Math. Forschungsinstitut Oberwolfach 35/81, p. 11 (1981)].
scientific article; zbMATH DE number 3995831