Arithmetically exceptional functions and elliptic curves (Q2712792)

Let \(f(X)\) be a rational function with coefficients in the rationals (many of the results extend to the case of number fields). We can reduce modulo all but finitely many primes and consider the reduced function as a rational function over the finite field \(F_p\) of \(p\) elements. The problem considered is to describe those \(f\) such that \(f\) is bijective modulo \(p\) for infinitely many prime \(p\). NEWLINENEWLINENEWLINEIf \(f\) is a polynomial, this problem was considered by \textit{I. Schur} [Berl. Ber. 1923, 123-134 (1923; JFM 49.0093.02)] and completely solved by \textit{M. Fried} [Mich. Math. J. 17, 41-55 (1970; Zbl 0169.37702)]. The method of proof used by Fried was to convert the problem to a group theoretic problem, then use a theorem of Schur to show that there were not many possibilities. Then one could easily show that such polynomial is a composition of cyclic and Dickson polynomials. Note that the reduction of \(f\) modulo infinitely many primes will in fact be a so called exceptional polynomial. NEWLINENEWLINENEWLINEIf \(f\) is only a rational function, then the problem is much more difficult. One can still translate the problem to group theory. The most important case is when \(f\) is indecomposable as a rational function -- clearly if \(f\) is a composition of rational functions and has the desired property so does each term. A complete list of group theoretic solutions to this problem was given by the reviewer, the author and \textit{J. Saxl} in [The rational functional analogue of a question of Schur and exceptionality of permutation representations, preprint]. The classification of finite simple groups plays an integral part in the proof (although it turns out that there are only two cases where the group involved is nonsolvable). NEWLINENEWLINENEWLINEOne then has to produce a curve that the monodromy group of the cover acts on appropriately. In almost all cases, this curve has genus zero or one. There are a handful of cases when the genus is greater than \(1\). The cases where \(g=0\) are quite easy to analyze. The cases where \(g=1\) involve some interesting questions involving elliptic curves (in particular, \textit{B. Mazur}'s result about isogenies of elliptic curves defined over the rationals is needed [Invent. Math. 44, 129-162 (1978; Zbl 0386.14009)]). Most of the situations are dealt with in the paper by Guralnick, Müller and Saxl mentioned above. In this paper, the author deals with the remaining cases. He also uses the methods developed here to answer a question posed by J. Thompson about the field of definition of a degree \(25\) genus zero extension found by the reviewer and \textit{J. G. Thompson} [J. Algebra 131, 303-341 (1990; Zbl 0713.20011)]. NEWLINENEWLINENEWLINEThere are some interesting methods developed here for computing fields of definitions for covers.NEWLINENEWLINEFor the entire collection see [Zbl 0941.00014].