Michael Zieve

Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve:
Linear relations between polynomial orbits,
Duke Math. J. 161 (2012), 1379–1410.

(Both the published version and the arXiv version are available online.)

We study the orbits of a polynomial g(X) in C[X], namely the sets {e, g(e), g(g(e)), …} with e in C. We prove that if nonlinear complex polynomials g and h have orbits with infinite intersection, then g and h have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.

In our previous paper, we proved the first result in case g and h have the same degree. The proof in the present paper combines this previous result with Siegel’s theorem on integral points on curves, the Bilu–Tichy classification of Diophantine equations G(x)=H(y) having infinitely many S-integer solutions, and several new results on polynomial decomposition, including a recent result of Müller and Zieve. We also give an alternate method for deducing our result from the result of our previous paper, which works in the non-isotrivial case (meaning that there is no linear change of variables after which both orbits are contained in a number field); this alternate approach replaces the use of Bilu–Tichy and polynomial decomposition with arguments involving canonical heights.

Robert M. Guralnick, Joel E. Rosenberg, and Michael E. Zieve:
A new family of exceptional polynomials in characteristic two,
Annals of Math.172 (2010), 1361–1390.

(Both the published version and the arXiv version are available online.)

A polynomial g over F_q is called exceptional if the map a→g(a) induces a bijection on F_qⁿ for infinitely many n. The composition of two polynomials is exceptional if and only if both polynomials are exceptional, so it suffices to study the indecomposable exceptional polynomials. Letting  p denote the characteristic of  F_q,  Fried, Guralnick and Saxl showed that any indecomposable exceptional polynomial has degree either prime or a power of  p, except perhaps when  p≤3  in which case they could not rule out the possibility that the degree is  p^r(p^r-1)/2  with r > 1 odd. In our previous paper, we determined all exceptional polynomials in this last situation, except for one ramification configuration; these polynomials were twists of exceptional polynomials discovered previously by Müller, Cohen–Matthews, and Lenstra–Zieve. In this paper we complete the classification of indecomposable exceptional polynomials of non-prime power degree, by addressing this final ramification configuration. It turns out that this yields a new family of indecomposable exceptional polynomials, which includes polynomials of degree 2^r-1(2^r-1) over F_2^s whenever r > 1 is odd and s > 1 is coprime to r.

The strategy of our proof is to identify the curves C which can occur as the Galois closure of the cover g : P¹→P¹ for a polynomial g satisfying the required properties. It turns out that any such C is geometrically isomorphic to the smooth plane curve y^q+1+z^q+1=T(yz)+c, where Q=2^r with r > 1 odd, and where T(X)=X^q/2+X^q/4+…+X. This family of curves is of independent interest, since each curve in the family has ordinary Jacobian and has automorphism group significantly larger than its genus. A key step in our proof is the computation of the automorphism groups of curves of the form v^q+v=h(w), with h varying over a two-parameter family of rational functions. Our method for this is rather general, and applies to many families of rational functions h.

Robert M. Guralnick and Michael E. Zieve:
Polynomials with PSL(2) monodromy,
Annals of Math.172 (2010), 1315–1359.

(Both the published version and the arXiv version are available online.)

Let k be a field of characteristic p≥0, and let g be a polynomial over k which is indecomposable (not a composition of lower-degree polynomials over k). Guralnick and Saxl showed that, if g decomposes over an extension of k, then the degree of g is either a power of  p or 21 or 55. We begin by determining all possibilities with the latter degrees. It turns out that there exists such a polynomial of degree 21 if and only if  p=7 and k contains nonsquares, in which case the polynomials can be presented explicitly. A similar conclusion holds for degree 55.

A polynomial g over F_q is called exceptional if the map a→g(a) induces a bijection on F_qⁿ for infinitely many n. The composition of two polynomials is exceptional if and only if both polynomials are exceptional, so it suffices to study the indecomposable exceptional polynomials. Letting  p denote the characteristic of  F_q,  Fried, Guralnick and Saxl showed that any indecomposable exceptional polynomial has degree either prime or a power of  p, except perhaps when  p ≤ 3 in which case they could not rule out the possibility that the degree is  p^r(p^r-1)/2  with r > 1 odd. Soon afterward, Müller, Cohen–Matthews, and Lenstra–Zieve produced indecomposable exceptional polynomials having all degrees allowed by the Fried–Guralnick–Saxl result. We determine all indecomposable exceptional polynomials of non-prime power degree, except for one ramification possibility which we treat in a companion paper with Joel Rosenberg. The conclusion is that, except for the new family of polynomials discovered in the latter paper, all such indecomposable exceptional polynomials are twists of the examples discovered previously.

We solve these two problems as consequences of a more general problem. Let k be a field of characteristic p>0, let q be a power of  p, and let t be transcendental over k. We determine all polynomials g in k[X] of degree q(q-1)/2 for which g(X)-t has simple roots and its Galois group over k(t) has a transitive normal subgroup isomorphic to PSL(2,q). These include the polynomials discussed in the previous two paragraphs. More generally, Guralnick and Saxl have produced a list of groups satisfying the known conditions necessary for occurring as the Galois group of g(X)-t over K(t), where g is a polynomial over an algebraically closed field K. Our result determines all such polynomials for one of the main families of groups on the Guralnick–Saxl list; this complements work of Abhyankar, who has exhibited polynomials realizing some of the other families of groups on the list.

Our strategy for proving these results is as follows. Let g be a polynomial of degree q(q-1)/2 over an algebraically closed field K of characteristic p, where q is a power of  p. Let Ω be the splitting field of g(X)-t over K(t), and suppose that G := Gal(Ω/K(t)) normalizes PSL(2,q). Then G is contained in the automorphism group PΓL(2,q) of PSL(2,q). We use the classification of subgroups of PΓL(2,q), together with Hilbert’s different formula, to determine all possibilities for the inertia and higher ramification groups in Ω/K(t) that are consistent with K(x) having genus zero. For each such possibility, we determine all candidates for Ω: this is the most difficult step, and involves various ingredients. We then determine the automorphism group of each of these (infinitely many) candidates for Ω, and use invariant theory to compute the subfields of Ω corresponding to K(t) and K(x), and finally to compute the polynomial g. This solves the geometric part of our problem; to solve the original arithmetic problems, we use the factorizations of g(X)-g(Y) determined previously by Cohen–Matthews and Zieve.

Our results should be viewed in the context of Abhyankar’s conjecture (the Raynaud/Harbater theorem). Given a smooth projective irreducible curve C over an algebraically closed field of characteristic p, and a finite set S of points on C, this conjecture describes the groups occurring as Galois groups of covers of C whose branch locus is contained in S. However, this work does not address the problem of determining the dimension of the moduli space of covers having prescribed Galois group and prescribed ramification data, let alone the problem of constructing the covers explicitly. The p=0 case of Abhyankar’s conjecture follows from Riemann’s existence theorem; however, even in that situation, the proof is nonconstructive. We have constructed all polynomial covers of  P¹ realizing certain permutation representations of PSL(2,q). In future work we hope to extend this to other classes of groups, which will provide data towards a conjecture on the dimension of the moduli space of covers having prescribed Galois group and prescribed ramification.

Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve:
Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture,
Inventiones Math.171 (2008), 463–483. MR 2008k:37099

(Both the published version and the arXiv version are available online.)

We study the orbits of a polynomial g(X) in C[X], namely the sets {e, g(e), g(g(e)), …} with e in C. We prove that if nonlinear complex polynomials g,h of the same degree have orbits with infinite intersection, then g and h have a common iterate. We also present a dynamical analogue of the Mordell–Lang conjecture, and deduce a special case of this conjecture from our result.

Our proof involves a dynamical analogue of Silverman’s specialization theorem, which we prove by means of the Tate/Call–Silverman theory of canonical heights of morphisms of varieties. This specialization result allows us to reduce to the case that both orbits are contained in a number field K, and hence in a ring R of S-integers of K. It follows that, for every n, the equation gⁿ(x)=hⁿ(y) has infinitely many solutions with x,y in R, where gⁿ denotes the n-th iterate of g. According to a result of Bilu and Tichy, this gives information about the functional decompositions of gⁿ and hⁿ. We obtain our result by combining the information deduced for all n with some additional arguments involving polynomial decomposition and the structure of the class group and unit group of the ring of integers of K.

Additional comment added July 2008:  See our subsequent paper for a generalization of the result in this paper, in which we need not assume that g and h have the same degree.

Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, and Michael E. Zieve:
Curves of every genus with many points, II: Asymptotically good families,
Duke Math. J.122 (2004), 399–422. MR 2005h:11123

(Both the published version and the arXiv version are available online.)

We resolve a 1983 question of Serre’s by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power  q  there is a positive constant  c_q  with the following property: for every integer  g≥0,  there is a genus-g  curve over  F_q  with at least  c_q g  rational points over  F_q.  We show also that there is a constant  c>0  such that for every  q  and every  n>0,  and for every sufficiently large  g,  there is a genus-g  curve over  F_q  that has at least  (c/n)g  rational points and whose Jacobian contains a subgroup of rational points isomorphic to  (Z/nZ)^r  for some  r > (c/n)g.

The strategy of our proof is to begin with some sequence of curves with many points that achieve `enough’ genera, and then to fill in the missing genera via degree-2 covers. More specifically, we start with a sequence of curves over  F_q  that have many points and whose genera grow at most exponentially. Then we show that for every curve  C  in this sequence, and for every integer  h  greater than some constant multiple of the genus of  C,  there exists a degree-2 cover  B→C  over  F_q  such that  B  has genus  h.  Then either  B  or its quadratic twist will have at least as many  F_q-rational points as does  C.

We produce the degree-2 covers by showing that, for any genus-g  curve  C  over  F_q,  there are degree-2 covers  B→C  over  F_q  in which the genus of  B  is any prescribed integer greater than  4g.  We produce the initial sequence of curves via class field towers.

If  q  is a square then we can prove better results by starting with a sequence of Shimura curves: we show that for every  g  there is a genus-g  curve over  F_q  having at least  g(q^½-1+o(1))/3  rational points. Conversely, Drinfeld and Vladut have shown that any genus-g  curve over  F_q  has at most  g(q^½-1+o(1))  rational points. Thus, over any prescribed field of square order, every large genus  g  behaves in roughly the same way as every other, in terms of the maximum value of  #C(F_q)/g  where  C  varies over the genus-g  curves over  F_q.

It is known that, over any square field, there are infinitely many curves achieving equality in the Drinfeld–Vladut bound (up to the  o(1)  error term). The known curves with this property are the modular curves and the analogous Shimura and Drinfeld modular curves, which have many rational points because all supersingular points are defined over a small square field. In particular, the classical modular curve  X₀(N)  achieves equality in the Drinfeld–Vladut bound over  F_p²  for any prime  p  coprime to  N.  Since the genus of  X₀(N)  is  N/12  plus lower-order terms, and on average the genus is less than  N,  it is natural to ask whether one can achieve equality in the Drinfeld–Vladut bound in every large genus by means of modular curves. However, it was shown by Csirik, Wetherell, and Zieve that the values occurring as genera of  X₀(N)  tend to occur in this way for many values  N,  so a random integer has probability zero of occurring as the genus of any curve  X₀(N).  In light of this, our approach via double covers does not seem farfetched.

The results of this paper improve the results of our previous paper, in which we used different methods to construct curves of every genus with many points.