Realizing alternating groups as monodromy groups of genus one covers by Mike Fried, Eric Klassen and Yaacov Kopeliovich Abstract: We prove that if n ≥ 4, a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of IP 1 ) of degree n such that every branch point has multiplicity 3, and the monodromy group is the alternating group A n . To prove this theorem, we construct a Hurwitz space and show that it maps (generically) onto the genus one moduli space. 1. Introduction Associated to any n-sheeted branched cover of IP 1 with branch set B ⊂ IP 1 is a homo- morphism π 1 (IP 1 − B) → S n (the symmetric group) called the monodromy representation of the branched cover. The image of this homomorphism in S n is simply called the mon- odromy group of the cover (this group is well-defined up to conjugacy in S n ). If Σ is a compact Riemann surface and φ is a nonconstant meromorphic function on Σ, then φ :Σ → IP 1 is a branched cover and so we may speak of the monodromy group of φ. In [GN], it is stated that “Thompson (private correspondence) has verified that A 4 is the monodromy group of the generic Riemann surface of genus 1 (as far as we are aware, this is the only known example of a cover of a generic genus g> 0 surface with monodromy group different from a symmetric group).” Our main result in this paper (Theorem 1, stated formally and proved in Section 4) states that this is true for all A n , where n ≥ 4. More precisely, Theorem 1 asserts that if n ≥ 4, then a generic Riemann surface of genus one admits a meromorphic function of degree n whose monodromy group is the alternating group A n and all of whose branch points have multiplicity 3. By generic, we mean that for a given n, all but a finite number of genus 1 Riemann surfaces admit such functions. It is amusing to note that there is only one Riemann surface of genus one which admits a meromorphic function with monodromy A 3 : it is the Fermat curve x 3 + y 3 + z 3 = 0, and the meromorphic function is projection onto any one of the three coordinate axes in IP 2 . To see that there is only one such curve, note that, first, the location of the three branch points in IP 1 is irrelevant to the moduli and, second, the combinatorics is completely determined by the monodromy requirements (since the only way to select three 3-cycles in A 3 whose product is 1 is to select the same 3-cycle three times). We now give a brief summary of our proof. Given a topological branched cover φ : Σ → IP 1 , one may form the corresponding Hurwitz space H , a moduli space whose points represent those branched covers Σ → IP 1 which may be obtained from φ by moving around the images of the branch points in IP 1 while holding constant the combinatorial branch structure over these points as they move. Each of these branched covers gives rise to a complex structure on Σ by pulling back the one on IP 1 . This defines a map Ψ : H → MΣ , where MΣ is the moduli space of complex structures on Σ. Under the assumption that φ :Σ → IP 1 is completely non-Galois (i.e., it has no non-trivial deck transformations), one 1