Abstract. We prove that if G is a finite simple group which is the unit group of a ring, then G is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order 2 k − 1 for some k; or (c) a projective special linear group PSLn(F 2 ) for some n ≥ 3. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups G with no non-trivial normal 2-subgroup.

Communicated by S. SehgalWe prove there is no ring with unit group isomorphic to Sn for n ≥ 5 and that there is no ring with unit group isomorphic to An for n ≥ 5, n = 8. To prove the non-existence of such a ring, we prove the non-existence of a certain ideal in the group algebra F 2 [G], with G an alternating or symmetric group as above. We also give examples of rings with unit groups isomorphic to S 1 , S 2 , S 3 , S 4 , A 1 , A 2 , A 3 , A 4 , and A 8 . Most of our existence results are well-known, and we recall them only briefly; however, we expect the construction of a ring with unit group isomorphic to S 4 to be new, and so we treat it in detail.

Abstract. Let Fq denote a finite field of characteristic p ≥ 5 and let d = q + 1. Let E d denote the elliptic curve over the function field F q 2 (t) defined by the equation y 2 + xy − t d y = x 3 . Its rank is q when q ≡ 1 mod 3 and its rank is q − 2 when q ≡ 2 mod 3. We describe an explicit method for producing points on this elliptic curve. In case q ≡ 11 mod 12, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on E d makes use of a dominant map from the degree d Fermat surface over F q 2 to the elliptic surface associated to E d . We in turn study lines on the Fermat surface F d using certain multiplicative character sums which are interesting in their own right. In particular, in the q ≡ 7 mod 12 case, a character sum argument shows that we can generate a full-rank subgroup using µ d -translates of a single rational point.

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