ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14, 15, 30 Streit, Manfred Field of Definition and Galois Orbits of some non–congruence Subgroups Torsion–free normal subgroups of cocompact triangle groups Δ define Riemann surfaces X with large automorphism groups, though they are finite of order at most 84(g − 1). This bound is attained for normal subgroups of Δ of signature (2, 3, 7). X then is called Hurwitz curve. Macbeath shows: 1. P SL_{2}(𝔽_{q}) is automorphism group of a Hurwitz curve for field characteristic p exactly for one p–power q. 2. The Hurwitz curve is uniquely determined by that group iff p = q = 7 or q = p^{3} and p ≡ ±2 or p ≡ ±3 mod 7. X then may be defined by equations with coefficients lying in ℚ.

For p = q ≡ ±1 mod 7 we have 3 Hurwitz curves X_{1}, X_{2}, X_{3}. One can show: 1. Coefficients of a defining equaiton lie in K := ℚ(ζ_{7} + ζ_{7}^{−1}), ζ_{7} := exp 2πi/7 as the smallest possible field. 2. All equations are equivalent under a coefficientwise action of Gal(\overline{ℚ}/ℚ). 3. They contain no K–rational points for p ≡ 1, 5, 9, 13, 17, 25 mod 28. A more general result for other non– congruence subgroups with a quotient equal to P SL_{2}(𝔽_{q}) can be obtained.