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On the orders of largest groups of automorphisms of compact Riemann surfaces
Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-04-09 , DOI: 10.1016/j.jpaa.2021.106758
Czesław Bagiński , Grzegorz Gromadzki

Let μ(g) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g2. The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that μ(g) ranges between 8(g+1) and 84(g1) and these two bounds are attained for infinitely many g. Here we prove that given a prime q>167 congruent to −1 modulo 3, not congruent +1 modulo 5 and a prime p84q, μ(g)=(8(q+4)/q)(g1) for g=qpm+1 for infinitely many integers m. Furthermore we also prove that for any prime q for which p=q+2 is a prime, μ(g)=(12(q+2)/q)(g1) for g=qpm+1 for infinitely many m. These mean that there exist an infinite rational sequence (rn) convergent to 8 and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence (sn) convergent to 12 together with infinite sets Rn and Sn of integers for whichμ(g)={rn(g1)ifgRnsn(g1)ifgSn and 8,12 are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of μ. Namely if instead of μ=μ(g) we consider its normalization μ˜(g)=μ(g)/g then {8}(Md)d{8,12}, where M stands for the set of values of μ˜ and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, 12(Md)d.



中文翻译:

紧致黎曼曲面的最大自同构的最大群的阶

μG 是属的紧致黎曼曲面的一组自同构的最大可能阶 G2个。结合Accola-Maclachlan和Hurwitz-Macbeath的著名成绩,他们说:μG 介于 8G+1个84G-1个并且这两个界限达到无限大的g。在这里我们证明给定素数q>167 等于-1模3,不等于+1模5和素数 p84qμG=8q+4/qG-1个 为了 G=qp+1个对于无限多个整数m。此外,我们还证明了对任意素数q为这p=q+2个 是素数 μG=12q+2个/qG-1个 为了 G=qp+1个无限数m。这意味着存在无限的有理序列[Rñ 收敛至8,并假设孪生素数猜想的真相,则一个无穷有理数列 sñ 收敛到12与无限集 [Rñ小号ñ 的整数μG={[RñG-1个如果G[RñsñG-1个如果G小号ñ812是可能发生的唯一编号。这些结果的结果是充分了解μ的渐近性。即如果不是μ=μG 我们考虑它的归一化 μG=μG/G 然后 {8}中号dd{812}, 在哪里 中号 代表的一组值 μ算子d为一组累积点,并假设Twin Prime Conjecture的真相,12中号dd

更新日期:2021-04-14
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