Moduli of Hyperelliptic Curves and Invariants of Binary Forms
Moduli of Hyperelliptic Curves and Invariants of Binary Forms
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发表时间：
2013
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通讯作者：
David W. Taylor
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作者：
David W. Taylor
对于特征不等于2的代数闭域k上的椭圆型和2属超椭圆曲线族，我们得到了k上具有不同线性因子的二元n型族参数化的光滑局部正规形式。我们还证明了作用于二元n型的一般二乘二矩阵线性群的不变性理论与作用于仿射k型的n元素置换群的不变性理论是同构的。我们证明了这个置换群不变量环在k与一个定义在整数上的多项式环的张量积上是纯不可分的。其次，我们证明了正特征的二元六分形式的GL_2(k)不变理论在以p为模的二元六分形式的经典不变理论给出的环上是纯不可分的。当k的特征不为2时，这些结果意味着对于除有限多个素数的所有素数，p，Frobenius自同态的幂次导出了2格曲线的粗模格式与取模p的二元六性经典不变理论构造的环谱之间的态射，并且这种态射是闭点水平上的双射。通过组合论证，给出了复数上定义的任意属超椭圆曲线周期所满足的picard - fuh方程的显式公式。这使我们可以直接得出Picard-Fuchs方程是正则奇异的结论。
For families of elliptic and genus 2 hyper-elliptic curves over an algebraically closed field k of characteristic not equal to 2, we derive smooth local normal forms parameterized by families of binary n-forms over k with distinct linear factors. We also prove that the invariant theory of the general linear group of two-by-two matrices acting on binary n-forms is isomorphic to the invariant theory of the permutation group of n elements acting on an affine k-variety. We show that this ring of permutation group invariants is purely inseparable over the tensor product of k with a polynomial ring defined over the integers. Next we show that the GL_2(k) invariant theory of binary sextic forms in positive characteristic is purely inseparable over a ring given by the classical invariant theory of binary sextic forms taken modulo p for all but finitely many primes p. When the characteristic of k is not 2 these results imply that for all but finitely many primes, p, a power of the Frobenius endomorphism induces a morphism between the coarse moduli scheme of genus 2 curves and the spectrum of a ring constructed using the classical invariant theory of binary sextics taken modulo p. Moreover, this morphism is a bijection at the level of closed points.We also give, via a combinatorial argument, explicit formulae for the Picard-Fuch equations satisfied by the periods of hyperelliptic curves of arbitrary genera defined over the complex numbers. This allows us to conclude directly that the Picard-Fuchs equations are regular singular.