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In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.
Abstract. In this article we study the inverse of the period map for the family F of complex algebraic curves of genus 6 equipped with an automorphism of order 5. This is a family with 2 parameters, and is fibred over a certain type of Del Pezzo surace. The period satisfies the hypergeometric differential equation for Appell’s F1( 3 5 , 3 5 , 2 5 , 6 5 ) of… (More)
The latter theorem shows the relation of the (coefficients of the realized) elliptic curves corresponding to two isogenous torus C/Z + τZ and C/Z + 2τZ. So in general this theorem is referred as the isogeny formula for the Jacobi theta constants. Any way these two theorems are telling us a very interesting story concerned with AGM, periods of algebraic… (More)
In this paper we study the theta contants appeared in [S] those induced the modular function for the family of Picard curves C(ξ) given by (1). Our theta constants θk(u, v) (k = 0, 1, 2) , given by (3), are ”Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball B, given by (2), with respect to a index finite subgroup Γθ of the… (More)
We show that the Prym map for 4-th cyclic étale covers of curves of genus 4 is a dominant morphism to a Shimura variety for a family of Abelian 6-folds of Weil type. According to the result of Schoen, this implies algebraicity of Weil classes for this family.
Possible existence of “hot-sector generations” above the well known 3 generation bound is investigated on the basis of a model of leptons and quarks, which is based on the Harari and Shupe’s one. Our model predicts the existence of 3 + 1 generations above the ordinary “cold-sector” 3 generations. Majorana neutrinos are introduced to realize the heavy… (More)
We explicitly identify infinitely many curves which are quotients of Fermat curves. We show that some of these have simple Jacobians with complex multiplication by a non-cyclotomic field. For a particular case we determine the local zeta functions with two independent methods. The first uses Jacobi sums and the second applies the general theory of complex… (More)