The 2-Primary Class Group of Certain Hyperelliptic Curves

The 2-primary Class Group of Certain Hyperelliptic Curves

In a letter to Dirichlet, dated May 30, 1828 ([5]), Gauß considered the divisibility of the class number of Q( √−p) (p prime, = 1 mod 4) by 8. Many variations on this theme can be found in the mathematical literature of the subsequent centuries, either using quadratic forms or class field theory ( Rédei [15], Barrucand-Cohn [3], Hasse [10], Kaplan [12], Stevenhagen [17]). Of these, the latter a...

Picard Group of Moduli of Hyperelliptic Curves

The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the non-existence of a tautological family over the coarse moduli space.

Hyperelliptic Curves in Characteristic 2

In this paper we prove that there are no hyperelliptic supersingular curves of genus 2n − 1 in characteristic 2 for any integer n ≥ 2. Let F be an algebraically closed field of characteristic 2, and let g be a positive integer. Write h = blog2(g + 1) + 1c, where b c denotes the greatest integer less than or equal to a given real number. Let X be a hyperelliptic curve over F of genus g ≥ 3 of 2-...

Class Numbers for Some Hyperelliptic Curves

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Let K be a field, a, b ∈ K and ab 6= 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation