Twists of the Burkhardt quartic threefold

On the smallest point on a diagonal quartic threefold

For the family a0x 4 = a1y +a2z +a3v +a4w 4, a0, . . . , a4 > 0, of diagonal quartic threefolds, we study the behaviour of the height of the smallest rational point versus the Tamagawa type number introduced by E. Peyre.

2 8 Ja n 20 05 Plane quartic twists of X ( 5 , 3 ) Julio

Given an odd surjective Galois representation ̺ : GQ → PGL2(F3) and a positive integer N, there exists a twisted modular curve X(N, 3)̺ defined over Q whose rational points classify the quadratic Q-curves of degree N realizing ̺. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N= 5.

Faster Pairing Computation on Jacobi Quartic Curves with High-Degree Twists

In this paper, we propose an elaborate geometric approach to explain the group law on Jacobi quartic curves which are seen as the intersection of two quadratic surfaces in space. Using the geometry interpretation we construct the Miller function. Then we present explicit formulae for the addition and doubling steps in Miller’s algorithm to compute Tate pairing on Jacobi quartic curves. Both the...

The Universal Kummer Threefold

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble’s quartic hypersu...

On the Stability of the Orthogonally Quartic Functional Equation