# ag.algebraic geometry – Infinitely many curves with isogenous Jacobians Answer

Hello expensive customer to our community We will proffer you an answer to this query ag.algebraic geometry – Infinitely many curves with isogenous Jacobians ,and the respond will breathe typical by documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

ag.algebraic geometry – Infinitely many curves with isogenous Jacobians

I consider I’ve an instance in genus 5:

Humbert curves (behold both Varley’s “Weddle’s Surfaces, Humbert’s Curves, and a Certain 4-Dimensional Abelian Variety“, or rehearse batch F in chapter 6 of ACGH) are in 1-1 correspondence with 5-tuples strains within the airplane — the airplane being $$I_C(2)$$ — as much as projective transformations of the airplane. These 5 strains are the picture of the double mask from $$W^1_4(C)$$ to rank < 4 quadrics enveloping the canonical picture of $$C$$. Each of the 5 strains is conspicuous with 4 factors – its intersection factors with the opposite 4 strains, and is the 2-quotient (underneath the map $$W^1_4to I_C(2)$$ ) of an elliptic round which lies contained in the Jacobian of $$C$$, the place 4 factors are the ramification factors. The Jacobian of C is isogenous to the product of those 5 elliptic curves.

Giving the airplane the projective coordinates $$x,y,z$$, we now deem the strains: $$x=0,quad y=0,quad x=z,quad y=z,quad x+y=az.$$
The $$y$$ (resp $$x$$) coordinate of the intersection factors on the primary (resp second) line are $$0,1,a,infty$$; whereas the $$x$$ (resp $$y$$) coordinate of the intersection factors on the third (resp fourth) strains are $$0,1, 1-a, infty$$. Since $$xmapsto 1-x$$ takes $$0to1,1to0,inftytoinfty, ato1-a$$, the 4 elliptic curves related to these 4 strains are isomorphic. Finally, The elliptic round related to fifth line has an spare involution, (flipping $$x,y$$), so it’s the identical round no matter $$a$$. Hence if we choose any $$a_1,a_2$$ which give isogenous elliptic curves, we get isogenous Humbert curves.

we’ll proffer you the answer to ag.algebraic geometry – Infinitely many curves with isogenous Jacobians query through our community which brings all of the solutions from a number of dependable sources.