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

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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.

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