Ability to represent curve groups

[whoami730]

Problem Description

I want to be able to create groups over general curves akin to Elliptic Curves.
As an example, consider the equation x^2 - y^2 = 1 modulo p for prime p. This is a hyperbolic equation, and we should be able to both represent groups of points on this curve as well as experiment on the structure of this curve - as an example, I wanted to know what would the abelian_group function provide for such a curve.

Proposed Solution

Maybe something like - Curve(Zp, x**2 - y**2 - 1) could be used to represent this.

Alternatives Considered

Really couldn't find anything similar.

Additional Information

No response

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

[vincentmacri]

The generalization of the elliptic curve group is the Jacobian/degree zero divisor class group/Picard group, which we do have in Sage: https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/jacobian_base.html.

Is that what you were looking for?

[user202729]

the issue isn't very meaningful as is, in order to specify a group you have to define the group operation.

Only in the (special) case of elliptic curve is there a 1-1 correspondence between points and degree zero divisor class group, given by point $P$ ⟺ divisor class $[P]-[O]$.

there's an interesting theorem that algebraic groups of dimension 1 are isomorphic to either additive group, multiplicative group or elliptic curve (assume certain conditions such as algebraically closed). Though the best reference I can find is https://ayoucis.wordpress.com/2019/11/19/classifying-one-dimensional-groups-ii/ .

[whoami730]

@user202729 Not sure I understood the latter details, but I agree that one needs to define the group operation and for the case of x^2 - y^2 = 1 % p, the group operation + can be defined as xR = (xP*xQ + yP*yQ) % p and yR = (xP*yQ + xQ*yP) % p and the resulting point R = P + Q indeed lies on the curve, thus yielding a group.