Elliptic Curves and Abelian Varieties

https://crypto.stackexchange.com/questions/35482/which-elliptic-curves-are-quantum-resistant

Leo, hyperelliptic cryptography or something is quantum secure?? check that

Curve theory references

Standard references: Hartshorne, Qing Liu, Vakil. Stichtenoth is a good reference for the function field perspective, including a simple argument for asymptotic Riemann Roch.

See Roy Smith's answers on MO and MSE. Other users to read?

MO: Elementary Proof of Riemann-Roch for Compact Riemann Surfaces; see Roy Smith and David Speyer answers in particular. Terry Tao blog post.

Joe Harris course notes; see e.g. Atanas Atanasov notes.

Mohan: $\operatorname{Sym}^d C \to \operatorname{Pic}^d C$ (dimension $d$ mapping to dimension $g$) supposed to give heuristic RR dimension count (even with corrections?). Observe that $\operatorname{Sym}^d C$ corresponds to effective divisors, so a fiber of this map will contain all effective divisors of the form $(f)+D$ with $(f)$ principal and $D$ fixed in the divisor class $[D]\in \operatorname{Pic}^d C$ being considered (so $D$ need not be an effective representative, which indeed might not exist; the only effectiveness restriction is on the $(f)+D \geq 0$ points in the fiber, which will be empty if the class $[D]$ has no effective representative).

Also Mohan: some direct way of seeing $\chi(D+p) = \chi(D)+1$, with an underlying geometric result (gluing line bundles? or sections? or total spaces?) of independent (perhaps greater) interest: maybe inquire about details. What is the geometric intuition for Euler characteristics? See MSE: Meaning of holomorphic Euler characteristics?; MO: Why is the Euler characteristic of powers of a line bundle a polynomial in the power?; MO: Effect of tensor product on euler characteristic of line bundles; MO: Euler characteristic of a curve;

MO: Intuition behind Riemann's bilinear relations

MO: What does the Jacobian of a curve tell us about the curve?

MO: How do you see the genus of a curve, just looking at its function field?

MO: Elliptic Curves with CM and Class Field Theory

MSE: Constructing one-forms on a Riemann surface using the uniformization theorem and MO: The existence of meromorphic functions on Riemann surfaces. See Tsimerman generals.