i'm a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack reducing the problem to a jacobian of a genus 4 curve. That doesn't mean anything but that got me really interested in applications of algebraic geometry to cryptography. In the end I leaned towards learning about the post-quantum elliptic curve algorithms such as SIDH, SQIsign, others and everything around. Specifically I started by going through the chapter I of Shafarevich's algebraic geometry book and chapter I-VI of Silverman's book and as an application I implemented Schoof's algorithm on counting points on elliptic curves. I also red some algebraic number theory (I did a small thesis on Chebotarev's theorem) and a good part of Cox book on the primes of the form $$x^2+ny^2$$ Continuing, I understood that the current understanding of elliptic curve based cryptosystems needed deeper understanding of abelian varieties/modular curves/class fields in general. So to motivate myself I attacked the class number one problem to learn about modular curves and class fields. I already know a bit about schemes and I'm not afraid of learning a bunch of stuff. I have a little complex analysis/sheaves/homology background, enough to read about riemann surfaces and currently, I'm going through modular forms until I can define properly modular curves and compute equations for them. The goal would be to read about isogeny volcanoes, some A. Sutherland papers, the SEA algorithm,etc... My questions are :
- How much should I read on modular forms ? What books/articles would be best to learn modular forms for what I want?
- How should I introduce myself to higher dimensional abelian varities, especially jacobians ?
- Should I read about Schemes ? Coherent cohomology ? I know the that characteristic $p$ part is where schemes get really important.
I know learning about all of this is very ambitious but If anyone could help me that would be very kind. Thanks for reading !