I am 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 sheaves(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 give me any hint for any part of my question that would be very kind. Thanks for reading !
