On the arithmetic of a classical family of elliptic curves with complex multiplication

https://hu-berlin.zoom.us/j/61686623112

I will study the family of elliptic curves C_N/Q  of the form x^3+y^3=Nz^3 for any cube-free positive integer N. They are cubic twists of the Fermat elliptic curve x3+y3=z^3, and they admit complex multiplication by the ring of integers of the imaginary quadratic field Q(sqrt{-3}). The celebrated conjecture of Birch and Swinnerton-Dyer is one of the most important open problems in number theory concerning elliptic curves. The p-part of the conjecture has been settled for these curves for all primes p not equal to 2 or 3  by K. Rubin using powerful techniques from Iwasawa theory. The aim of this talk is to study the conjecture at the remaining primes. First, I will establish a lower bound for the 3-adic valuation of the algebraic part of their central L-values in terms of the number of distinct prime divisors of N. I will then show that the bound is sometimes sharp, which gives us the 3-part of the conjecture for C_N/Q in certain special cases. In addition, I will study the non-triviality and growth of the 2-part and the 3-part of their Tate-Shafarevich group.