In this talk, I formulate and prove a new Rubin-type Iwasawa main conjecture for imaginary quadratic fields in which p is inert or ramified, as well as a Perrin-Riou type Heegner point main conjecture for certain supersingular CM elliptic curves. These main conjectures and their proofs are related to p-adic L-functions that I have previously constructed, and have applications to two classical problems of arithmetic. First, I prove the 1879 conjecture of Sylvester stating that if p = 4,7,8 mod 9, then x^3 + y^3 = p has a solution with x,y rational numbers. Second, combined with previous Selmer distribution results, I show that 100% of squarefree d = 5,6,7 mod 8 are congruent numbers, thus establishing Goldfeld's conjecture for the family y^2 = x^3 - d^2x, and solving the congruent number problem in 100% of cases.