Congruences of modular forms and Selmer groups

We show that the congruence modulo 11 between the normalized cusp form Delta of weight 12 and the normalized cusp form of weight 2 and level 11 'descends' to a congruence between forms of weights 13/2 and 3/2. Combining Waldspurger's theorem with the Bloch-Kato conjecture we predict the existence of elements of order 11 in Selmer groups for certain quadratic twists of Delta. These are then constructed using rational points on twists of the elliptic curve X-0(11), assuming the Birch and Swinnerton-Dyer conjecture on the rank. Everything generalizes to forms of weights 2 + 10s in an 11-adic family, to congruences modulo higher powers of 11, and to other elliptic curves over Q of prime conductor p equivalent to 3 (mod 4) such that L(E-p,1) not equal 0 and p inverted iota ord(p) (j(E)).