We show that the congruence modulo 11 between the normalized cusp form Delt
a of weight 12 and the normalized cusp form of weight 2 and level 11 'desce
nds' to a congruence between forms of weights 13/2 and 3/2. Combining Walds
purger's theorem with the Bloch-Kato conjecture we predict the existence of
elements of order 11 in Selmer groups for certain quadratic twists of Delt
a. These are then constructed using rational points on twists of the ellipt
ic 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 fami
ly, 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)).