The effect of torsion on the distribution of Ш among quadratic twists of an elliptic curve

Let E be an elliptic curve of rank zero defined over Q and ℓ an odd prime number. For E of prime conductor N, in Quattrini (2006) [Qua06], we remarked that when ℓ||E(Q)Tor|, there is a congruence modulo ℓ among a modular form of weight 3/2 corresponding to E and an Eisenstein series. In this work we relate this congruence in weight 3/2, to a well-known one occurring in weight 2, which arises when E(Q) has an ℓ torsion point. For N prime, we prove that this last congruence can be lifted to one involving eigenvectors of Brandt matrices Bp(N) in the quaternion algebra ramified at N and infinity. From this follows the congruence in weight 3/2. For N square free we conjecture on the coefficients of a weight 3/2 Cohen–Eisenstein series of level N, which we expect to be congruent to the weight 3/2 modular form corresponding to E.