Yoshida lifts and the Bloch-Kato conjecture for the convolution \(L\)-function (Q2444993)

The paper studies a special case of the Bloch-Kato conjecture, where the motive in question is \(\rho_{f_1}\otimes\rho_{f_2}(-k/2-1)\), \(\rho_{f_i}\) are \(\ell-\)adic Galois representations attached to \(f_i\), two new forms of prime level and weight \(2\) and \(k+2\) respectively. Here \(k\) has to be either \(8\) or \(12\). By looking at the congruences satisfied by the Yoshida lift of \((f_1,f_2)\), the paper establishes a lower bound of the \(\ell-\)valuation of the order of the Bloch-Kato Selmer group of this motive by a special value of the fuction \(L(s,f_1\times f_2)\). The result is conditional upon a conjecture on the formula for the Petersson norm of Yoshida lift and some other assumptions.