Congruences of Saito-Kurokawa lifts and divisibility of degree-8 L-values

In this thesis, we study the arithmeticity of critical values of degree-8 tensor product L-functions attached to Siegel modular forms of genus 1 and 2. We show that the congruence between the Hecke eigenvalues of two cuspidal Siegel Hecke eigenforms of genus 2 implies a similar congruence between certain suitably normalised critical values of the associated degree-8 L-functions. This phenomenon is predicted by the Bloch-Kato conjecture, for which we therefore provide further evidence in this particular setting. We prove this by employing integral representation formulae, due to Saha, and B¨ocherer and Heim, linking critical L-values to iterated Pe- tersson inner products against diagonally restricted non-holomorphic Eisenstein series.