On the Cuspidality of Pullbacks of Siegel Eisenstein Series and Applications to the Bloch-Kato Conjecture

Abstract

Abstract. Let k > 3 be an integer and p a prime with p > 2k − 2. Let f be a newform of weight 2k − 2 and level 1 so that f is ordinary at p and ρ f is irreducible. Under some additional hypotheses we prove that ordp(L alg (k, f )) ≤ ordp(#S) where S is the Pontryagin dual of the Selmer group associated to ρ f ⊗ ε 1−k with ε the p-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of f and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

“…The proof of this result follows along the same lines as the main result of [10]. The main difference is that allowing for non-trivial level here adds several obstacles that must be overcome.…”

“…We now recall the results from [10] which involve the construction of a lattice which we will need to give a lower bound on the size of the desired Selmer group. In [10] the second author generalizes a result in [43] wherein Urban presents the case where the Galois representationρ ss splits into two irreducible representations. We will limit ourselves to recalling the results here and refer the reader to [10] for more details.…”

“…Finally, we give an exposition of the Bloch-Kato conjecture in our setting and relate our results to the conjecture. Such a discussion was absent in [10], so this can serve as the relation of the results given there to the Bloch-Kato conjecture as well.…”

“…The work for this is primarily contained in [10]. The main difference here is in the relaxation of the conditions at the primes dividing the level of f .…”

“…The main difference here is in the relaxation of the conditions at the primes dividing the level of f . We cite the relevant results from [10] and give their adapted statements in our setting. Finally, we give an exposition of the Bloch-Kato conjecture in our setting and relate our results to the conjecture.…”