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

Abstract

Abstract. Let κ ≥ 6 be an even integer, M an odd square-free integer, and f ∈ S2κ−2(Γ0(M )) a newform. We prove that under some reasonable assumptions that half of the λ-part of the Bloch-Kato conjecture for the near central critical value L(κ, f ) is true. We do this by bounding the ℓ-valuation of the order of the appropriate Bloch-Kato Selmer group below by the ℓ-valuation of algebraic part of L(κ, f ). We prove this by constructing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel modular form.

Irreducibility of limits of Galois representations of Saito–Kurokawa type

“…Such congruences for Saito-Kurokawa lifts have been proven by Brown, Agarwal and Li [1,12,14] for holomorphic Siegel modular forms of congruence level 2 0 (N ) and paramodular level para (N ) for weights k larger than 6 (see [14] Corollary 6.15). With this new result [8] Theorem 10.2 can be generalized to allow ramification at a squarefree level N , and establishes a so-called R = T result and the modularity of Fontaine-Laffaille representations that residually are of Saito-Kurokawa type (with an elliptic f of weight 2k − 2 for k ≥ 6).…”

Congruence primes for Ikeda lifts and the Ikeda ideal

“…Thus, we have infinitely many characters to choose from so it is reasonable to expect that one can often find such a χ. In the case n = 2, i.e., when one considers Saito-Kurokawa lifts, one can find computational evidence supporting the existence of such a χ in [1]. One can use the same methods outlined there to produce computational evidence for n > 2.…”

Hermitian Maass lift for general level

“…See section 2 for our definition and notation of the spaces and groups mentioned. In [18], Krieg generalized Kojima's result to general imaginary quadratic fields K. He established the maps (1.2) M + k−1 (D, χ) / / J * k,1 (1) / / M k,2 (1) Moreover, Krieg defined the Hermitian Maaß space M * k,2 (1) ⊂ M k,2 (1) and he showed that the image of the second map is M * k,2 (1). Also, J * k,1 (1) → M + k−1 (D, χ) is an isomorphism.…”

“…where (i) the lift from S k−1 (4, χ) to S + k−1 (4, χ) is an analogue of Shimura-Shitani isomorphism (but it is not an isomorphism here though and unlike Shitani's map, this map is not obtained by certain cycle integrals) and S + k−1 (4, χ) is an analogue of Kohnen's plus space [16]; (ii) the lift from S + k−1 (4, χ) to the space of special Hermitian Jacobi forms J * k,1 (1); and finally, (iii) the Hermitian analogue of the original Maaß lift from J * k,1 (1) to M k,2 (1).…”

“…For general level N such that (D, N) = 1, Berger and Klosin, in their recent work [6], generalized the definition of Krieg's Maaß space M * k,2 (1) to that of higher level M * k,2 (N). Using a modification of Kojima and Krieg's result and Ibukiyama's construction of the classical Maaß lift for general level, they then constructed the Hermitian analogue of the Maaß lift J * k,1 (N) → M k,2 (N) i.e.…”