Yoshida lifts and Selmer groups

Abstract

We prove an explicit inner product formula for vector-valued Yoshida lifts by an explicit calculation of local zeta integrals in the Rallis inner product formula for O(4) and Sp(4). As a consequence , we obtain the non-vanishing of Yoshida lifts.

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

“…One should note that we could rephrase our results in terms of X Σ (W f,λ (κ − 2)) and L alg (κ, f ) by adding in the condition that for no prime p | M is f congruent modulo λ to a newform of weight 2κ − 2, trivial character, and level dividing M/p. For the arguments needed to change our results to align with this, one can see [6,15]. We chose to phrase our results as above to avoid this extra congruence condition.…”

“…The argument used here originates in the work of Ribet in his proof of the converse of Herbrand's theorem ( [29]), but has now appeared in many forms. One can see [4,6,7,8,22,48] for some other examples. We now give a brief outline of the argument focusing on points of divergence from the arguments in the level one case.…”

Inner product formula for Yoshida lifts

“…In addition to the application to the non-vanishing of Yoshida lifts, our main motivation for the explicit Petersson norm formula for Yoshida lifts of type (I) originates from the study on the congruences between Hecke eigen-systems of Yoshida lifts and stable forms on GSp(4), the so-called Yoshida congruence as well as its application to the Bloch-Kato conjecture for special values of Asai L-functions. The Yoshida congruence was first investigated by the independent works [BDSP12] and [AK13], where the Petersson norm formula was used to relate the congruence primes of Yoshida lifts of type (I) to special values of the Rankin-Selberg L-functions. More precisely, in [BDSP12, Corollary 9.2] and [AK13, Theorem 6.6], the authors proved that if a prime p divides the algebraic part of the L-values L(f 1 ⊗ f 2 , k 1 + k 2 + 2), then p is a congruence prime for Yoshida lifts attached to a pair of elliptic newforms (f 1 , f 2 ) of weight (2k 1 + 2, 2k 2 + 2) under some restricted hypotheses.…”

Period of the adelic Ikeda lift for U(m, m)

“…Here, we say that F is a lift of an elliptic modular form f if F or the adelization of F is a Hecke eigenform in the space of Siegel cusp forms or Hermitian cusp forms whose certain L-function is expressed in terms of L-functions related to f. There are several results concerning this problem in the Siegel modular form case (cf. [2], [19]). This type of period relation sometimes gives rise to congruence between the lift and non-lift, and are important also from the view point of arithmetic geometry (cf.…”

“…This type of period relation sometimes gives rise to congruence between the lift and non-lift, and are important also from the view point of arithmetic geometry (cf. [2], [4], [12]). In [16], we proved a conjecture on the period of the Duke-Imamoglu-Ikeda lift (DII lift) proposed by Ikeda [9].…”