On the Bloch--Kato conjecture for the symmetric cube

We prove one inclusion of the Iwasawa Main Conjecture, and the Bloch-Kato conjecture in analytic rank 0, for the symmetric cube of a level 1 modular form.


Introduction
In this short note, we study the arithmetic of the p-adic symmetric cube Galois representation associated to a level 1 modular cusp form. We prove that the Bloch-Kato conjecture holds in analytic rank 0 for the critical twists of this representation (corresponding to the critical values of the L-function); and we prove one inclusion of the cyclotomic Iwasawa Main Conjecture for this representation, showing that the characteristic ideal of the Selmer group divides the p-adic L-function. We deduce these results from the results of [LZ20] on the arithmetic of automorphic representations of GSp(4), using the existence of a symmetric cube lifting from GL(2) to GSp(4).
• The results of this paper were inspired by the striking work of Haining Wang [Wan20], where he proves many cases of the Bloch-Kato conjecture for symmetric cubes of modular forms via arithmetic level-raising on triple products of Shimura curves. Our results are complementary to those of Wang, since we restrict to level 1 (and hence weight 12), while he considers weight 2 and general level. • The converse implication of the Bloch-Kato conjecture in this setting -i.e. that the vanishing of the central critical L-value implies the existence of a non-zero Selmer class -will be treated in forthcoming work of Samuel Mundy, using a functorial lifting to the exceptional group G 2 . • The restriction to modular forms of level 1 is carried over from the results in op.cit., which rely on forthcoming work of Barrera, Dimitrov and Williams regarding interpolation of p-adic L-functions for GL(2n) in finite-slope families. At present these results are only available for level 1, but we are confident that it should be possible to relax this assumption. In this case, all of the results of this paper will generalize without change to modular forms of higher level. ⋄ In forthcoming work, we will prove the analogous results for the Galois representations attached to quadratic Hilbert modular forms, using the twisted Yoshida lift to GSp(4).

Large Galois image
Let f be a cuspidal, new, non-CM eigenform of weight k 2 and level Γ 1 (N ) for some N 1, and write F for the coefficient field of f . Throughout this section, let p be a prime > 3. Let v be a prime of F above p, and denote by V f,v the 2-dimensional F v -linear Galois representation associated to f . We are interested in the representation Sym 3 V f,v .
(1) Our conventions are such that the trace of geometric Frobenius at a prime ℓ ∤ pN is the Hecke eigenvalue a ℓ (f ).
Π with the following properties: The authors are grateful to acknowledge financial support from the European Research Council (ERC Consolidator Grant "Euler Systems and the Birch-Swinnerton-Dyer conjecture") and the Royal Society (University Research Fellowship "L-functions and Iwasawa theory").
In order to prove this proposition, we quote the following result due to Ribet [Rib85]: Lemma 2.3. If p is sufficiently large, then there exists a Gal(Q/Q)-stable O Fv -lattice T of V * f,v such that the image of the homomorphism contains SL 2 (Z p ). Here, the last isomorphism is given by a basis of T .
We now prove Proposition 2.2: Proof. Let τ ∈ Gal Q/Q(µ p ∞ ) such that its image in GL 2 (O Fv ) is equal to 1 1 0 1 . Then the matrix of τ with respect to the natural basis of T Π = Sym 3 T is given by Remark 2.4. We can clearly choose τ ∈ Gal Q/Q ab ) . Hence we have that for every Dirichlet character χ of prime-to-p-conductor. ⋄

Functoriality
Notation 3.1. Let π be the associated automorphic representation of GL 2 (A f ), and write Sym 3 π for the image of π under the Langlands functoriality transfer from GL 2 to GL 4 attached to Sym 3 : GL 2 (C) → GL 4 (C).
Remark 3.3. In more classical terms, Π (or more precisely its L-packet) corresponds to a holomorphic vector-valued Siegel modular form taking values in the representation det k+1 ⊗ Sym 3k−3 of GL 2 (C). ⋄

Application to the Bloch-Kato conjecture
Theorem 4.1. Assume that f is a cuspidal eigenform of weight k and level 1, and let V = Sym 3 V f,v (2k− 2). Let p be a prime > 3, and assume that f is ordinary at p. Let 0 ≤ j ≤ k − 2, and let ρ be a finite order character of The assumption on f being ordinary at p implies that the representation Π is Borel ordinary at p. Moreover, Note 2.4 implies that the big image assumption Hyp(Q(µ p ∞ ), T Π (χ) (c.f. [LZ20, §. 2]) is satisfied for all Dirichlet characters χ of prime-to-p conductor. We are hence in the situation where can apply [LZ20, Theorem D].

Application to the Iwasawa main conjecture
We similarly obtain applications to the cyclotomic Iwasawa Main Conjecture. We preserve the notations and hypotheses of the previous section. Let Γ = Gal(Q(µ p ∞ )/Q), and write Λ = O Fv [[Γ]] for the Iwasawa algebra. 5.1. P-adic L-functions for the symmetric cube. We recall the following two known results: Theorem 5.1 (Algebraicity of Sym 3 L-values). There exist constants Ω + Π , Ω − Π ∈ C, well-defined up to multiplication by F × , such that for every Dirichlet character χ and every 0 j k − 2, the quantity lies in F (χ), and depends Gal(F /F )-equivariantly on χ. Here ε = (−1) j ρ(−1).
Theorem 5.2 (Existence of the Sym 3 p-adic L-function). Let α be the unit root of the Hecke polynomial of f at p. Then there exists an element L v,α (Sym 3 f ) ∈ Λ with the following interpolation property: for any Dirichlet character ρ of p-power conductor, and any 0 j k − 2, we have L v,α (Sym 3 f )(j + ρ) = j!(j + k − 1)! R p (Sym 2 f, ρ, j) L alg (Sym 3 f ⊗ρ, j + k), These results are due to Dimitrov, Januszewski and Raghuram [DJR18], as a special case of general theorems applying to any automorphic representation Π of GL(4) admitting a Shalika model. An alternative proof using coherent cohomology of Siegel Shimura varieties is given in [LPSZ19, Theorem A].