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We work in the stable homotopy category of p p -local spectra for a fixed prime number p p . Let E E be a spectrum and L E \mathcal {L}_E denote the stable homotopy category of localized spectra with respect to E E in the sense of Bousfield. Then, M. Hopkins introduced a Picard group P i c ( L E ) Pic(\mathcal {L}_E) of the category L E \mathcal {L}_E . If the spectra E E and F F satisfy the relation ⟨ E ⟩ ≥ ⟨ F ⟩ \left \langle {E}\right \rangle \ge \left \langle F\right \rangle of the Bousfield classes, then we have a homomorphism ℓ : P i c ( L E ) → P i c ( L F ) \ell \colon Pic(\mathcal {L}_E)\to Pic(\mathcal {L}_F) . We consider the spectra K m n = E ( n ) ∧ M J m K_m^n=E(n)\wedge MJ_m for the n n -th Johnson-Wilson spectrum E ( n ) E(n) and a type m m generalized Moore spectrum M J m MJ_m for 0 ≤ m ≤ n 0\le m\le n . For E = K m n E=K_m^n , we have a subgroup P i c 0 ( L E ) Pic^0({\mathcal {L}_E}) of P i c ( L E ) Pic(\mathcal {L}_E) consisting of exotic elements. In this paper, we study the homomorphism ℓ : P i c 0 ( L E ( n ) ) → P i c 0 ( L K m n ) \ell \colon Pic^0({\mathcal {L}_{E(n)}})\to Pic^0({\mathcal {L}_{K_m^n}}) , and give conditions under which it is an isomorphism. This is a generalization of the result P i c 0 ( L 2 ) ≅ κ 2 Pic^0(\mathcal {L}_2)\cong \kappa _2 (P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk [J. Topol. 8 (2015), 267–294, Remark. 6.5]) for ( p , n , m ) = ( 3 , 2 , 2 ) (p,n,m)=(3,2,2) .

Let S ( p ) 0 S^0_{(p)} denote the sphere spectrum localized at an odd prime p p . Then we have the first beta element β 1 ∈ π 2 p 2 − 2 p − 2 ( S ( p ) 0 ) \beta _1\in \pi _{2p^2-2p-2}(S^0_{(p)}) , whose cofiber we denote by W W . We also consider a generalized Smith-Toda spectrum V r V_r characterized by B P ∗ ( V r ) = B P ∗ / ( p , v 1 r ) BP_*(V_r)=BP_*/(p,v_1^r) . In this note, we show that an element of π ∗ ( V r ∧ W ) \pi _*(V_r\wedge W) gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of β 9 t + 3 \beta _{9t+3} at the prime three to complete a conjecture of Ravenel’s: β s ∈ π 16 s − 6 ( S ( 3 ) 0 ) \beta _{s}\in \pi _{16s-6}(S^0_{(3)}) exists if and only if s s is not congruent to 4 4 , 7 7 or 8 8 mod 9 9 .

Let L 2 L_2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E ( 2 ) E(2) . A spectrum L 2 X L_2X is called invertible if there is a spectrum Y Y such that L 2 X ∧ Y = L 2 S 0 L_2X\wedge Y=L_2S^0 . Hovey and Sadofsky, Invertible spectra in the E ( n ) E(n) -local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E ( 2 ) E(2) -local sphere L 2 S 0 L_2S^0 at a prime p > 3 p>3 . At the prime 3 3 , it is shown, A relation between the Picard group of the E ( n ) E(n) -local homotopy category and E ( n ) E(n) -based Adams spectral sequence, that there exists an invertible spectrum X X that is not homotopy equivalent to a suspension of L 2 S 0 L_2S^0 . In this paper, we show the homotopy equivalence v 2 3 : Σ 48 L 2 V ( 1 ) ≃ V ( 1 ) ∧ X v_2^3\colon \Sigma ^{48}L_2V(1)\simeq V(1)\wedge X for the Smith-Toda spectrum V ( 1 ) V(1) . In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V ( 1 ) → L 2 V ( 1 ) \beta \colon \Sigma ^{144}L_2V(1)\to L_2V(1) that induces v 2 9 v_2^9 on the E ( 2 ) ∗ E(2)_* -homology.

In this paper, we try to compute the homotopy groups of the L 2 L_2 -localized Toda-Smith spectrum V ( 1 ) V(1) at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials d 5 d_5 and d 9 d_9 of the Adams-Novikov spectral sequence, different from the case p > 3 p>3 . We also determine the homotopy groups of some L 2 L_2 -localized finite spectra relating to V ( 1 ) V(1) . We further show some of the non-trivial differentials on elements relating so-called β \beta -elements in the Adams-Novikov spectral sequence for π ∗ ( S 0 ) \pi _*(S^0) .