TITLE: Cokernel of the stable J-homomorphism at odd primes
QUESTION [8 upvotes]: Where can one learn about odd-primary components of the cokernel of the stable J-homomorphism? 
According to wonderful Wikipedia article on Homotopy groups of spheres, the "hard" part of the stable stem is the cokernel of $J$. I am not an expert and have trouble finding what is known. After an extensive search all I found was 

some low-dimensional computations (in $k$th stem for $k\le 17$) which do not work for my current purpose.
Theorem 1.1.14 in Ravenel's "green book" book which gives some infinite families. Embarassingly, I do not even understand the statement of the theorem and cannot locate its proof (which is probably implicit in Section 4 of Chapter 4). 

Theorem 1.1.14 says in particular "For $p\ge 3$ the $p$-component of $\mathrm{coker}\, J$ has the following generators in dimensions $\le 3pq − 6$ (where $q = 2p − 2$), each with order $p$", and then it goes on to list two generators in $(pq-2)$ stem and $(pq+q-3)$ stem, and six of their products.
Does this mean that the $p$-component of $\mathrm{coker}\, J$ in the $k$th stem (with $k\le 3pq-6$)
is zero unless $\pi_k^S$ contains one of the either elements mentioned above?
Or does this merely mean that these eight elements are nonzero?
Is there a more comprehensive account of what is known (preferably with proofs or references)? Again, I am after infinite families of nonzero elements.

REPLY [2 votes]: I wish to record some other families of elements in $\mathrm{coker}(J)$ arising from computations of Toda and of Oka of stable homotopy groups at odd primes. These computations aren't merely for small stems (as I incorrectly believed from reading Toda's book and looking at tables in Hatcher or Ravenel texts). 
Since the image of the stable J-homomorphism is known explicitly at any prime $p$, we can determine $\mathrm{coker}(J)$ in Toda-Oka range.
As is explained in Appendix B of Milnor-Stasheff, the image of the J-homomorphism in the $k$th stem is a cyclic group $\mathrm{Im}(J_k)$ whose $p$-component $\mathrm{Im}(J_k: p)$ is as follows for an odd prime $p$
(the case $p=2$ is slighly different but is just as easy to describe): 

If $\frac{k+1}{2(p-1)}\notin\mathbb N$, then $\mathrm{Im}(J_k: p)$  is zero. 
If $\frac{k+1}{2(p-1)}\in\mathbb N$, then $\mathrm{Im}(J_k: p)$ is isomorphic to $\mathbb Z_{p^{r+1}}$ where
$p^r$ is the largest power of $p$ that divides $\frac{k+1}{2(p-1)}$. 

Toda  computed  the $p$-component of the $k$th stem for $k<2p^2(p-1)-3$. I won't analyse $\mathrm{coker}(J_k)$ for his range except for one obvious example:
If $k=2p(p-1)^2-1$, Toda shows that the $k$th stem has $p$ component $\mathbb Z_p\times\mathbb Z_{p^2}$, which is non-cyclic, and hence $\mathrm{coker}(J_k)$
is nontrivial; in fact $\mathrm{coker}(J_k: p)$ is $\mathbb Z_p$.
Oka in a series of papers, see  here  and references therein, extended Toda's range and constructed for each $p>3$, some elements (in his notations $\phi$, $\mu$, $\beta$) such that $\mathrm{coker}(J_k: p)$ is $\mathbb Z_{p^2}$. In particular, for the $\beta$-elements the degree $k$ is even, and for $\phi$-elements $\frac{k+3}{2(p-1)}\in\mathbb N$, so in these cases the image of $J$ is zero at $p$.
I do not know any examples where $\mathrm{coker}(J_k: p)$ has elements of order $>p^2$, and wonder if this is due to natural limitations of Toda-Oka range, or is there some other explanation of this phenomenon?