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Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive, i.e. $2p$ is the GCD of all the values of $log$.

By the way, I think the cleanest way to perform the above calculations is to fix the generator $g = exp(\frac{2p}{p-1})$ for the group $Z_p/\pm$ of p-adic Adams operations acting on p-adic $KO$. Then you get to write down the fiber sequence $$L_{K/p}S \rightarrow KO\overset{1-\psi^g}{\rightarrow} KO,$$ and the result is that $log\circ J$ identifies with $ko\rightarrow KO\rightarrow L_{K/p}S^{1}$, where the last map is the boundary map in the above sequence. (Again my signs might be off, sorry for that.)

Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive. That is, the set of all $(p-1)^{st}$ powers of $p$-adic units is exactly the domain of convergence of $log$, and, also, $2p$ is the GCD of all the values of $log$.

By the way, I think the cleanest way to perform the above calculations is to fix the generator $g = exp(\frac{2p}{p-1})$ for the group $Z_p^\times/\pm$ of p-adic Adams operations acting on p-adic $KO$. Then you get to write down the fiber sequence $$L_{K/p}S \rightarrow KO\overset{1-\psi^g}{\rightarrow} KO,$$ and the result is that $log\circ J$ identifies with $ko\rightarrow KO\rightarrow L_{K/p}S^{1}$, where the last map is the boundary map in the above sequence. (Again my signs might be off, sorry for that.)

Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive, i.e. $2p$ is the GCD of all the values of $log$. When you get into it, the above calculation sees this as the ``same fact'' that the homotopy groups of KO have exact period 8.

Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive, i.e. $2p$ is the GCD of all the values of $log$.

The motivation is that, as Neil says, the spectrum $pic(S^0)$ (or $gl_1(S^0)$) is kind of mysterious. That's too bad for us, since apparently we need to understand it to prove the above claim. However, the Bousfield-Kuhn functor tells you that for any prime p, the "part of a spectrum which mod p K-theory understands", meaning the K/p-localization of the spectrum, only depends on the n^{th} space of the spectrum, for any n you like. This is some amazing consequence of periodicity. The upshot is that the K/p-localization doesn't care that $pic(S^0)$ has this exotic infinite loop structure, and you get a natural map $$pic(S^0)\rightarrow L_{K/p}S^{-1}$$ which exhibits the latter as the K/p-localization of the former.

Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{-1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive, i.e. $2p$ is the GCD of all the values of $log$. When you get into it, the above calculation sees this as the ``same fact'' that the homotopy groups of KO have exact period 8.

We deduce from this that the composition $log\circ J:ko\rightarrow L_{K/p}S^{-1}$ sends the unit class in $\pi_0ko$ to that same homomorphism. Now the point is that $K/p$-local homotopy is computationally friendly. Also, $ko$ is nearly $K/p$-local: its $K/p$-localiztion is the $p$-adic $KO$, via the connective cover map $ko\rightarrow KO$. Thus it's not hard to calculate that a homotopy class of maps $ko\rightarrow L_{K/p}S^{-1}$ is completely determined by its effect on $\pi_0$. So actually the above minimal information tells us exactly what $log\circ J$ is. Then if we look on higher homotopy groups, we see that the image of $log\circ J$ has size given by Adams' upper bound. This implies the claim.

By the way, I think the cleanest way to perform the above calculations is to fix the generator $g = exp(\frac{2p}{p-1})$ for the group $Z_p/\pm$ of p-adic Adams operations acting on p-adic $KO$. Then you get to write down the fiber sequence $$L_{K/p}S \rightarrow KO\overset{1-\psi^g}{\rightarrow} KO,$$ and the result is that $log\circ J$ identifies with $ko\rightarrow KO\rightarrow L_{K/p}S^{-1}$, where the last map is the boundary map in the above sequence. (Again my signs might be off, sorry for that.)

The motivation is that, as Neil says, the spectrum $pic(S^0)$ (or $gl_1(S^0)$) is kind of mysterious. That's too bad for us, since apparently we need to understand it to prove the above claim. However, the Bousfield-Kuhn functor tells you that for any prime p, the "part of a spectrum which mod p K-theory understands", meaning the K/p-localization of the spectrum, only depends on the n^{th} space of the spectrum, for any n you like. This is some amazing consequence of periodicity. The upshot is that the K/p-localization doesn't care that $pic(S^0)$ has this exotic infinite loop structure, and you get a natural map $$pic(S^0)\rightarrow L_{K/p}S^{1}$$ which exhibits the latter as the K/p-localization of the former.

Now, a priori this map is somewhat mysterious on the level of homotopy groups. But actually Rezk made a fantastic study of it in his paper on logarithmic cohomology operations (which the above is essentially an instance of.) One consequence of Rezk's work is a calculation of the effect of the above map on $\pi_0$. It is as follows. First, recall that $\pi_{-1} L_{K/p}S$ identifies with $Hom(Z_p^\times,Z_p)$ (noncanonically, this is just $Z_p$, the $p$-adic integers.) Then, the above map $log:pic(S^0)\rightarrow L_{K/p}S^{1}$ sends the class of the $1$-sphere to the homomorphism $Z_p^\times\rightarrow Z_p$ given by $$x\mapsto \frac{1}{2p}log(x^{p-1}).$$ (I might have a sign wrong, but that doesn't matter for our purposes.) Here $log$ stands for the $p$-adic logarithm. It's an interesting thing that the above expression makes sense and is primitive, i.e. $2p$ is the GCD of all the values of $log$. When you get into it, the above calculation sees this as the ``same fact'' that the homotopy groups of KO have exact period 8.

We deduce from this that the composition $log\circ J:ko\rightarrow L_{K/p}S^{1}$ sends the unit class in $\pi_0ko$ to that same homomorphism. Now the point is that $K/p$-local homotopy is computationally friendly. Also, $ko$ is nearly $K/p$-local: its $K/p$-localiztion is the $p$-adic $KO$, via the connective cover map $ko\rightarrow KO$. Thus it's not hard to calculate that a homotopy class of maps $ko\rightarrow L_{K/p}S^{1}$ is completely determined by its effect on $\pi_0$. So actually the above minimal information tells us exactly what $log\circ J$ is. Then if we look on higher homotopy groups, we see that the image of $log\circ J$ has size given by Adams' upper bound. This implies the claim.

By the way, I think the cleanest way to perform the above calculations is to fix the generator $g = exp(\frac{2p}{p-1})$ for the group $Z_p/\pm$ of p-adic Adams operations acting on p-adic $KO$. Then you get to write down the fiber sequence $$L_{K/p}S \rightarrow KO\overset{1-\psi^g}{\rightarrow} KO,$$ and the result is that $log\circ J$ identifies with $ko\rightarrow KO\rightarrow L_{K/p}S^{1}$, where the last map is the boundary map in the above sequence. (Again my signs might be off, sorry for that.)