2.1. Topological André–Quillen homology

For a map of connective commutative ring spectra $i \;:\; A \rightarrow B$ , we use the connectivity of the map to determine the bottom homotopy group of $\textsf{TAQ}^A(B)$ [Reference Basterra6].

2.1.2. The connective Adams summand

Let $\ell$ denote the Adams summand of connective $p$ -localized topological complex K-theory, $ku_{(p)}$ . Here, $p$ is an odd prime.

The inclusion $i \;:\; \ell \rightarrow ku_{(p)}$ induces an isomorphism on $\pi _0$ and $\pi _1$ . Thus, by the Hurewicz theorem for topological André–Quillen homology [Reference Basterra6, Lemma 8.2], [Reference Baker, Gilmour and Reinhard4, Lemma 1.2], we get that $\pi _2\textsf{TAQ}^{\ell }(ku_{(p)})$ is the bottom homotopy group and is isomorphic to the second homotopy group of the cone of $i$ , and this in turn can be determined by the long exact sequence:

\begin{equation*} \cdots \rightarrow \pi _2(\ell ) = 0 \rightarrow \pi _2(ku_{(p)}) \rightarrow \pi _2(\text {cone}(i)) \rightarrow \pi _1(\ell ) = 0 \rightarrow \cdots \; .\end{equation*}

Hence, we have $\pi _2 \textsf{TAQ}^{\ell }(ku_{(p)}) \cong \mathbb{Z}_{(p)}$ .

We know from [Reference Dundas, Lindenstrauss and Richter19] that $\ell \rightarrow ku_{(p)}$ shows features of a tamely ramified extension of number rings, and Sagave shows [Reference Sagave50, Theorem 6.1] that $\ell \rightarrow ku_{(p)}$ is log-étale.

2.1.3. Real and complex connective topological K-theory

The complexification map $c \;:\; ko \rightarrow ku$ induces an isomorphism on $\pi _0$ and an epimorphism on $\pi _1$ , so it is a $1$ -equivalence. Hence, again $\pi _2\text{cone}(c) \cong \pi _2(\textsf{TAQ}^{ko}(ku))$ is the bottom homotopy group, but here we obtain an extension:

\begin{equation*} 0 \rightarrow \pi _2ku = \mathbb {Z} \rightarrow \pi _2\text {cone}(c) \rightarrow \pi _1(ko)=\mathbb {Z}/2\mathbb {Z} \rightarrow 0. \end{equation*}

In order to understand $\pi _2\text{cone}(c)$ , we consider the cofiber sequence:

\begin{equation*} { \Sigma KO \stackrel {\eta }{\longrightarrow} KO \stackrel {c}{\longrightarrow } KU \stackrel {\delta }{\longrightarrow } \Sigma ^2 KO }\end{equation*}

and the commutative diagram on homotopy groups:

Here, $\tau _e \;:\; e \rightarrow E$ denotes the map from the connective cover $e$ of $E$ to $E$ . The middle vertical map $g$ is the map induced by the cofiber sequences. By the five lemma, $g$ is an isomorphism hence

\begin{equation*} \pi _2 \text {cone}(c) \cong \pi _2\Sigma ^2 KO \cong \mathbb {Z}. \end{equation*}

So this group is also torsion-free. We will later see that $ko \rightarrow ku$ is not log-étale, and we will see some other indicators for wild ramification, but the bottom homotopy group of $\textsf{TAQ}^{ko}(ku)$ does not detect that.

2.1.4. Connective topological modular forms with level structure (case $n=3$ )

We consider $\textsf{tmf}_1(3)$ . Its homotopy groups are $\pi _*(\textsf{tmf}_1(3)) \cong \mathbb{Z}[\frac{1}{3}][a_1, a_3]$ with $|a_i| = 2i$ . See [Reference Hill and Lawson23] for some background. There is a $C_2$ -action on $\textsf{tmf}_1(3)$ coming from the permutation of elements of exact order three and one denotes by $\textsf{tmf}_0(3)$ the connective cover of the homotopy fixed points, $\textsf{tmf}_1(3)^{hC_2}$ . There is a homotopy fixed point spectral sequence that was studied in detail in [Reference Mahowald and Rezk33] for the periodic versions. In [Reference Hill and Lawson23, p. 407], it is explained how to adapt this calculation to the connective variants: the terms in the spectral sequence with $s \gt t-s \geqslant 0$ can be ignored. The $C_2$ -action on the $a_i$ ’s is given by the sign-action, so if $\tau$ generates $C_2$ , then $\tau (a_i^n) =(\!-\!1)^na_i^n$ .

This implies that only $H^0(C_2;\; \pi _0(\textsf{tmf}_1(3))) \cong \mathbb{Z}[\frac{1}{3}]$ survives to $\pi _0(\textsf{tmf}_0(3))$ . For $\pi _1$ , we get a contribution from $H^1(C_2;\; \pi _2(\textsf{tmf}_1(3)))$ , giving a $\mathbb{Z}/2\mathbb{Z}$ generated by the class of $a_1$ (this detects an $\eta$ ). For $\pi _2(\textsf{tmf}_0(3))$ , the class of $a_1^2$ generates a copy of $\mathbb{Z}/2\mathbb{Z}$ .

Hence, the map $j \;:\; \textsf{tmf}_0(3)_{(2)} \rightarrow \textsf{tmf}_1(3)_{(2)}$ is $1$ -connected, so $\pi _2\textsf{TAQ}^{\textsf{tmf}_0(3)_{(2)}}(\textsf{tmf}_1(3)_{(2)})$ is the bottom homotopy group and is isomorphic to $\pi _2(\text{cone}(j))$ which sits in an extension. We can use the commutative diagram of commutative ring spectra from [Reference Hill and Lawson23, Theorem 6.3]

in order to determine to $\pi _2(\text{cone}(j))$ . By [Reference Lawson and Naumann30, Theorem 1.2], there is a cofiber sequence of $\textsf{tmf}_1(3)_{(2)}$ -modules

\begin{equation*} {\Sigma ^6\textsf{tmf}_1(3)_{(2)} \stackrel {v_2}{\longrightarrow } \textsf{tmf}_1(3)_{(2)} \longrightarrow ku_{(2)}} \end{equation*}

and hence $\pi _2(\textsf{tmf}_1(3)_{(2)}) \cong \pi _2(ku_{(2)})$ .

The diagram

commutes and the $5$ -lemma implies that $\pi _2(\text{cone}(j)) \cong \mathbb{Z}_{(2)}$ .

2.1.5. Connective topological modular forms with level structure (case $n=2$ , $p=3$ )

Forgetting a $\Gamma _0(2)$ -structure yields a map $f\;:\; \textsf{tmf}_{(3)} \rightarrow \textsf{tmf}_0(2)_{(3)}$ such that $f$ is a $3$ -equivalence. We will recall more details about these spectra at the beginning of Section 2.2. Again, we obtain that the bottom nontrivial homotopy group of the spectrum of topological André–Quillen homology is $\pi _4(\textsf{TAQ}^{\textsf{tmf}_{(3)}}(\textsf{tmf}_0(2)_{(3)})) \cong \pi _4(\text{cone}(f))$ . There is a short exact sequence

\begin{equation*} 0 = \pi _4\textsf{tmf}_{(3)} \rightarrow \pi _4\textsf{tmf}_0(2)_{(3)} = \mathbb {Z}_{(3)} \rightarrow \pi _4\text {cone}(f) \rightarrow \pi _3\textsf{tmf}_{(3)} \cong \mathbb {Z}/3\mathbb {Z} \rightarrow 0\end{equation*}

so a priori $\pi _4 \text{cone}(f)$ could be isomorphic to $\mathbb{Z}_{(3)}$ or to $\mathbb{Z}_{(3)} \oplus \mathbb{Z}/3\mathbb{Z}$ .

There is an equivalence:

\begin{equation*} \textsf{tmf}_{(3)} \wedge T \simeq \textsf{tmf}_0(2)_{(3)}\end{equation*}

where $T = S^0 \cup _{\alpha _1} e^4 \cup _{\alpha _1} e^8$ with $\alpha _1$ denoting the generator of $\pi _3^s$ at $3$ , [Reference Behrens10, Lemma 2, p. 382], [Reference Mathew36, Theorem 4.15]. Thus, $T$ is part of a cofiber sequence:

\begin{equation*} S^0 \rightarrow T \rightarrow \Sigma ^4 \text {cone}(\alpha _1) \end{equation*}

and we obtain a cofiber sequence:

\begin{equation*} \textsf{tmf}_{(3)} = \textsf{tmf}_{(3)} \wedge S^0 \rightarrow \textsf{tmf}_{(3)} \wedge T \simeq \textsf{tmf}_0(2)_{(3)} \rightarrow \textsf{tmf}_{(3)} \wedge \Sigma ^4 \text {cone}(\alpha _1)\end{equation*}

and thus

\begin{equation*} \pi _4(\text {cone}(f)) \cong \pi _4 (\textsf{tmf}_{(3)} \wedge \Sigma ^4 \text {cone}(\alpha _1)) \cong \pi _0 (\textsf{tmf}_{(3)} \wedge \text {cone}(\alpha _1)). \end{equation*}

But as we have a short exact sequence:

\begin{equation*} 0 = \pi _0(\Sigma ^3 \textsf{tmf}_{(3)}) \rightarrow \pi _0 (\textsf{tmf}_{(3)}) \cong \mathbb {Z}_{(3)} \rightarrow \pi _0 (\textsf{tmf}_{(3)} \wedge \text {cone}(\alpha _1)) \rightarrow 0 \end{equation*}

we obtain

\begin{equation*} \pi _4(\textsf{TAQ}^{\textsf{tmf}_{(3)}}(\textsf{tmf}_0(2)_{(3)})) \cong \mathbb {Z}_{(3)}. \end{equation*}

2.2. Relative topological Hochschild homology

In [Reference Dundas, Lindenstrauss and Richter19] (see also [Reference Dundas, Lindenstrauss and Richter20] for a correction), we show that the relative topological Hochschild homology spectra $\textsf{THH}^{\ell }(ku_{(p)})$ and $\textsf{THH}^{ko}(ku)$ have highly nontrivial homotopy groups. Here, we extend these results to the relative $\textsf{THH}$ -spectra of $\textsf{tmf}_{(3)} \to \textsf{tmf}_0(2)_{(3)}$ , $\textsf{Tmf}_{(3)} \to \textsf{Tmf}_0(2)_{(3)}$ and $\textsf{tmf}_0(2)_{(3)} \to \textsf{tmf}(2)_{(3)}$ . For formulas concerning the coefficients of elliptic curves, we refer to [Reference Deligne16].

Recall that we have $\textsf{tmf}_0(2)_{(3)} \simeq \tau _{\geqslant 0}\textsf{tmf}(2)_{(3)}^{hC_2}$ . By [Reference Stojanoska53, §7], we know that $\pi _*\textsf{tmf}(2)_{(3)} \cong \mathbb{Z}_{(3)}[\lambda _1, \lambda _2]$ with $|\lambda _i| = 4$ and with $C_2$ -action given by $\lambda _1 \mapsto \lambda _2$ and $\lambda _2 \mapsto \lambda _1$ [Reference Stojanoska53, Lemma 7.3]. Since $|C_2|$ is invertible in $\pi _*\textsf{tmf}(2)_{(3)}$ , the $E^2$ -page of the homotopy fixed point spectral sequence is given by:

\begin{equation*} H^*(C_2, \pi _*\textsf{tmf}(2)_{(3)}) = H^0(C_2, \pi _*\textsf{tmf}(2)_{(3)}) = \pi _*(\textsf{tmf}(2)_{(3)})^{C_2}. \end{equation*}

Thus, we have

\begin{equation*} \pi _*\textsf{tmf}_0(2)_{(3)} = \mathbb {Z}_{(3)}[\lambda _1 + \lambda _2, \lambda _1\lambda _2] = \mathbb {Z}_{(3)}[a_2, a_4] \end{equation*}

with $a_2 = -(\lambda _1 + \lambda _2)$ and $a_4 = \lambda _1 \lambda _2$ . Recall the following facts about the homotopy of $\textsf{tmf}_{(3)}$ (see for instance [Reference Douglas, Francis, Henriques and Michael Hill18, p. 192]), we have

\begin{equation*} \pi _*\textsf{tmf}_{(3)} = \begin {cases} \mathbb {Z}_{(3)}\{1\}, & * = 0; \\[5pt] \mathbb {Z}/3\mathbb {Z}\{\alpha _1\} & * = 3; \\[5pt] \mathbb {Z}_{(3)}\{c_4\}, & * = 8; \\[5pt] \mathbb {Z}_{(3)}\{c_6\}, & * = 12; \\[5pt] 0, & * = 4,5,6,7; \end {cases} \end{equation*}

where $\alpha _1$ is the image of $\alpha _1 \in \pi _3(S_{(3)})$ under $\pi _3(S_{(3)}) \to \pi _3\textsf{tmf}_{(3)}$ . By [Reference Stojanoska53, Proof of Proposition 10.3], we have that the map $\pi _*\textsf{tmf}_{(3)} \to \pi _*\textsf{tmf}_0(2)_{(3)}$ satisfies $c_4 \mapsto 16a_2^2 -48 a_4$ and $c_6 \mapsto -64 a_2^3 + 288 a_2a_4$ . (There is a discrepancy between our sign for $c_6$ and that in [Reference Stojanoska53].)

We know from personal communication with Mike Hill that there is a fiber sequence of $\textsf{tmf}_0(2)_{(3)}$ -modules:

\begin{equation*} \textsf{Tmf}_0(2)_{(3)} \longrightarrow \textsf{tmf}_0(2)_{(3)}[a_2^{-1}] \times \textsf{tmf}_0(2)_{(3)}[a_4^{-1}] \buildrel f \over \longrightarrow \textsf{tmf}_0(2)_{(3)}[(a_2a_4)^{-1}]. \end{equation*}

See [Reference Hill and Meier24, Proposition 4.24] for the analogous statement at $p = 2$ . The kernel of $\pi _*(f)$ has $\mathbb{Z}_{(3)}$ -basis:

\begin{equation*} \left\{\left(a_2^n a_4^m, -a_2^n a_4^m\right) | n, m \in \mathbb {N} \right\} \end{equation*}

and the cokernel has $\mathbb{Z}_{(3)}$ -basis:

\begin{equation*} \left\{ \frac {1}{a_2^n a_4^m} \mid n \geqslant 1, m \geqslant 1 \right\}. \end{equation*}

We get that in negative degrees $\pi _*\textsf{Tmf}_0(2)_{(3)}$ is given by:

\begin{equation*} \bigoplus _{n,m \geqslant 1} \mathbb {Z}_{(3)}\left\{ \frac {1}{a_2^na_4^m}\right\}, \end{equation*}

where $\frac{1}{a_2^n a_4^m}$ has degree $-4n-8m-1$ . The $\pi _*\textsf{tmf}_0(2)_{(3)}$ -action is given by:

\begin{equation*} a_2 \cdot \frac {1}{a_2^na_4^m} = \begin {cases} \dfrac{1}{a_2^{n-1}a_4^m}, & \text { if } n \geqslant 2 \\[12pt] 0, & \text { otherwise}, \end {cases} \end{equation*}

and analogously for $a_4$ .

By the gap theorem (see for instance [Reference Konter28]), we have $\pi _*\textsf{Tmf}_{(3)} \cong 0$ for $-21 \lt * \lt 0$ .

Lemma 2.1.

\begin{equation*} \pi _*(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}) \cong \mathbb {Z}_{(3)}[a_2,a_4,r]/r^3+a_2r^2 +a_4 r \;=\!:\; \mathbb {Z}_{(3)}[a_2,a_4,r]/I\end{equation*}

where $r$ has degree $4$ and is mapped to 0 under the multiplication map.

Proof. As above, we use that we have an equivalence of $\textsf{tmf}_{(3)}$ -modules $\textsf{tmf}_{(3)} \wedge T \simeq \textsf{tmf}_0(2)_{(3)}$ . Here, $T$ is defined by the cofiber sequences:

\begin{equation*}S^3_{(3)} \stackrel {\alpha _1}{\longrightarrow } S^0_{(3)} \longrightarrow \text {cone}(\alpha _1) \longrightarrow S^4_{(3)}\end{equation*}

and

\begin{equation*} S^7_{(3)} \stackrel {\phi }{\longrightarrow } \text {cone}(\alpha _1) {\longrightarrow } T {\longrightarrow } S^8_{(3)},\end{equation*}

where $S^7_{(3)} \stackrel {\phi }{\longrightarrow } \text{cone}(\alpha _1) \to S^4_{(3)}$ is equal to $\alpha _1$ . We get an equivalence of left $\textsf{tmf}_0(2)_{(3)}$ -modules:

\begin{equation*}\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)} \simeq \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} (\textsf{tmf}_{(3)} \wedge T) \simeq \textsf{tmf}_0(2)_{(3)} \wedge T. \end{equation*}

Smashing the above cofiber sequences with $\textsf{tmf}_0(2)_{(3)}$ gives cofiber sequences of $\textsf{tmf}_0(2)_{(3)}$ -modules:

\begin{equation*} \Sigma ^3 \textsf{tmf}_0(2)_{(3)} \stackrel {\bar {\alpha }_1} {\longrightarrow} \textsf{tmf}_0(2)_{(3)} \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1) \stackrel {\delta } {\longrightarrow} \Sigma ^4 \textsf{tmf}_0(2)_{(3)} \end{equation*}

and

\begin{equation*} \Sigma ^7 \textsf{tmf}_0(2)_{(3)} \stackrel {\bar {\phi }}{\longrightarrow} \textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1) \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge T \stackrel {\Delta }{\longrightarrow} \Sigma ^8 \textsf{tmf}_0(2)_{(3)}. \end{equation*}

The map $\bar{\alpha }_1$ is zero in the derived category of $\textsf{tmf}_0(2)_{(3)}$ -modules, because $\pi _*(\textsf{tmf}_0(2)_{(3)})$ is concentrated in even degrees. We therefore get an equivalence of $\textsf{tmf}_0(2)_{(3)}$ -modules:

\begin{equation*} \textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1) \simeq \textsf{tmf}_0(2)_{(3)} \vee \Sigma ^4 \textsf{tmf}_0(2)_{(3)}.\end{equation*}

This implies that $\textsf{tmf}_0(2)_{(3)} \wedge \text{cone}(\alpha _1)$ has nontrivial homotopy groups only in even degrees, and therefore that $\bar{\phi }$ is zero in the derived category of $\textsf{tmf}_0(2)_{(3)}$ -modules. We get an equivalence of $\textsf{tmf}_0(2)_{(3)}$ -modules:

\begin{equation*} \textsf{tmf}_0(2)_{(3)} \wedge T \simeq \textsf{tmf}_0(2)_{(3)} \vee \Sigma ^4\textsf{tmf}_0(2)_{(3)} \vee \Sigma ^8 \textsf{tmf}_0(2)_{(3)}.\end{equation*}

We can assume that the map $\textsf{tmf}_{(3)} \to \textsf{tmf}_0(2)_{(3)}$ factors in the derived category of $\textsf{tmf}_{(3)}$ -modules as:

\begin{equation*} \textsf{tmf}_{(3)} \longrightarrow {} \textsf{tmf}_{(3)} \wedge \text {cone}(\alpha _1) \longrightarrow {} \textsf{tmf}_{(3)} \wedge T \stackrel {\simeq } {\longrightarrow} \textsf{tmf}_0(2)_{(3)} \end{equation*}

This implies that the inclusion in the first smash factor:

\begin{equation*}\eta _L\;:\; \textsf{tmf}_0(2)_{(3)} \to \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)} \end{equation*}

is given by:

\begin{equation*} \textsf{tmf}_0(2)_{(3)} \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1) \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge T \stackrel {\simeq } {\longrightarrow} \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(2)}} \textsf{tmf}_0(2)_{(3)}. \end{equation*}

We obtain that the map:

\begin{equation*} \textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1) \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge T \simeq \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)} \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \end{equation*}

is a left inverse for $\textsf{tmf}_0(2)_{(3)} \to \textsf{tmf}_0(2)_{(3)} \wedge \text{cone}(\alpha _1)$ . It is also clear that the inclusion in the second smash factor $\eta _R \;:\; \textsf{tmf}_0(2)_{(3)} \to \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}$ is given by:

\begin{equation*} \textsf{tmf}_0(2)_{(3)} \stackrel {\simeq } {\longrightarrow} \textsf{tmf}_{(3)} \wedge T \longrightarrow {} \textsf{tmf}_0(2)_{(3)} \wedge T \stackrel {\simeq } \longrightarrow \textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}. \end{equation*}

We claim that

\begin{equation*} \eta _R(a_2) \in \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge T) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1)) \end{equation*}

maps to three times a unit under:

\begin{equation*} \delta _4 \;:\; \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1)) \to \pi _4(\Sigma ^4 \textsf{tmf}_0(2)_{(3)}) \cong \mathbb {Z}_{(3)}.\end{equation*}

By commutativity of the diagram:

it suffices to show that $a_2 \in \pi _4(\textsf{tmf}_{(3)} \wedge T)$ maps to three times a unit under the bottom map. This follows by the exact sequence:

We define $r$ to be the unique element in $\pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text{cone}(\alpha _1))$ that maps to that unit under $\delta _4$ and that is in the kernel of the composition of $\pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text{cone}(\alpha _1)) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge T)$ and the multiplication map

\begin{equation*} \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge T) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}) \to \pi _4(\textsf{tmf}_0(2)_{(3)}). \end{equation*}

We have that $3r- \eta _R(a_2)$ is in the image of $\pi _4(\textsf{tmf}_0(2)_{(3)}) \to \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text{cone}(\alpha _1))$ and thus can be written as $3r- \eta _R(a_2) = n \cdot a_2$ for an $n \in \mathbb{Z}_{(3)}$ . Applying the map

\begin{equation*} \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge \text {cone}(\alpha _1)) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge T) \cong \pi _4(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}) \to \pi _4(\textsf{tmf}_0(2)_{(3)}) \end{equation*}

gives $n = -1$ .

We claim that $\eta _R(a_4) \in \pi _8(\textsf{tmf}_0(2)_{(3)} \wedge T)$ maps to three times a unit under

\begin{equation*}\Delta _8 \;:\; \pi _8(\textsf{tmf}_0(2)_{(3)} \wedge T) \to \pi _8(\Sigma ^8\textsf{tmf}_0(2)_{(3)}). \end{equation*}

As above one sees that it suffices to show that $a_4$ maps to three times a unit under the map $\pi _8(\textsf{tmf}_{(3)} \wedge T) \to \pi _8(\Sigma ^8 \textsf{tmf}_{(3)})$ . For this, we consider the exact sequence:

Using that $\pi _4(\textsf{tmf}_{(3)}) = 0 = \pi _5(\textsf{tmf}_{(3)})$ , one gets that $\pi _8(\textsf{tmf}_{(3)}) \cong \pi _8(\textsf{tmf}_{(3)} \wedge \text{cone}(\alpha _1))$ , and under this isomorphism the first map in the exact sequence identifies with

\begin{equation*}\pi _8(\textsf{tmf}_{(3)}) \cong \mathbb {Z}_{(3)}\{c_4\} \to \pi _8(\textsf{tmf}_0(2)_{(3)}) \cong \mathbb {Z}_{(3)}\{a_2^2\} \oplus \mathbb {Z}_{(3)}\{a_4\}, \quad c_4 \mapsto 16a_2^2- 48a_4.\end{equation*}

As $\pi _6(\textsf{tmf}_{(3)}) = 0 = \pi _7(\textsf{tmf}_{(3)})$ , one gets that $\pi _7(\textsf{tmf}_{(3)} \wedge \text{cone}(\alpha _1)) \cong \pi _7(\Sigma ^4 \textsf{tmf}_{(3)})$ , and under this isomorphism the third map in the exact sequence identifies with

\begin{equation*} \pi _8(\Sigma ^8\textsf{tmf}_{(3)}) \cong \mathbb {Z}_{(3)} \to \pi _3(\textsf{tmf}_{(3)}) \cong \mathbb {Z}/{3\mathbb {Z}}\{\alpha _1\}, \quad 1 \mapsto \alpha _1.\end{equation*}

One obtains that the second map in the exact sequence maps $a_4$ to $3 \cdot m$ and $a_2^2$ to $9 \cdot m$ for a unit $m \in \mathbb{Z}_{(3)}$ .

Since the map $\pi _*(\textsf{tmf}_{(3)}) \to \pi _*(\textsf{tmf}_0(2)_{(3)})$ maps $c_4$ to $16a_2^2 -48 a_4$ , we have the equation:

\begin{equation*} 16 \cdot a_2^2 - 48 \cdot a_4 = 16 \cdot \eta _R(a_2)^2- 48 \cdot \eta _R(a_4) \end{equation*}

in $\pi _*(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)})$ . Replacing $\eta _R(a_2)$ by $3r + a_2$ and using torsion-freeness, one gets the equation:

\begin{equation*}\eta _R(a_4) = a_4 + 3r^2 + 2 a_2 r.\end{equation*}

We apply the map $\Delta _8\;:\; \pi _8(\textsf{tmf}_0(2)_{(3)} \wedge T) \to \pi _8(\Sigma ^8 \textsf{tmf}_0(2)_{(3)})$ to this equation and obtain by torsion-freeness of $\pi _*(\textsf{tmf}_0(2)_{(3)})$ :

\begin{equation*} \Delta _8(r^2) = m. \end{equation*}

We thus have an isomorphism of left $\pi _*(\textsf{tmf}_0(2)_{(3)})$ -modules:

\begin{equation*} \pi _*(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)}) \cong \pi _*(\textsf{tmf}_0(2)_{(3)}) \oplus \pi _*(\textsf{tmf}_0(2)_{(3)})\{r\} \oplus \pi _*(\textsf{tmf}_0(2)_{(3)}) \{r^2\}.\end{equation*}

Since the map $\pi _*(\textsf{tmf}_{(3)}) \to \pi _*(\textsf{tmf}_0(2)_{(3)})$ maps $c_6$ to $-64 a_2^3 + 288 a_2 a_4$ , we have

\begin{equation*} -64 \cdot a_2^3 + 288 \cdot a_2\cdot a_4 = -64 \cdot \eta _R(a_2)^3 + 288 \cdot \eta _R(a_2)\cdot \eta _R(a_4) \end{equation*}

in $\pi _*(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)})$ . Replacing $\eta _R(a_2)$ by $3r + a_2$ and $\eta _R(a_4)$ by $a_4 + 3r^2 + 2a_2 r$ and using torsion-freeness, one gets

\begin{equation*} r^3 + a_2 r^2 + a_4 r = 0.\end{equation*}

This implies the lemma.

Theorem 2.3. The canonical map $\textsf{tmf}_0(2)_{(3)} \rightarrow \textsf{THH}^{\textsf{tmf}_{(3)}}(\textsf{tmf}_0(2)_{(3)})$ is far from being an equivalence. More precisely,

\begin{align*} \textsf{THH}_*^{\textsf{tmf}_{(3)}}(\textsf{tmf}_0(2)_{(3)}) & \cong \mathbb{Z}_{(3)}[a_2,a_4] \oplus \bigoplus _{i \geqslant 0} \Sigma ^{14i +5} \mathbb{Z}_{(3)}[a_2] \\[5pt] & \cong \pi _*\textsf{tmf}_0(2)_{(3)} \oplus \bigoplus _{i \geqslant 0} \Sigma ^{14i +5} \pi _*\textsf{tmf}_0(2)_{(3)}/(a_4). \end{align*}

Proof. We use the Tor spectral sequence

\begin{equation*} E^2_{*,*} = \textsf{Tor}_{*,*}^{\pi _*(\textsf{tmf}_0(2)_{(3)} \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)})}(\pi _*\textsf{tmf}_0(2)_{(3)},\pi _*\textsf{tmf}_0(2)_{(3)}) \Rightarrow \pi _*\textsf{THH}^{\textsf{tmf}_{(3)}}(\textsf{tmf}_0(2)_{(3)}) \end{equation*}

in order to calculate relative topological Hochschild homology. For determining

\begin{equation*} \textsf{Tor}_{*,*}^{\mathbb {Z}_{(3)}[a_2,a_4,r]/I}(\mathbb {Z}_{(3)}[a_2,a_4], \mathbb {Z}_{(3)}[a_2,a_4])\end{equation*}

we consider the free resolution of $\mathbb{Z}_{(3)}[a_2,a_4]$ as a $\mathbb{Z}_{(3)}[a_2,a_4,r]/I$ -module:

\begin{equation*}{\cdots \longrightarrow \Sigma ^{12}\mathbb {Z}_{(3)}[a_2,a_4,r]/I {\stackrel {r^2+a_2r+a_4} \longrightarrow} \Sigma ^4\mathbb {Z}_{(3)}[a_2,a_4,r]/I {\stackrel {r} \longrightarrow} \mathbb {Z}_{(3)}[a_2,a_4,r]/I.} \end{equation*}

Applying $(\!-\!) \otimes _{\mathbb{Z}_{(3)}[a_2,a_4,r]/I} \mathbb{Z}_{(3)}[a_2,a_4]$ yields

\begin{equation*}\cdots \longrightarrow {}\Sigma ^{12}\mathbb {Z}_{(3)}[a_2,a_4] {\stackrel {a_4} \longrightarrow} \Sigma ^4\mathbb {Z}_{(3)}[a_2,a_4] {\stackrel {0} \longrightarrow} \mathbb {Z}_{(3)}[a_2,a_4] \end{equation*}

and hence we get

\begin{equation*} E^2_{n,*} = \begin {cases} \pi _*(\textsf{tmf}_0(2)_{(3)}), & n = 0; \\[5pt] \Sigma ^{4 +12k} \mathbb {Z}_{(3)}[a_2], &n = 2k+1, k \geqslant 0; \\[5pt] 0, & \text {otherwise}. \end {cases} \end{equation*}

We note that all nontrivial classes in positive filtration degree have an odd total degree. Since the edge morphism $\pi _*(\textsf{tmf}_0(2)_{(3)}) \to \textsf{THH}^{\textsf{tmf}_{(3)}}_*(\textsf{tmf}_0(2)_{(3)})$ is the unit, the classes in filtration degree zero cannot be hit by a differential and the spectral seqence collapses at the $E^2$ -page. Since $E^2_{n,m} = E^\infty _{n,m}$ is a free $\mathbb{Z}_{(3)}$ -module for all $n,m$ , there are no additive extensions.

As for the connective covers, we have an equivalence of $\textsf{Tmf}_{(3)}$ -modules $\textsf{Tmf}_{(3)} \wedge T \simeq \textsf{Tmf}_0(2)$ [Reference Mathew36, §4.6] such that the map $\textsf{Tmf}_{(3)} \to \textsf{Tmf}_0(2)_{(3)}$ factors in the derived category of $\textsf{Tmf}_{(3)}$ -modules as:

\begin{equation*} \textsf{Tmf}_{(3)} \to \textsf{Tmf}_{(3)} \wedge \text {cone}(\alpha _1) \to \textsf{Tmf}_{(3)} \wedge T \simeq \textsf{Tmf}_0(2)_{(3)}. \end{equation*}

Using the gap theorem, one can argue analogously to the proof of Lemma 2.1 to show that

\begin{equation*} \pi _*(\textsf{Tmf}_0(2)_{(3)} \wedge _{\textsf{Tmf}_{(3)}} \textsf{Tmf}_0(2)_{(3)}) \cong \pi _*\textsf{Tmf}_0(2)_{(3)}[r]/{(r^3 + a_2r^2 + a_4r)}.\end{equation*}

Theorem 2.4. There is an additive isomorphism:

\begin{equation*} \textsf{THH}^{\textsf{Tmf}_{(3)}}(\textsf{Tmf}_0(2)_{(3)}) \cong \pi _*\textsf{Tmf}_0(2)_{(3)} \oplus \bigoplus _{i \geqslant 0}\Sigma ^{14i + 5} \mathbb {Z}_{(3)}[a_2] \oplus \bigoplus _{i \geqslant 1}\Sigma ^{14i} \mathbb {Z}_{(3)}\left\{\frac {1}{a_2^i a_4}\right\}. \end{equation*}

Proof. As above we have the following free resolution of $\pi _*(\textsf{Tmf}_0(2)_{(3)})$ as a module over

\begin{equation*} C_* = \pi _*(\textsf{Tmf}_0(2)_{(3)} \wedge _{\textsf{Tmf}_{(3)}} \textsf{Tmf}_0(2)_{(3)}): \end{equation*}

\begin{equation*}\cdots {\stackrel {r} {\longrightarrow}} \Sigma ^{12}C_* {\stackrel {r^2 + a_2 r + a_4} \longrightarrow} \Sigma ^4 C_* {\stackrel {r} \longrightarrow} C_* \longrightarrow {}\pi _*\textsf{Tmf}_0(2)_{(3)} \longrightarrow {} 0.\end{equation*}

We get that the $E^2$ -page of the Tor spectral sequence:

\begin{equation*} E^2_{*,*} = \textsf{Tor}^{C_*}_{*,*}(\pi _*\textsf{Tmf}_0(2)_{(3)}, \pi _*\textsf{Tmf}_0(2)_{(3)}) \longrightarrow {} \pi _*\textsf{THH}^{\textsf{Tmf}_{(3)}}(\textsf{Tmf}_0(2)_{(3)}) \end{equation*}

is given by:

\begin{align*} E^2_{n, *} & = \begin{cases} \pi _*\textsf{Tmf}_0(2)_{(3)}, & n = 0; \\[5pt] \Sigma ^{4+12k} \pi _*\textsf{Tmf}_0(2)_{(3)}/{a_4}, & n = 2k + 1, k \geqslant 0; \\[5pt] \ker \bigl ( \Sigma ^{12k} \pi _*\textsf{Tmf}_0(2)_{(3)} {\stackrel {\cdot a_4} \longrightarrow} \Sigma ^{12(k-1) +4} \pi _*\textsf{Tmf}_0(2)_{(3)}\bigr ), & n = 2k, k \gt 0 \end{cases} \\[5pt] & = \begin{cases} \pi _*\textsf{Tmf}_0(2)_{(3)}, & n = 0; \\[5pt] \Sigma ^{4+12k} \mathbb{Z}_{(3)}[a_2], & n = 2k+1, k \geqslant 0; \\[5pt] \Sigma ^{12k} \bigoplus _{n \geqslant 1} \mathbb{Z}_{(3)}\{\dfrac{1}{a_2^n a_4}\}, & n = 2k, k \gt 0. \end{cases} \end{align*}

Since all nontrivial classes in positive filtration have an odd total degree, the spectral sequence collapses at the $E^2$ -page. There are no additive extensions, because the $E^{\infty } = E^2$ -page is a free $\mathbb{Z}_{(3)}$ -module.

Theorem 2.5. We have an additive isomorphism:

\begin{equation*} \pi _*\textsf{THH}^{\textsf{tmf}_0(2)_{(3)}}(\textsf{tmf}(2)_{(3)}) \cong \mathbb {Z}_{(3)}[\lambda _1, \lambda _2] \oplus \bigoplus _{i \geqslant 0} \Sigma ^{10i + 5}\mathbb {Z}_{(3)}[\lambda _1] .\end{equation*}

Proof. The map $\pi _*\textsf{tmf}_0(2)_{(3)} \to \pi _*\textsf{tmf}(2)_{(3)}$ is given by $\mathbb{Z}_{(3)}[\lambda _1 + \lambda _2, \lambda _1\lambda _2] \to \mathbb{Z}_{(3)}[\lambda _1, \lambda _2]$ . One easily sees that

\begin{equation*} \mathbb {Z}_{(3)}[\lambda _1, \lambda _2] \cong \mathbb {Z}_{(3)}[\lambda _1 + \lambda _1, \lambda _1\lambda _2] \oplus \mathbb {Z}_{(3)}[\lambda _1 + \lambda _2, \lambda _1\lambda _2]\lambda _1,\end{equation*}

so $\pi _*\textsf{tmf}(2)_{(3)}$ is a free $\pi _*\textsf{tmf}_0(2)_{(3)}$ -module. We get

\begin{align*} \pi _*(\textsf{tmf}(2)_{(3)} \wedge _{\textsf{tmf}_0(2)_{(3)}} \textsf{tmf}(2)_{(3)}) & = \pi _*\textsf{tmf}(2)_{(3)} \otimes _{\pi _*\textsf{tmf}_0(2)_{(3)}} \pi _*\textsf{tmf}(2)_{(3)} \\[5pt] & = \mathbb{Z}_{(3)}[\lambda _1, \lambda _2, a]/{a^2 + \lambda _1 a - \lambda _2 a}, \end{align*}

where $a = \eta _R(\lambda _1) - \lambda _1$ . Let $C_* = \mathbb{Z}_{(3)}[\lambda _1, \lambda _2, a]/{a^2 + \lambda _1 a - \lambda _2 a}$ . We have the following free resolution of $\pi _*\textsf{tmf}(2)_{(3)}$ as a $C_*$ -module:

\begin{equation*}\cdots {\stackrel {\cdot a} \longrightarrow} \Sigma ^{8}C_* {\stackrel {\cdot (a + \lambda _1 - \lambda _2)} \longrightarrow} \Sigma ^4C_* {\stackrel {\cdot a} \longrightarrow} C_* \longrightarrow {} \pi _*\textsf{tmf}(2)_{(3)} \longrightarrow {} 0\end{equation*}

Thus, the $E^2$ -page of the Tor spectral sequence:

\begin{equation*} E^2_{*,*} = \textsf{Tor}^{C_*}_{*,*}(\pi _*\textsf{tmf}(2)_{(3)}, \pi _*\textsf{tmf}(2)_{(3)}) \longrightarrow {} \pi _*\textsf{THH}^{\textsf{tmf}_0(2)_{(3)}}(\textsf{tmf}(2)_{(3)}) \end{equation*}

is given by:

\begin{equation*} E^2_{n, *} = \begin {cases} \pi _*\textsf{tmf}(2)_{(3)}, & n = 0; \\[5pt] \Sigma ^{8k+4}\pi _*\textsf{tmf}(2)_{(3)}/{(\lambda _1- \lambda _2)}, & n = 2k+1, k \geqslant 0; \\[5pt] 0, & \text { otherwise}. \end {cases} \end{equation*}

Since the nontrivial classes in positive filtration have odd total degree, the spectral sequence collapses at the $E^2$ -page. There are no additive extensions, because the $E^2 = E^{\infty }$ -page is a free $\mathbb{Z}_{(3)}$ -module.

2.3. The discriminant map

Let $A \to B$ be a map of commutative ring spectra and $G$ be a finite group that acts on $B$ through $A$ -algebra maps such that $A \to B^{hG}$ is an equivalence. One then can define the discriminant map $\mathfrak{d}_{B|A} \;:\; B \rightarrow F_A(B,A)$ [Reference Rognes44, Definition 6.4.5]. The map $\mathfrak{d}_{B|A}$ is right adjoint to the trace pairing:

\begin{equation*} B \wedge _A B {\stackrel {\mu } \longrightarrow} B {\stackrel {tr} \longrightarrow} A. \end{equation*}

Here, $\mu$ is the multiplication of $B$ , and $tr$ is defined to be the composite:

\begin{equation*} B \longrightarrow {} B_{hG} \stackrel {N} \longrightarrow B^{hG} \xleftarrow {\simeq } A,\end{equation*}

where $N$ is the norm map. One has that $(A \to B) \circ tr$ is homotopic to $\sum _{g \in G} g$ . If $A \rightarrow B$ is a $G$ -Galois extension, then $\mathfrak{d}_{B|A}$ is a weak equivalence [Reference Rognes44, Proposition 6.4.7]. Rognes proposes that the deviation of $\mathfrak{d}_{B|A}$ from being a weak equivalence might be used for measuring ramification. Note that if $A$ and $B$ are connective, then $tr$ and $\mathfrak{d}_{B|A}$ are defined even if only $A \simeq \tau _{\geqslant 0} B^{hG}$ . We show in the examples of $\ell _p \rightarrow ku_p$ and $ko \rightarrow ku$ that $\mathfrak{d}$ does notice the ramification, but it does not give any information about the type of ramification.

Proposition 2.6. There is a cofiber sequence:

\begin{equation*} ku_p \stackrel {\mathfrak {d}_{ku_p|\ell _p}} \longrightarrow F_{\ell _p}(ku_p, \ell _p) \longrightarrow {} \bigvee _{i=1}^{p-2} \Sigma ^{-2p+2i+2} H\mathbb {Z}_p. \end{equation*}

Proof. We know that $F_{\ell _p}(ku_p, \ell _p)$ can be decomposed as:

\begin{equation*} F_{\ell _p}(ku_p, \ell _p) \simeq F_{\ell _p}\left(\bigvee _{i=0}^{p-2} \Sigma ^{2i} \ell _p, \ell _p\right) \cong \prod _{i=0}^{p-2} \Sigma ^{-2i} \ell _p \simeq \bigvee _{i=0}^{p-2} \Sigma ^{-2i} \ell _p \end{equation*}

and $\mathfrak{d}_{ku_p|\ell _p}$ can be identified with a map:

\begin{equation*} \bigvee _{i=0}^{p-2} \Sigma ^{2i} \ell _p \rightarrow \bigvee _{i=0}^{p-2} \Sigma ^{-2i} \ell _p. \end{equation*}

As $\pi _*ku_p$ is a free graded $\pi _*\ell _p$ -module, we can calculate the effect of $\mathfrak{d}_{ku_p|\ell _p}$ algebraically via the trace pairing: the element $\Sigma ^{2i}1 \in \pi _*\Sigma ^{2i} \ell _p$ corresponds to $u^i$ , and it maps an element $u^j$ to $tr(u^i \cdot u^j)$ . Since $(\ell _p \to ku_p) \circ tr$ is $\sum _{g \in C_{p-1}} g$ , the sum of $(p-1)$ $\ell _p$ -algebra maps, we have

\begin{equation*} tr(u^{i+j}) = \begin {cases} 0, & (p-1) \nmid i+j, \\[5pt] (p-1)u^{i+j}, & (p-1) \mid i+j.\end {cases}\end{equation*}

Hence on the level of homotopy groups, $\mathfrak{d}_{ku_p|\ell _p}$ maps $1 \in \pi _0 \Sigma ^0 \ell _p$ to $(p-1)\cdot 1 \in \pi _*\Sigma ^0\ell _p = \pi _*\ell _p$ and for $0 \lt i \leqslant p-2$ it maps $\Sigma ^{2i}1 \in \pi _*\Sigma ^{2i} \ell _p$ via multiplication with $(p-1)v_1 = (p-1)u^{p-1}$ to $\pi _*\Sigma ^{-2p+2i+2} \ell _p$ . On the summands $\Sigma ^{2i}\ell _p$ , we get the following maps:

As $(p-1)$ is a unit in $\pi _0(\ell _p)$ the cofiber of

\begin{equation*} { \Sigma ^{2i} \ell _p {\stackrel {(p-1)v_1} \longrightarrow} \Sigma ^{-2p+2i+2} \ell _p}\end{equation*}

is $\Sigma ^{-2p+2i+2} H\mathbb{Z}_p$ .

Note that $ko \simeq \tau _{\geqslant 0}ku^{hC_2}$ , but as the trace map $tr \;:\; ku \rightarrow ku^{hC_2}$ has the connective spectrum $ku$ as a source, it factors through $\tau _{\geqslant 0}ku^{hC_2}\simeq ko$ , and we obtain a discriminant $\mathfrak{d}_{ku|ko} \;:\; ku \rightarrow F_{ko}(ku,ko)$ . We fix notation for $\pi _*ko$ as:

\begin{equation*} \pi _*ko = \mathbb {Z}[\eta, y, \omega ]/(2\eta,\eta ^3,\eta y, y^2-4\omega )\end{equation*}

with $|\eta | = 1$ , $|y|=4$ , and $|\omega |=8$ .

Proof. The cofiber sequence $\Sigma ko {\stackrel {\eta } \longrightarrow} ko {\stackrel {c} \longrightarrow} ku {\stackrel {\delta } \longrightarrow} \Sigma ^2ko {\stackrel {\eta } \longrightarrow} \Sigma ko$ induces a cofiber sequence:

\begin{equation*} {F_{ko}(\Sigma ko, ko) {\stackrel {\eta } \longrightarrow} F_{ko}(\Sigma ^2 ko, ko) {\stackrel {\delta } \longrightarrow} F_{ko}(ku, ko) {\stackrel {c} \longrightarrow} F_{ko}(ko, ko) {\stackrel {\eta } \longrightarrow} F_{ko}(\Sigma ko, ko)} \end{equation*}

which is equivalent to

\begin{equation*} {\Sigma ^{-1} ko {\stackrel {\eta } \longrightarrow} \Sigma ^{-2} ko {\stackrel {\delta } \longrightarrow} F_{ko}(ku, ko) {\stackrel {-{c}}\longrightarrow} ko {\stackrel {-{\eta }} \longrightarrow} \Sigma ^{-1}ko.} \end{equation*}

This is the twofold desuspension of the cofiber sequence of and hence,

\begin{equation*} F_{ko}(ku, ko) \simeq \Sigma ^{-2}ku. \end{equation*}

We consider the composition $c_* \circ \mathfrak{d}_{ku|ko} \;:\; ku \rightarrow F_{ko}(ku,ko) \rightarrow F_{ko}(ku,ku)$ . As $c_*$ is part of the cofiber sequence

\begin{equation*} {F_{ko}(ku,\Sigma ko) {\stackrel {\eta _*} \longrightarrow} F_{ko}(ku,ko) {\stackrel {c_*} \longrightarrow} F_{ko}(ku,ku) } \end{equation*}

and as $\eta$ is trivial on $ku$ , we know that $c_*$ induces a monomorphism on the level of homotopy groups.

As $\mathfrak{d}_{ku|ko}$ is adjoint to the trace pairing, the composite

\begin{equation*} {\pi _*ku \longrightarrow {} \pi _*F_{ko}(ku,ko) \longrightarrow {} \pi _*F_{ko}(ku,ku)}\end{equation*}

can be identified with

\begin{equation*} {\pi _*ku \longrightarrow {} \pi _*F_{ko}(ku,ku) {\stackrel {(\textrm{id} +t)_*} \longrightarrow} F_{ko}(ku,ku)}\end{equation*}

where $t$ denotes the generator of $C_2$ , and the first map is adjoint to the multiplication $ku \wedge _{ko} ku \rightarrow ku$ .

The target of $c_*$ is $F_{ko}(ku,ku) \simeq F_{ku}(ku \wedge _{ko} ku, ku)$ , and we know by work of the first author, documented in [Reference Dundas, Lindenstrauss and Richter20, Proof of Lemma 0.1] that

\begin{equation*} \pi _*F_{ku}(ku \wedge _{ko} ku, ku) \cong \textsf{Hom}_{ku_*}(ku_*[s]/(s^2-su), \Sigma ^{-*}ku_*), \end{equation*}

so we can control the effect of $c_* \circ \mathfrak{d}_{ku|ko}$ on homotopy groups.

Note that $t$ induces a $ku$ -linear map $t_* \;:\; ku \rightarrow t^*ku$ , where $t^*ku$ is the $ku$ -module given by restriction of scalars along $t$ .

As $t^2=\textrm{id}$ , we therefore obtain

\begin{equation*} F_{ko}(ku,ku) {\stackrel {t_*} \longrightarrow} F_{ko}(ku, t^*ku)\end{equation*}

and a commutative diagram

Here, $\beta$ induces the map on $\pi _*$ that sends an $f \;:\; (ku \wedge _{ko} ku)_* \rightarrow \Sigma ^{-i}ku_*$ to

\begin{equation*} {(ku \wedge _{ko} ku)_* {\stackrel {(t \wedge \textrm{id})_*} \longrightarrow} (ku \wedge _{ko} ku)_* {\stackrel {f} \longrightarrow} \Sigma ^{-i}ku_* {\stackrel {t} \longrightarrow} \Sigma ^{-i}ku_*}. \end{equation*}

If we denote the right unit $\eta _R \;:\; ku \rightarrow ku \wedge _{ko} ku$ applied to $u$ by $u_r$ , then we have the relation $2s + u_r = u$ . As $(t \wedge \textrm{id})_*(u) = -u$ and $(t \wedge \textrm{id})_*(u_r) = u_r$ , this implies that

\begin{equation*} (t \wedge \textrm{id})_*(2s) = 2s-2u. \end{equation*}

Torsion-freeness then yields $ (t \wedge \textrm{id})_*(s) = s-u$ .

The adjoint of the multiplication map $\pi _*ku \rightarrow \pi _*F_{ko}(ku, ku)$ maps $u^i$ to the map that sends $1$ to $\Sigma ^{-2i}u^i$ and $s$ to zero. Therefore, the composite $c_* \circ \mathfrak{d}_{ku|ko}$ maps $u^i$ to the map with values $1 \mapsto \Sigma ^{-2i}(u^i+(\!-\!1)^iu^i)$ and

\begin{equation*} s \mapsto (s-u)u^i \mapsto -t(u^{i+1}) = (\!-\!1)^iu^{i+1}. \end{equation*}

In order to understand the effect of $\mathfrak{d}_{ku|ko}$ , we consider the diagram

where we can identify $c^* \;:\; \pi _*F_{ko}(ku, ko) \cong \pi _{*+2}(ku) \rightarrow \pi _*(ko)$ with $\pi _*\Sigma ^{-2}\delta$ .

The application of $c^*$ gives the restriction to the unit $c \;:\; ko \rightarrow ku$ . Say $(\mathfrak{d}_{ku|ko})_*(u^{2}) = x \in \pi _{6}(ku)$ . Then $\pi _*\Sigma ^{-2}\delta (x) = \lambda y$ , and as $c_*(y) = 2u^2$ , we obtain that $c^* (\mathfrak{d}_{ku|ko})_*(u^{2}) = \Sigma ^{-4}y$ and therefore $(\mathfrak{d}_{ku|ko})_*(u^{2}) = u^3$ . Similarly $c^*(\mathfrak{d}_{ku|ko})_*(u^{4}) = \Sigma ^{-8}2\omega$ and $(\mathfrak{d}_{ku|ko})_*(u^{4}) = u^5$ . By $\pi _*(ko)$ -linearity, these calculations yield

\begin{equation*} (\mathfrak {d}_{ku|ko})_*(u^{2i}) = u^{2i+1} \end{equation*}

for all $i \geqslant 0$ .

Restriction to the unit of the odd powers of $u$ gives zero.

All the $u^i$ send $s$ to $\pm u^{i+1}$ under $c_* \circ (\mathfrak{d}_{ku|ko})_*$ , so also the odd powers of $u$ have to hit a generator under $(\mathfrak{d}_{ku|ko})_*$ , so as a map from $ku$ to $\Sigma ^{-2}ku$ the map $\mathfrak{d}_{ku|ko}$ has cofiber $\Sigma ^{-2}H\mathbb{Z}$ .

3.1. Log-étaleness

It is shown in [Reference Rognes, Sagave and Schlichtkrull49] and [Reference Sagave50] that $\ell \rightarrow ku_{(p)}$ is log-étale with respect to the log structures that are generated by $v_1$ and by $u$ . We will use the class $u \in \pi _2ku_{(2)}$ in order to define a pre-log structure for $ko_{(2)} \rightarrow ku_{(2)}$ and show that $ko_{(2)} \rightarrow ku_{(2)}$ is not log-étale with respect to this pre-log structure. This indicates that the map is not tamely ramified. We use the notation from [Reference Sagave50].

Let $\omega$ denote the Bott element $\omega \in \pi _8ko_{(2)}$ . The complexification map sends $\omega$ to $u^4$ .

By [Reference Sagave50, Lemma 6.2], we have an exact sequence

Here, $D(u)$ and $D(\omega )$ are the pre-log structures for the elements $u$ and $\omega$ as in [Reference Sagave50, Construction 4.2], and $\textsf{TAQ}^{(-,-)}(-,-)$ is log topological André–Quillen homology [Reference Sagave50, Definition 5.20]. The commutative ring spectrum $C$ is given by $ko_{(2)} \wedge _{S^{\mathcal{J}} D(w)}S^{\mathcal{J}} D(u)$ , where $S^{\mathcal{J}}$ is the functor from commutative $\mathcal{J}$ -space monoids to commutative ring spectra defined in [Reference Sagave and Schlichtkrull52, p. 2139]. For the definition of the functor $\gamma (\!-\!)$ , see [Reference Sagave51, Section 3] and [Reference Sagave50, p. 457]. Using [Reference Sagave50, Lemma 4.6], it follows that $\gamma (D(w))$ and $\gamma (D(u))$ have the homotopy type of the sphere and that $\gamma (D(w)) \to \gamma (D(u))$ is multiplication by $4$ . Therefore, we get

\begin{equation*}\pi _0\Bigl (ku_{(2)} \wedge \gamma (D(u))/{\gamma (D(w))\Bigr )} = \mathbb {Z}/4\mathbb {Z}.\end{equation*}

We want to show that $\pi _1\textsf{TAQ}^C(ku_{(2)}) = 0 = \pi _0\textsf{TAQ}^C(ku_{(2)})$ . By [Reference Basterra6, Lemma 8.2], it suffices to show that $C \to ku_{(2)}$ is an $1$ -equivalence. Since $\pi _1(ku_{(2)}) = 0$ , it is enough to show that the map is an isomorphism on $\pi _0$ . Since $S^{\mathcal{J}} D(w)$ and $S^{\mathcal{J}} D(u)$ are concentrated in nonnegative $\mathcal{J}$ -space degrees by [Reference Rognes, Sagave and Schlichtkrull48, Example 6.8], they are connective. Thus, it is enough to show that $S^{\mathcal{J}} D(w) \to S^{\mathcal{J}} D(u)$ induces an isomorphism on $\pi _0$ . For this, we only have to prove that $H_0( S^{\mathcal{J}} D(w), \mathbb{Z}) \to H_0(S^{\mathcal{J}} D(u), \mathbb{Z})$ is an isomorphism. Since this map is a ring map, we only need to know that both sides are $\mathbb{Z}$ . This follows from [Reference Rognes, Sagave and Schlichtkrull49, Proposition 5.2, Corollary 5.3]. Hence, we obtain the following result:

One could try to distinguish between tame and wild ramification by testing for log-étaleness. In many examples, however, it is less obvious what a suitable log structure would be. Calculations with log structures that are generated by more than one element are challenging because the methods above do not work. For a thorough investigation of log-étaleness and for related calculations, see Lundemo [Reference Lundemo31].

3.2. Ramification and Tate cohomology

In the algebraic context of Galois extensions of number fields and corresponding extension of number rings, tame ramification yields a normal basis and a surjective trace map. Both facts are actually also sufficient in order to distinguish tame from wild ramification. For structured ring spectra, it does not work to impose these properties on the level of homotopy groups, because even for finite faithful Galois extensions these would not hold. Instead, we propose to use the Tate construction in order to understand ramification.

Remark 3.2. Let $G$ be a finite group. Usually one calls a $G$ -module $M$ cohomologically trivial, if $\hat{H}^i(H;\;M) =0$ , for all $i \in \mathbb{Z}$ and all $H \lt G$ . If $M$ is a commutative ring $S$ , however, it suffices to require $\hat{H}^i(G;\;S) =0$ for all $i \in \mathbb{Z}$ : In particular, $\hat{H}^0(G;\;S) =0$ , and hence the norm map $N_G \;:\; S_G \rightarrow S^G$ (resp. the trace map $tr_G \;:\; S \rightarrow S^G$ ) is surjective. Thus, $1_{S^G}$ is in the image of the norm, say $N_G[x] = 1_{S^G}$ for $[x] \in S_G$ . If $H \lt G$ , then we consider the diagram:

and therefore we can express can express $1_{S^H}$ as:

\begin{equation*} 1_{S^H} = i^*(1_{S^G}) = i^*N_G[x] = N_H tr^G_H[x],\end{equation*}

so $1_{S^H}$ is in the image of $N_H$ and $\hat{H}^0(H;\;S) =0$ . But $\hat{H}^*(H;\;S)$ is a graded commutative ring with unit $[1_{S^H}]=0$ , and thus $\hat{H}^*(H;\;S) = 0$ .

The same argument shows that the surjectivity of the trace map suffices for being cohomologically trivial.

In particular, if the Tate cohomology is nontrivial, then the trace map is not surjective and this indicates wild ramification. In the following, we transfer this relationship to structured ring spectra.

We need the following generalization of Tate cohomology; for background, see [Reference Greenlees and May21]. If $E$ is a spectrum with an action of a finite group $G$ , then there is a norm map $N \;:\; E_{hG} \rightarrow E^{hG}$ from the homotopy coinvariants of $E$ with respect to $G$ , $E_{hG}$ , to the homotopy fixed points, $E^{hG}$ . Its cofiber is the Tate construction of $E$ with respect to $G$ , $E^{tG}$ . If $E$ is an Eilenberg–MacLane spectrum $E = HD$ for some abelian group $D$ , then $\pi _*((HD)^{tG}) \cong \hat{H}^{-*}(G;\; D)$ .

Even if $A \rightarrow B$ is a $G$ -Galois extension of ring spectra in the sense of Rognes [Reference Rognes44, Definition 4.1.3], it is not true that this implies that $B$ is faithful as an $A$ -module [Reference Rognes44, Definition 4.3.1]. An example due to Wieland is the $C_2$ -Galois extension $F((BC_2)_+, H\mathbb{F}_2) \rightarrow F((EC_2)_+, H\mathbb{F}_2) \simeq H\mathbb{F}_2$ which is not faithful: the $F((BC_2)_+, H\mathbb{F}_2)$ -module spectrum $(H\mathbb{F}_2)^{tC_2}$ is not trivial, but $H\mathbb{F}_2 \wedge _{F((BC_2)_+, H\mathbb{F}_2)} (H\mathbb{F}_2)^{tC_2} \sim *$ . In fact, a $G$ -Galois extension $A \to B$ is faithful if and only if $B^{tG}$ is contractible [Reference Rognes44, Proposition 6.3.3].

In the following, we denote by $\tau _{\geqslant 0} X$ the connective cover of a spectrum $X$ . Note that for a map $A \rightarrow B$ between connective commutative ring spectra with a finite group $G$ acting on $B$ via commutative $A$ -algebra maps it makes sense to replace the usual homotopy fixed point condition by the condition that $A$ is weakly equivalent to $\tau _{\geqslant 0}B^{hG}$ . In many examples, $B^{hG}$ won’t be connective. The map $A \rightarrow B$ factors through $A \rightarrow B^{hG} \rightarrow B$ , but as $A$ is connective, we can consider the induced map on connective covers and obtain a map of commutative ring spectra:

\begin{equation*} \tau _{\geqslant 0}A = A \rightarrow \tau _{\geqslant 0}B^{hG} \rightarrow \tau _{\geqslant 0}B = B, \end{equation*}

that turns $\tau _{\geqslant 0}B^{hG}$ into a commutative $A$ -algebra spectrum.

For any spectrum $X$ , we denote by $\tau _{\lt 0}X$ the cofiber of the map $\tau _{\geqslant 0} X \rightarrow X$ .

Lemma 3.3. Let $G$ be a finite group and let $e$ be a naive connective $G$ -spectrum. Then,

\begin{equation*} \tau _{\geqslant 0} e^{hG} \rightarrow e^{hG} \rightarrow \tau _{\lt 0}e^{tG} \end{equation*}

is a cofiber sequence and in particular, $\tau _{\lt 0}e^{tG} \simeq \tau _{\lt 0}e^{hG}$ .

Proof. We consider the norm sequence:

\begin{equation*} {e_{hG} {\stackrel {N} \longrightarrow} e^{hG} \longrightarrow {} e^{tG}}. \end{equation*}

As $e_{hG}$ is a connective spectrum, we have that $\pi _{-1}e_{hG} = 0$ . Hence, applying $\tau _{\geqslant 0}$ still gives rise to a cofiber sequence:

We combine the norm cofiber sequences with the defining cofiber sequence of $\tau _{\lt 0}$ and obtain

Thus, $\tau _{\lt 0} e^{hG} \simeq \tau _{\lt 0} e^{tG}$ and the cofiber sequence in the second row then yields the claim.

Remark 3.4. In many cases, if $B^{tG} \not \simeq *$ , then $\pi _*(B^{tG})$ is actually periodic. As the canonical Künneth map:

\begin{equation*} \pi _*(B^{tG}) \otimes _{\pi _*(B^{hG})} \pi _*(B) \rightarrow \pi _*(B^{tG} \wedge _{B^{hG}} B) \end{equation*}

is a map of graded commutative rings and as $\pi _*(B^{tG}) \cong \pi _*(B^{hG})$ in negative degrees, a periodicity generator in a negative degree would map to zero in $\pi _*B$ for connective $B$ and hence $\pi _*(B^{tG}) \otimes _{\pi _*(B^{hG})} \pi _*(B)$ is the zero ring. But then also $\pi _*(B^{tG} \wedge _{B^{hG}} B) \cong 0$ and

\begin{equation*} B^{tG} \wedge _{B^{hG}} B \simeq *. \end{equation*}

Therefore, $B$ would not be a faithful $B^{hG}$ -module in these cases. This emphasizes the importance of replacing the condition that $A$ be weakly equivalent to $B^{hG}$ by the requirement that $A \simeq \tau _{\geqslant 0}(B^{hG})$ .

From Lemma 3.3, we also know that in order to show that $B^{tG} \not \simeq *$ for connective $B$ it is sufficient to show that $\tau _{\lt 0}B^{hG}$ is not trivial.

We recall the following result from [Reference Rognes44]:

Proposition 3.5. [Reference Rognes44, Proposition 6.3.3] Assume that $G$ is a finite group, $B$ is a cofibrant commutative $A$ -algebra on which $G$ acts via maps of commutative $A$ -algebras. If $B$ is dualizable and faithful as an $A$ -module and if

\begin{equation*} h \;:\; {B \wedge _A B {\stackrel {\sim } \longrightarrow} F(G_+, B)}, \end{equation*}

then $B^{tG} \simeq *$ .

Rognes assumes that $A \simeq B^{hG}$ , but that assumption is not needed. A referee actually noted that it follows from the remaining assumptions in the Proposition: smashing the map $A \rightarrow B^{hG}$ with $B$ over $A$ yields $B \rightarrow B \wedge _A B^{hG}$ . Dualizability of $B$ as an $A$ -module identifies the latter with $(B \wedge _A B)^{hG} \simeq (F(G_+, B))^{hG} \simeq B$ . As $B$ is faithful as an $A$ -module, this implies $A \simeq B^{hG}$ .

In the following, we study the Tate constructions in several examples. To compute the homotopy of $B^{tG}$ , we use the Tate spectral sequence:

\begin{equation*} E^2_{n,m} = \hat {H}^{-n}(G;\; \pi _m(B)) \Longrightarrow \pi _{n+m}B^{tG}\end{equation*}

which is of standard homological type, multiplicative, and conditionally convergent. In particular by [Reference Boardman11, Theorem 8.2], it converges strongly if it collapses at a finite stage.

If the spectrum $B$ is connective, then the vanishing of $\pi _*B^{tG}$ is equivalent to the fact that $1 \in \pi _0B$ is in the image of the norm map $N \;:\; \pi _0(B_{hG}) \rightarrow \pi _0B$ . We learned a proof of this fact from one of the referees of an earlier version: one direction follows directly because the Postnikov section $B \rightarrow H(\pi _0(B))$ is a $G$ -equivariant map of commutative ring spectra. For the reverse note that the vanishing of $\hat{H}^{*}(G;\; \pi _0(B))$ implies the vanishing of $\hat{H}^{*}(G;\; \pi _n(B))$ for all $n$ . Thus, the $E^2$ -page of the Tate spectral sequence is trivial and hence $B^{tG} \simeq *$ .

We will now investigate the Tate construction in examples. First, we establish faithfulness:

Proof. For the map $\textsf{tmf}_0(2)_{(3)} \rightarrow \textsf{tmf}(2)_{(3)}$ , we know that $C_2$ acts on $\textsf{tmf}(2)_{(3)}$ via commutative $\textsf{tmf}_0(2)_{(3)}$ -algebra maps and that $\textsf{tmf}_0(2)_{(3)} \simeq \tau _{\geqslant 0}(\textsf{tmf}(2)_{(3)}^{hC_2})$ . The trace map $tr \;:\; \textsf{tmf}(2)_{(3)} \rightarrow \textsf{tmf}(2)_{(3)}^{hC_2}$ factors through $\tau _{\geqslant 0}(\textsf{tmf}(2)_{(3)}^{hC_2}) \simeq \textsf{tmf}_0(2)_{(3)}$ , because $\textsf{tmf}(2)_{(3)}$ is connective. As in [Reference Rognes44, Lemma 6.4.3], one can show that the composite:

\begin{equation*} {\textsf{tmf}_0(2)_{(3)} \simeq \tau _{\geqslant 0}(\textsf{tmf}(2)_{(3)}^{hC_2}) \longrightarrow {} \textsf{tmf}(2)_{(3)} {\stackrel {(0.3){tr}} \longrightarrow} \tau _{\geqslant 0}(\textsf{tmf}(2)_{(3)}^{hC_2}) \simeq \textsf{tmf}_0(2)_{(3)} }\end{equation*}

is homotopic to the map that is the multiplication by $|C_2|=2$ . As $2$ is invertible in $\pi _0\textsf{tmf}_0(2)_{(3)}$ , the trace map $tr \;:\; \textsf{tmf}(2)_{(3)} \rightarrow \textsf{tmf}_0(2)_{(3)}$ is a split surjective map of $\textsf{tmf}_0(2)_{(3)}$ -modules and hence $\textsf{tmf}_0(2)_{(3)} \rightarrow \textsf{tmf}(2)_{(3)}$ is faithful.

Alternatively, faithfulness also follows from the fact that $\pi _*\textsf{tmf}(2)_{(3)}$ is a free $\pi _*\textsf{tmf}_0(2)_{(3)}$ -module (see the proof of Theorem 2.5).

This result also follows from [Reference Meier40, Proposition 4.15].

Proof. We already mentioned the identification $\textsf{tmf}_0(2)_{(3)} \simeq \textsf{tmf}_{(3)} \wedge T$ where $T = S^0 \cup _{\alpha _1} e^4 \cup _{\alpha _1} e^8$ with $\alpha _1 \in (\pi _3S)_{(3)}$ , [Reference Behrens10, Lemma 2, p. 382], [Reference Mathew36, Theorem 4.15]. Note that $\alpha _1$ is nilpotent of order $2$ because $(\pi _6S)_{(3)} = 0$ .

Assume that $M$ is a $\textsf{tmf}_{(3)}$ -module with

\begin{equation*} * \simeq M \wedge _{\textsf{tmf}_{(3)}} \textsf{tmf}_0(2)_{(3)} \simeq M \wedge T. \end{equation*}

Then, the cofiber sequences

\begin{equation*} {S^0 \longrightarrow {} T \longrightarrow {} \Sigma ^4\text {cone}(\alpha _1)} \text { and } {\text {cone}(\alpha _1) \longrightarrow {} T \longrightarrow {} S^8} \end{equation*}

imply that $\Sigma ^4\text{cone}(\alpha _1) \wedge M \simeq \Sigma M$ and $\Sigma ^8 M \simeq \Sigma \text{cone}(\alpha _1) \wedge M$ and therefore,

\begin{equation*} \Sigma ^{10} M \simeq M. \end{equation*}

The equivalence is induced by a class in $\pi _{10}S_{(3)} \cong \mathbb{Z}/{3\mathbb{Z}}\{\beta _1\}$ . As this is nilpotent, we get that $M \simeq *$ .

We show that the extensions $\textsf{tmf}_0(3)_{(2)} \rightarrow \textsf{tmf}_1(3)_{(2)}$ and $\textsf{tmf}_{(3)} \rightarrow \textsf{tmf}(2)_{(3)}$ have nontrivial Tate spectra. For $ku$ , the Tate spectrum with respect to the complex conjugation $C_2$ -action satisfies

\begin{equation*} ku^{tC_2} \simeq \bigvee _{i \in \mathbb {Z}} \Sigma ^{4i} H\mathbb {Z}/2\mathbb {Z}. \end{equation*}

This result is due to Rognes (compare [Reference Rognes44, §5.3]). As $ku_{(p)}^{tC_2} \simeq *$ for odd primes $p$ , $2$ is the only wildly ramified prime.

Theorem 3.11. For $\textsf{tmf}_1(3)_{(2)}$ with its $C_2$ -action, we obtain an equivalence of spectra:

\begin{equation*} \textsf{tmf}_1(3)_{(2)}^{tC_2} \simeq \bigvee _{i \in \mathbb {Z}} \Sigma ^{8i} H\mathbb {Z}/2\mathbb {Z}. \end{equation*}

Proof. We use the calculations in [Reference Mahowald and Rezk33]. They compute the homotopy fixed point spectral sequence:

\begin{equation*} E^2_{n,m} = H^{-n}(C_2;\; \pi _m\textsf{TMF}_1(3)_{(2)}) \longrightarrow {} \pi _{n+m}\textsf{TMF}_0(3)_{(2)}, \end{equation*}

where $\pi _*\textsf{TMF}_1(3)_{(2)} = \mathbb{Z}_{(2)}[a_1, a_3][\Delta ^{-1}]$ with $\Delta = a_3^3(a_1^3-27a_3)$ . From their computations, we deduce the following behavior of the Tate spectral sequence:

(3.1) \begin{equation} E^2_{n,m} = \hat{H}^{-n}(C_2;\; \pi _m\textsf{TMF}_1(3)_{(2)}) \longrightarrow{} \pi _{n+m}\textsf{TMF}_1(3)_{(2)}^{tC_2}: \end{equation}

Let $R_{n,m}$ be the bigraded ring $\mathbb{Z}/2[a_1, a_3][\Delta ^{-1}][\zeta ^{\pm }]$ with $|\zeta | = (\!-\!1, 0)$ . If we assign odd weight to $a_1$ , $a_3$ , and $\zeta$ , then the $E^2$ -page of the Tate spectral sequence is the even part of $R_{n,m}$ . Alternatively, it is given by:

\begin{equation*} E^2_{*,*} = S_*[\Delta ^{-1}][x^{\pm }], \end{equation*}

where $S_*$ is the subalgebra of $\mathbb{Z}/{2\mathbb{Z}}[a_1,a_3]$ generated by $a_1^2, a_1a_3, a_3^2$ , and where $x = \zeta a_3^3 \in E_{-1,18}$ . Note that $a_3^2$ is invertible in this ring with $a_3^{-2} = ((a_1a_3)a_1^2-27a_3^2)\Delta ^{-1}$ . By Mahowald–Rezk’s computations, the first nontrivial differential is $d^3$ and we have

\begin{align*} d^3(a_1^2) = & (x (a_1a_3) a_3^{-4})^3, & d^3(a_1a_3) = 0, & \qquad d^3(a_3^2) = x^3 (a_1a_3) a_3^{-8}, \\[5pt] d^3(x) = & 0, & d^3(\Delta ^{-1}) = 0. & \end{align*}

Using the Leibniz rule, we get that the class $c_{n,m,k,l,i} = (a_1^2)^n(a_1a_3)^m(a_3^2)^k\Delta ^{-l}x^i$ with $n,m,k,l \in \mathbb{N}$ and $i \in \mathbb{Z}$ has differential

\begin{equation*} d^3(c_{n,m,k,l,i}) = (n+k) x^3 (a_1a_3) a_3^{-10} c_{n,m,k,l,i}. \end{equation*}

It follows that $\ker d^3$ is generated as an $\mathbb{F}_2$ -vector space by the classes $c_{n,m,k,l,i}$ with $n+k = 0$ in $\mathbb{F}_2$ . We claim that

\begin{equation*} E^4_{*,*} \cong \mathbb {F}_2[x^{\pm }, \Delta ^{\pm }]. \end{equation*}

To see this, note the following: If $n+k = 0$ in $\mathbb{F}_2$ and $m \gt 0$ , then $c_{n,m,k,l,i}$ is zero in $E^4_{*,*}$ because

\begin{equation*} d^3(c_{n,m-1,k+5,l,i-3}) = c_{n,m,k,l,i}.\end{equation*}

If $n+k = 0$ in $\mathbb{F}_2$ and $n, k \gt 0$ , then we have $c_{n,0,k,l,i} = c_{n-1, 2, k-1,l,i}$ . This is in the image of $d^3$ , because $n-1 + k-1 = 0$ in $\mathbb{F}_2$ and $ 2 \gt 0$ . If $n = 0$ in $\mathbb{F}_2$ and $n \gt 0$ , then

\begin{align*} c_{n, 0, 0,l,i} & = (a_1^2)^n \Delta ^{-l} x^i \\[5pt] & = (a_1 a_3)^2 (a_1^2)^{n-1} a_3^{-2} \Delta ^{-l} x^i \\[5pt] & = (a_1a_3)^2(a_1^2)^{n-1}((a_1a_3)a_1^2 + a_3^2)\Delta ^{-1}\Delta ^{-l} x^i \\[5pt] & = c_{n,3,0,l+1,i} + c_{n-1,2,1,l+1,i}, \end{align*}

and both of these summands are in the image of $d^3$ . Furthermore, note that in $E^4_{*,*}$ we have

\begin{equation*} \Delta = (a_1 a_3)^3 + a_3^4 = c_{0,3,0,0,0} + a_3^4 = a_3^4.\end{equation*}

This implies that for $k = 0$ in $\mathbb{F}_2$ we have

\begin{equation*} c_{0,0,k,l,i} = (a_3^{4})^{\frac {k}{2}} \Delta ^{-l} x^i \equiv \Delta ^ {-l + \frac {k}{2}} x^i \end{equation*}

in $E^4_{*,*}$ . We thus get a surjective map $\mathbb{F}_2[x^{\pm }, \Delta ^{\pm }] \to E^4_{*,*}$ , which is injective, because the classes $\Delta ^lx^i$ for $l,i \in \mathbb{Z}$ are not divisible by $(a_1a_3)$ in $S_*[\Delta ^{-1}][x^{\pm }]$ .

From Mahowald–Rezk’s computations, we get that the next nontrivial differential is $d^7$ and that we have

\begin{equation*} d^7(x) = 0 \text { and } d^7(\Delta ) = x^7\Delta ^{-4}. \end{equation*}

This gives $E^8_{*,*} = 0$ .

We now want to determine the behavior of the Tate spectral sequence:

(3.2) \begin{equation} E^2_{n,m} = \hat{H}^{-n}(C_2;\; \pi _m \textsf{tmf}_1(3)_{(2)}) \Longrightarrow \pi _{n+m} \textsf{tmf}_1(3)_{(2)}^{tC_2}. \end{equation}

If we assign again odd weight to $a_1$ , $a_3$ , and $\zeta$ , then the $E^2$ -page is the even part of

\begin{equation*} \mathbb {Z}/{2\mathbb {Z}}[a_1, a_3][\zeta ^{\pm }], \end{equation*}

and one sees that the map of spectral sequences from (3.2) to (3.1) is injective. We get that $d^3$ is the first nontrivial differential in (3.2) and that we have

\begin{align*} d^3(a_1a_3) & = 0, & d^3(a_1^2) & = (a_1 \zeta )^3, \\[5pt] d^3(a_3^2 ) & = a_1 a_3^2 \zeta ^3, & d^3(a_3\zeta ) & = (a_1a_3) \zeta ^4, \\[5pt] d^3(a_1 \zeta ) & = 0, & d^3(\zeta ^2) & = a_1 \zeta ^5. \end{align*}

Note that an $\mathbb{F}_2$ -basis of the $E^3$ -page is given by the classes:

\begin{align*} d_{n,m,i} = & (a_1^2)^n (a_3^2)^m (\zeta ^2)^i, \\[5pt] e_{n,m,i} = & (a_1^2)^n(a_1a_3)(a_3^2)^m (\zeta ^2)^i, \\[5pt] f_{n,m,i} = & (a_1^2)^n (a_3^2)^m (a_1 \zeta ) (\zeta ^2)^i, \\[5pt] g_{n,m,i} = & (a_1^2)^n(a_3^2)^m (a_3 \zeta ) (\zeta ^2)^i, & \end{align*}

for $n,m \in \mathbb{N}$ , and $i \in \mathbb{Z}$ .

The $d^3$ -differential on these classes is given by:

\begin{align*} d^3(d_{n,m,i}) & = (n+m+i) \cdot f_{n,m,i+1}, \\[5pt] d^3(e_{n,m,i}) & = (n+m+i) \cdot g_{n+1, m, i+1}, \\[5pt] d^3(f_{n,m,i}) & = (n+m+i) \cdot d_{n+1,m, i+2}, \\[5pt] d^3(g_{n,m,i}) & = (n+m+i+1) \cdot e_{n,m,i+2}. \end{align*}

We get

\begin{equation*} E^4_{*,*} = \bigoplus _{\substack {m \in \mathbb {N}, i \in \mathbb {Z}\\ m+i = 0 \; \text {in} \; \mathbb {F}_2}} \mathbb {F}_2\{d_{0,m,i} \} \oplus \bigoplus _{\substack {m \in \mathbb {N}, i \in \mathbb {Z} \\ m + i+ 1 = 0 \; \text {in} \; \mathbb {F}_2} }\mathbb {F}_2\{g_{0, m, i}\}. \end{equation*}

The map of spectral sequences from (3.2) to (3.1) satisfies

\begin{equation*} d_{0,m,i} \mapsto \Delta ^{\frac {m-3i}{2}}x^{2i}, \qquad g_{0, m,i} \mapsto \Delta ^{\frac {m-3i-1}{2}} x^{2i+1}. \end{equation*}

In particular, one sees that it is injective on $E^4$ -pages. We conclude that the next nontrivial differential in spectral sequence (3.2) is $d^7$ and that we have

\begin{equation*} d^7(d_{0,m,i}) = \frac {m-3i}{2} g_{0, m, i+3}, \qquad d^7(g_{0,m,i}) = \frac {m-3i-1}{2} d_{0,m+1,i+4}. \end{equation*}

We obtain that

\begin{equation*} E^8_{*,*} = \bigoplus _{i \in \mathbb {Z}} \mathbb {F}_2\{d_{0,0,4i}\} = \bigoplus _{i \in \mathbb {Z}} \mathbb {F}_2 \{\zeta ^{8i}\}. \end{equation*}

Since the $E^8$ -page is concentrated in the zeroth row, the spectral sequence collapses at this stage. This gives the answer on the level of homotopy groups. As $\textsf{tmf}_1(3)^{tC_2}$ is an $E_\infty$ -ring spectrum [Reference McClure39], it is in particular an $E_2$ -ring spectrum and therefore a result by Hopkins–Mahowald (see [Reference Mathew, Naumann and Noel38, Theorem 4.18]) implies that $\textsf{tmf}_1(3)^{tC_2}$ receives a map from $H\mathbb{F}_2$ and therefore is a generalized Eilenberg–MacLane spectrum of the claimed form.

Theorem 3.12. The $\Sigma _3$ -action on $\textsf{tmf}(2)_{(3)}$ yields

\begin{equation*} \textsf{tmf}(2)_{(3)}^{t\Sigma _3} \simeq \bigvee _{i \in \mathbb {Z}} \Sigma ^{12 i} H\mathbb {Z}/3\mathbb {Z}. \end{equation*}

Proof. We use the calculation of [Reference Stojanoska53]. She proves that $\textsf{Tmf}(2)_{(3)}^{t\Sigma _3} \simeq *$ via the Tate spectral sequence:

\begin{equation*} E^2_{n.m} = \hat {H}^{-n}\bigl (\Sigma _3;\; \pi _m(\textsf{Tmf}(2)_{(3)})\bigr ) \Longrightarrow \pi _{n+m}(\textsf{Tmf}(2)_{(3)}^{t\Sigma _3}).\end{equation*}

The $E^2$ -page is given by:

\begin{equation*} \mathbb {Z}/{3\mathbb {Z}}[\alpha, \beta ^{\pm }, \Delta ^{\pm }]/{\alpha ^2}\end{equation*}

with $| \alpha | = (\!-\!1,4)$ , $| \beta | = (\!-\!2,12)$ , and $| \Delta | = (0,24)$ , and the differentials are determined by:

\begin{equation*} d^5(\Delta ) = \alpha \beta ^2 \quad \text {and} \quad d^9(\alpha \Delta ^2) = \beta ^5.\end{equation*}

Since $\textsf{tmf}(2)_{(3)}$ is the connective cover of $\textsf{Tmf}(2)_{(3)}$ , the $E^2$ -page of the Tate spectral sequence:

\begin{equation*} \bar {E}^2_{n,m} = \hat {H}^{-n}\bigl (\Sigma _3;\; \pi _m(\textsf{tmf}(2)_{(3)})\bigr ) \Longrightarrow \pi _{n+m}(\textsf{tmf}(2)_{(3)}^{t\Sigma _3}) \end{equation*}

is the $\mathbb{Z}/{3\mathbb{Z}}$ -module

\begin{equation*} \bigoplus _{\substack {k,l \in \mathbb {Z} \\ k + 2l \geqslant 0}} \mathbb {Z}/{3\mathbb {Z}}\{\beta ^k \Delta ^l\} \oplus \bigoplus _{\substack {k, l \in \mathbb {Z} \\ 1 + 3k + 6l \geqslant 0}} \mathbb {Z}/{3\mathbb {Z}}\{ \alpha \beta ^k \Delta ^l\}. \end{equation*}

Using the map of Tate spectral sequences $\bar{E}^*_{*,*} \to E^*_{*,*}$ one sees that

\begin{equation*} \bar {E}^6_{*,*} = \bigoplus _{\substack {k,l \in \mathbb {Z} \\ k+6l \geqslant 0}} \mathbb {Z}/{3 \mathbb {Z}} \{\beta ^k (\Delta ^{3})^l\} \oplus \bigoplus _{\substack {k, l \in \mathbb {Z} \\ 13 + 3k + 18l \geqslant 0}} \mathbb {Z}/{3 \mathbb {Z}}\{ (\alpha \Delta ^2)\beta ^k (\Delta ^{3})^l\}.\end{equation*}

Since $E^6_{*,*} = \mathbb{Z}/{3\mathbb{Z}}[\alpha \Delta ^2, \beta ^{\pm }, \Delta ^{\pm 3}]/{(\alpha \Delta ^2)^2}$ the map $\bar{E}^6_{*,*} \to E^6_{*,*}$ is injective. Thus, $\bar{d}^9$ is determined by $d^9$ and one gets

\begin{equation*} \bar {E}^{10}_{*,*} = \bigoplus _{k \in \mathbb {Z}} \mathbb {Z}/{3\mathbb {Z}}\{(\beta ^{-6} \Delta ^3)^k\}. \end{equation*}

The class $\beta ^{-6}\Delta ^3$ has bidegree $(12,0)$ , and so $\bar{E}^{10}_{*,*}$ is concentrated in line zero and the spectral sequence collapses at this stage.

So the extensions $ko_{(2)} \rightarrow ku_{(2)}$ , $\textsf{tmf}_{(3)} \rightarrow \textsf{tmf}(2)_{(3)}$ , and $\textsf{tmf}_0(3)_{(2)} \rightarrow \textsf{tmf}_1(3)_{(2)}$ have nontrivial Tate constructions.

In contrast, $KO \rightarrow KU$ is a faithful $C_2$ -Galois [Reference Rognes44, §5], and $\textsf{TMF}_0(3) \rightarrow \textsf{TMF}_1(3)$ and $\textsf{Tmf}_0(3) \rightarrow \textsf{Tmf}_1(3)$ are both faithful $C_2$ -Galois extensions [Reference Mathew and Meier37, Theorem 7.12]. In general, $\textsf{TMF}[1/n] \rightarrow \textsf{TMF}(n)$ is a faithful $GL_2(\mathbb{Z}/n\mathbb{Z})$ -Galois extension [Reference Mathew and Meier37, Theorem 7.6] and the Tate spectrum $\textsf{Tmf}(n)^{tGL_2(\mathbb{Z}/n\mathbb{Z})}$ is contractible [Reference Mathew and Meier37, Theorem 7.11].

For general $n \gt 1$ , constructions of $\textsf{tmf}_1(n)$ and $\textsf{tmf}_0(n)$ are tricky: for some large $n$ , $\pi _1\textsf{Tmf}_1(n)$ is nontrivial. Lennart Meier constructs a connective version of $\textsf{Tmf}_1(n)$ with trivial $\pi _1$ as an $E_\infty$ -ring spectrum in [Reference Meier40, Theorem 1.1] so that there are $E_\infty$ -models of $\textsf{tmf}_1(n)$ for all $n$ .

We cannot determine the homotopy type of the $GL_2(\mathbb{Z}/n\mathbb{Z})$ -Tate construction of $\textsf{tmf}(n)$ for arbitrary $n \gt 1$ , but we can identify cases where it is nontrivial.

Proof. Since $\textsf{tmf}(n)_{hGL_2(\mathbb{Z}/{n\mathbb{Z}})}$ is connective, the defining cofiber sequence of $\textsf{tmf}(n)^{tGL_2(\mathbb{Z}/{n\mathbb{Z}})}$ gives an exact sequence:

\begin{equation*}\cdots \longrightarrow {} \pi _0\textsf{tmf}(n)_{hGL_2(\mathbb {Z}/{n\mathbb {Z}})} {\stackrel {N} \longrightarrow} \pi _0\textsf{tmf}(n)^{hGL_2(\mathbb {Z}/{n\mathbb {Z}})} \longrightarrow {} \pi _0\textsf{tmf}(n)^{t GL_2(\mathbb {Z}/{n\mathbb {Z}})} \longrightarrow {} 0. \end{equation*}

We have that $\pi _0(\textsf{tmf}(n)) \cong \mathbb{Z}[\frac{1}{n}, \zeta _n]$ , where $\zeta _n$ is a primitive $n$ th root of unity. Consider the commutative diagram

By the homotopy orbit spectral sequence, we have that the left-hand vertical map is an isomorphism. By [Reference Katz and Mazur27, p. 282], an element $A \in GL_2(\mathbb{Z}/{n\mathbb{Z}})$ acts on $\zeta _n^i$ as:

\begin{equation*} A \zeta _n^i = \zeta _n^{\det A \cdot i }.\end{equation*}

This implies that the ring in the lower right corner is $\mathbb{Z}\left[\frac{1}{n}\right]$ . Since we also have

\begin{equation*}\pi _0\textsf{tmf}(n)^{hGL_2(\mathbb {Z}/{n\mathbb {Z}})} \cong \pi _0\textsf{tmf}\left[\frac{1}{n}\right] = \mathbb {Z}\left[\frac{1}{n}\right]\end{equation*}

and since the right-hand vertical map in the diagram is a map of rings, it follows that it is an isomorphism. We thus have to compute the cokernel of the algebraic norm map:

\begin{equation*}N\;:\; \mathbb {Z}\left[\frac{1}{n}, \zeta _n\right]_{GL_2(\mathbb {Z}/{n\mathbb {Z})}} \longrightarrow {} \mathbb {Z}\left[\frac{1}{n}, \zeta _n\right]^{GL_2(\mathbb {Z}/{n\mathbb {Z})}} = \mathbb {Z}\left[\frac{1}{n}\right]. \end{equation*}

We claim that its image is $|SL_2(\mathbb{Z}/{n\mathbb{Z}})| \mathbb{Z}\left[\frac{1}{n}\right]$ so that $\pi _0(\textsf{tmf}(n)^{tGL_2(\mathbb{Z}/{n\mathbb{Z}})}) \cong \mathbb{Z}\left[\frac{1}{n}\right]/|{SL_2(\mathbb{Z}/{n\mathbb{Z}}})|$ .

Let $\varphi (\!-\!)$ denote the Euler $\varphi$ -function and let $\mu (\!-\!)$ denote the Möbius function. If $d$ is the order of a power $\zeta _n^i$ , then the norm map $N$ sends $\zeta _n^i$ to

\begin{align*} \sum _{A \in GL_2(\mathbb{Z}/{n\mathbb{Z}})} \zeta _n^{i \det (A)} & = |SL_2(\mathbb{Z}/{n \mathbb{Z}})| \cdot \sum _{r \in (\mathbb{Z}/{n\mathbb{Z}})^\times } \zeta _n^{ir} \\[5pt] & = |SL_2(\mathbb{Z}/{n \mathbb{Z}})| \cdot \frac{\varphi (n)}{\varphi (d)} \cdot \mu (d) \end{align*}

For the second equality note that the canonical map $(\mathbb{Z}/{n \mathbb{Z}})^\times \to (\mathbb{Z}/{d\mathbb{Z}})^\times$ is a surjection whose kernel has order $\frac{\varphi (n)}{\varphi (d)}$ . Now, let $d$ be the maximal number which is square-free and divides $n$ . Then, since $d$ is square-free, we have $\mu (d) \in \{1, -1\}$ . If

\begin{equation*} n = \prod _{\substack {p \mid n \\ [ p \;\text {prime}}}p^{k_p},\end{equation*}

we have

\begin{equation*} \varphi (n) = \prod _{\substack {p \mid n \\ p \;\text {prime}}}p^{k_p-1}(p-1)\end{equation*}

and

\begin{equation*} \frac {\varphi (n)}{\varphi (d)} = \prod _{\substack {p \mid n \\ p \;\text {prime}}} p^{k_p-1}. \end{equation*}

Since the latter is a unit in $\mathbb{Z}\left[\frac{1}{n}\right]$ , we get that the image of the norm is $|SL_2(\mathbb{Z}/{n\mathbb{Z}})|\mathbb{Z}\left[\frac{1}{n}\right]$ . Therefore, we have $\textsf{tmf}(n)^{tGL_2(\mathbb{Z}/{n\mathbb{Z}})}\simeq *$ if and only if $|SL_2(\mathbb{Z}/{n\mathbb{Z})}|$ is a unit in $\mathbb{Z}\left[\frac{1}{n}\right]$ . Since

\begin{equation*} |SL_2(\mathbb {Z}/{n\mathbb {Z}})| = n^3\prod _{\substack {p \mid n \\ p \;\text {prime}}}\frac {1}{p^2}(p+1)(p-1) \end{equation*}

and $|GL_2(\mathbb{Z}/{n\mathbb{Z}})| = \varphi (n) \cdot |SL_2(\mathbb{Z}/{n\mathbb{Z}})|$ , we see that $|SL_2(\mathbb{Z}/{n\mathbb{Z}})|$ is invertible in $\mathbb{Z}\left[\frac{1}{n}\right]$ if and only if $|GL_2(\mathbb{Z}/{n\mathbb{Z}})|$ is invertible in $\mathbb{Z}\left[\frac{1}{n}\right]$ .

If $n \geqslant 2$ and $2 \nmid n$ , then $|GL_2(\mathbb{Z}/n\mathbb{Z})|$ and $|SL_2(\mathbb{Z}/n\mathbb{Z})|$ are not units in $\mathbb{Z}\left[\frac{1}{n}\right]$ : let $q$ be an odd prime factor of $n$ . Then in

\begin{equation*} n^3\prod _{\substack {p \mid n \\ p \;\text {prime}}}\frac {1}{p^2}(p+1)(p-1) \end{equation*}

we have a factor of $q-1$ and this is even, but $2$ is not invertible in $\mathbb{Z}\left[\frac{1}{n}\right]$ .

If $n = 2^k$ for some $k \geqslant 1$ , we obtain

\begin{equation*} |SL_2(\mathbb {Z}/n\mathbb {Z})| = 2^{3k}\frac {3}{4} \end{equation*}

which contains $3$ as a non-invertible factor.

Note that Meier shows [Reference Meier40, Theorem 4.4 and Proposition 4.15] that $\textsf{tmf}(n)$ is dualizable and faithful as a $\textsf{tmf}[1/n]$ -module.

We close with a periodic example. For a fixed prime $p$ , let $E_n$ denote the Lubin-Tate spectrum whose coefficient ring is

\begin{equation*} \pi _*(E_n) = W(\mathbb {F}_{p^n})[[u_1,\ldots,u_{n-1}]][u^{\pm 1}], \end{equation*}

where $u$ is an element of degree $2$ and the $u_i$ s have degree 0. For a perfect field $k$ , $W(k)$ denotes the ring of Witt vectors of $k$ . The ring $W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]] = \pi _0(E_n)$ represents deformations of the height $n$ Honda formal group law over $\mathbb{F}_{p^n}$ . The quotient $E_n/(p,u_1,\ldots,u_{n-1}) =K_n$ is a $2$ -periodic version of Morava K-theory whose coefficient ring is the graded field $\pi _*(K_n) = \mathbb{F}_{p^n}[u^{\pm 1}]$ .

For any finite group $G$ , $F(BG_+, E_n) \rightarrow F(EG_+, E_n) \simeq E_n$ is faithful in the $K_n$ -local category [Reference Baker and Richter5, Theorem 4.4]. At the moment, we don’t know whether $E_n$ is a dualizable $F(BG_+, E_n)$ -module for any finite group $G$ .

In [Reference Baker and Richter5, Theorem 5.1], it is shown that $F((BC_{p^r})_+, E_n) \rightarrow E_n$ is ramified and one can also consider more general groups than $C_{p^r}$ . The corresponding Tate constructions are not trivial:

Lemma 3.15. For all $r \geqslant 1$ and $n \geqslant 1$

\begin{equation*} E_n^{tC_{p^r}} \not \simeq \ast. \end{equation*}

Proof. The Tate spectral sequence

\begin{equation*} E_2^{s,t} = \hat {H}^{-s}(C_{p^r};\; \pi _tE_n) \Rightarrow \pi _{s+t}(E_n^{tC_{p^r}}) \end{equation*}

has as $E^2$ -term

\begin{equation*} \hat {H}^{-s}(C_{p^r};\; \pi _tE_n) \cong \begin {cases} \pi _tE_n^{C_{p^r}}/p^r = \pi _tE_n/p^r, & \text { for } s \text { even}, \\[5pt] \ker (N)/\text {im}(t-1) = 0, & \text { for } s \text { odd}.\end {cases} \end{equation*}

As $\pi _*(E_n)$ is concentrated in even degrees, the whole $E_2$ -term is concentrated in bidegrees $(s,t)$ where $s$ and $t$ are even. Therefore, all differentials have to be trivial and $E_2 = E_\infty$ . Thus, $\pi _*(E_n^{tC_{p^r}})$ is highly nontrivial.

Proof. The assumption implies that $G$ has $C_p$ as a subgroup. The restriction map induces a map on Tate constructions $E_n^{tG} \rightarrow E_n^{t{C_{p}}}$ . For the remainder of the proof, we use the notation from [Reference Greenlees and May21], denoting the Tate construction $E_n^{tG}$ by the $G$ -fixed points $t((E_n)_G)^G$ of a $G$ -spectrum $t((E_n)_G)$ . McClure [Reference McClure39] shows that the $E_\infty$ -structure on Tate constructions $t((E_n)_G)^G$ is compatible with inclusions of subgroups and Greenlees–May show [Reference Greenlees and May21, Proposition 3.7] that for any subgroup $H \lt G$ the $H$ -spectrum $t((E_n)_G)$ is equivalent to $t((E_n)_H)$ . Therefore, the inclusion of fixed points $t((E_n)_G)^G \rightarrow t((E_n)_G)^H$ is a map of $E_\infty$ -ring spectra. As we know that $E_n^{tC_{p}} = t((E_n)_G)^{C_{p}}$ is nontrivial by Lemma 3.15, $E_n^{tG}$ cannot be trivial, either.