added to *modular form* a brief paragraph with a minimum of information on modular forms *As automorphic forms*. Needs to be expanded.

I have created an entry *modular equivariant elliptic cohomology*.

The subject barely exists, for the moment the entry is to serve two purposes:

first, to highlight that by results of Mahowald-Rezk, Lawson-Naumann, Hill-Lawson this exists as a rather compelling generalization of KR-theory;

second, that the close the relation of KR-theory to type II string theory on orientifolds has previously been conjectured to correspond in the lift of the latter to F-theory to a modular equivariant universal elliptic cohomology.

So while the subject hasn’t been studied yet (it seems), both its construction and plenty of motivation for it already exists. And now also an $n$Lab entry for it does. :-)

]]>in order toput things in perspective, I created a table

and included it into relevant entries (under “Related concepts”)

]]>I have been adding some stuff to *j-invariant*, but it’s not really good yet (this here just in case you are watching the logs and are wondering what’s happening)

started something at *elliptic fibration*

created an entry for *Tmf(n)*

some basics at *modular curve*

added to *nodal curve* a brief paragraph *over the complex numbers*

started *level structure on an elliptic curve* with an Idea-section on what it means over the complex numbers.

started an entry *cubic curve*,

For the moment I wanted to record (see the entry) a pointer to Akhil Mathew’s identification of that eight-fold cover of $\mathcal{M}_{cub}$ (hence of $\mathcal{M}_{ell}$) which is analogous to the 2-fold cover of the “moduli stack of formal tori” $B \mathbb{Z}_2$ that ends up being the reason for the $\mathbb{Z}_2$-action on $KU$.

So here is the question that I am after: that cover is classified by a map $\mathcal{M}_{ell} \to B \mathbb{Z}/8\mathbb{Z}$, hence we get a double cover of the moduli space of elliptic curves $d \colon \mathcal{M}_{ell} \to B\mathbb{Z}/2\mathbb{Z}$.

Accordingly there is a spectrum $Q \coloneqq d_\ast(\mathcal{O}^{top})$ equipped with a $\mathbb{Z}_2$-action whose homotopy fixed points is $tmf$, I suppose: $tmf \simeq Q^{\mathbb{Z}_2}$. (Hm, maybe I need to worry about the compactification…).

I’d like to say that $Q$ is to $tmf$ as $KU$ is to $KO$. This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?

]]>started *Eisenstein series* with some formulas.

Given the series of entries lately, I naturally came to the point that I started to want a “floating context” table of contents. So I started one and included it into relevant entries:

But this needs more work still, clearly.

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