## Thoughts on Fractional Cohomology

Before I get into this post, allow me to give a bit of back story.

A close friend of mine, Aaron Slipper, mentioned the notion of a set $$S$$ with $$i$$ elements, that is, $$S \times S \cup {*} = \emptyset$$, ($$\sqrt{-1} \times \sqrt{-1} + 1 = 0$$). The cardinality of the automorphism group of a set with n elements is n!, but n doesn’t have to be a natural number, we can also evaluate the factorial when n is a complex number, such as $$i!$$, using the Gamma function.

I responded that we could similarly construct a “manifold” M which squared to the n-sphere (and call it a radical manifold).

We then extended the category of manifolds by appending radical manifolds, then asking for “Grothendieck completion wrt cartesian product and disjoint union,” in this case, formally constructing the -n-dimensional manifold, $$M^{-n}$$.

• $$M \coprod -M = \emptyset$$ (gives us a group, with disjoint union as the operation, $$-M$$ is the same dimension as M)
• $$M^n \times M^{-n} = *$$ (gives us negative dimensional spaces)
• take the algebraic closure, the $$n$$th root of an m-dimensional manifold should be $$m/n$$ dimensional (gives us a field)

Then, looking at polynomials with coefficients in this strange field.

There are a few immediate and natural questions:

• What are ideals of this ring?
• What is the cell decomposition of a “manifold” with a $$\sqrt{2}$$ dimensional cell? How do we compute the $$\sqrt{2}$$ cohomology group?

The following thoughts resulted from a dinner-time discussion (between Aaron Slipper, Alex Mennen and I) on potentially computing homology in fractional dimensions.

Sometimes we have a graded module $$H^*$$ indexed by the natural numbers, equipped with a filtration:

\$$F^0H^* \supset F^1H^* \supset … \supset F^nH^* \supset F^{n+1}H^* \supset … {0}\$$

such that $$F^kH^* \supset F^{k + 1}H^* \supset … \supset {0} = \bigoplus_{k \geq n} H^n$$

Why restrict ourselves to indexing by the naturals? What if we index by the reals? What is the continuous version of a direct sum, a “direct integral”?

We’ll get back to that, first, recall that a formal power series can be thought of as an indexed collection of coefficients, for example, $${4, 0, 5, 6, 0, 0, …}$$ corresponds to $$4 + 5x^2 + 6x^3$$

What about an indexed collection of abelian groups? For example $${Z/2, Z/2, 0, Z, 0, 0, 0, Z, …}$$?

Well, this certainly looks like a collection of coefficients to me, albeit strange ones.

It corresponds to $$Z/2 + Z/2 x + Z x^3 + Zx^8 …$$, this is perfectly reasonable to evaluate when $$x$$ is a group, multiplication is tensor product and sum is direct sum.

Analogous to a polynomial, taking us from $$Z \to Z$$, we’re going from $$Z$$-mod $$\to Z$$-mod.

We’re familiar with the Poincare polynomial, the polynomial which keeps track of the rank of homology groups, for example, the sphere, which has one 0-dimensional contractible component and one 2-dimensional contractible component, and nothing more, is $$f(x) = 1 + x^2$$.

But sometimes we care about more than Betti numbers, and Euler characteristics — that just captures the rank of these finitely generated abelian groups. Sometimes we care about keeping track of their torsion, not just their rank.

So what about the following, where each $$H^i$$ is an abelian group indexed by $$i$$.

$$f(x) = H^0 + H^1x +H^2x^2 + H^3x^3 + …. + H^nx^n + …$$

Why limit ourselves to the natural numbers as our indexing set. What happens if we index over the reals? What is the 1/2th cohomology group?

## What happens when we take the fractional derivative of our (integer indexed) group-valued polynomial?

There’s a notion of taking the $$1/2$$-derivative of any function $$f(x)$$:

\$$D^\alpha f(x) = 1/\Gamma(1-\alpha) \frac{\partial}{\partial x} \int \frac{f(x)}{(x-t)^{\alpha}}dt\$$

Let’s test that this makes sense, using $$e^x$$, a nice measuring stick for your sanity in the land of integration. The Gamma function, $$\Gamma(1-\alpha)$$, when $$\alpha = 1/2$$ is $$\sqrt{\pi}$$.

\$$D^{1/2} e^x = 1/\sqrt{\pi} \frac{\partial}{\partial x} \int \frac{f(x)}{(x-t)^{\alpha}}dt\$$

Hark! It is so! $$D^{1/2} e^x = e^x$$

Does this give us a reasonable notion of the “1/2th cohomology group”?

Let’s try out something silly to dip our toe in: $$f(x) = H^4x^4 + H^3x^3 + H^2x^2$$, already we start to get strange things:

First, the integral $$\int f(t)/(x-t)^{1/2} dt$$

Now let’s take the derivative wrt x:

Unfortunately, I find this whole approach to reindexing via derivatives unsatisfying, it’s not a very meaningful notion of interpolation. It seems most natural to first understand how to build a fractional dimensional space on the level of a CW-complex.

## What is a CW-complex with fractional-dimensional cells?

An admittedly ill-developed example, is a “fuzzy torus”:
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Edit: I’ve since learned that there’s something called Shape Theory.

“The intuitive idea of shape theory is to define invariants of quite general topological spaces by approximating them with ‘good’ spaces, either by embedding them into good spaces, and looking at open or polyhedral neighborhoods of them, or by considering abstract inverse systems of good spaces. The two approaches are closely related.” – nLab

I wonder if the fuzzy torus could be made equivalent to the cartesian product of two Warsaw circles! The Warsaw circle is the made from the curve $${(x, \sin(\frac{1}{x})) | x \in (0, 1)} \cup {(0,0)}$$ closed by a line:
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## What is a direct integral of groups?

Recap: We’ve been toying with the notion of re-indexing our cohomology theories, rather than using for, using or to get “fractional cohomology theories” and work with “fractional CW-complexes” and more generally “fractifolds”. I’m very curious wrt how von Neumann’s direct integral of operators (defined below) could be redefined to take in groups.
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Does this help us understand, say, given a fractal rather than a space, how to compute a meaningful long exact sequence of groups?

Alternative interpretation of “integrating over groups to get a group”: Since finitely generated abelian groups are classified by rank and torsion. Maybe we can have a notion of expected value of a group assigning a probability that at a real number, the group will be (rank, torsion) for some choice of (rank, torsion).

Over each point, we have a group that can be expressed uniquely as. Now pick a particular p (including 0), and look at the value of over each point. This gives us a function from $$R$$ to $$Z$$, and we can take the expected value of that function. Again, this doesn’t quite make sense as a notion of interpolation. I’m not sure that it’s meaningful, but it’s kind of neat.

## Where does this fractional line of thought fit into the mathematical narrative?

Aaron mentioned that this might be connected to Zagier’s result on the “orbifold Euler characteristic” (on I assume of some complex of modules?) of a “mapping class group.” His description of the mapping class group sounded to me like homotopy classes of self-maps of CW-complexes.

All I really know about orbifolds is that an orbifold point gives us a notion of a $$1/n$$-dimensional point, which is delightful; looking at the fundamental group of such a point: if we loop around it $$n$$ times, our loop contracts to a point.

Afternote on stacks: Nlab tells me that orbifolds are the stack object in the category Diff. A stack is a groupoid minus an atlas, just as a manifold is a collection of open subsets of Euclidean space along with the data of how they glue together but then you have to be willing to forget the original atlas (e.g., in case the local computation you’re trying to do moves outside of any single coordinate chart). In this analogy, the collection of opens is the groupoid, the atlas is the atlas, and the underlying manifold is the stack.

If that didn’t parse for you, keeping the following phrases in mind as you go through life may be of use: “sheaf valued in $$\text{Grpd}$$” and (wrt moduli stacks) “moduli space (parameterizing family of objects) + keeping track of automorphisms of objects.”

## Dirichlet Series and Homology Theories

Euler was a swashbuckler. He considered the following series, nevermind that it sums to infinity, let’s see what we can do with it!

I highly recommend that you follow along with a pen and paper to convince yourself that everything cancels as I say it will, if you’re a bit too lazy for that, or already familiar with the L-function, feel free to scroll ahead.

$$x = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + …$$

Let’s multiply by 1/2 and see what happens:

$$\frac{1}{2}x = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + …$$

Then subtract this from the original for funsies.

$$x -\frac{1}{2}x$$

$$=\frac{1}{1}$$ + 1/2 $$+ \frac{1}{3}$$ + 1/4 $$+ \frac{1}{5} + …$$

$$= \frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + …$$

$$= (1-\frac{1}{2})x$$

Huh, we’ve taken out all of the multiples of 2.

Hmm, what happens if we multiply the these by 3?

$$\frac{1}{3}(1-\frac{1}{2})x = \frac{1}{3} + \frac{1}{9} + \frac{1}{15} + \frac{1}{21} + \frac{1}{27} + …$$

Then subtract:

$$(1-\frac{1}{2})x – \frac{1}{3}(1-\frac{1}{2})x$$

$$= \frac{1}{1} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19} + \frac{1}{23} + …$$

Notice that $$(1-\frac{1}{2})x – \frac{1}{3}(1-\frac{1}{2})x = (1-\frac{1}{2})(1-\frac{1}{3})$$

That’s interesting, we’ve gotten rid of all of the multiples of 2 and all of the multiples of 3.

Let’s take out the multiples of $$5$$ that remain, that is, $$-\frac{1}{5}((1-\frac{1}{2})-\frac{1}{3}(1-\frac{1}{2}))x$$:

$$(1-\frac{1}{2})x – \frac{1}{3}(1-\frac{1}{2})x – \frac{1}{5}((1-\frac{1}{2})-\frac{1}{3}(1-\frac{1}{2}))x$$

$$= \frac{1}{1} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19} + \frac{1}{23} + …$$

Note that we can rewrite this mess: $$(1-1/2)x – 1/3(1-1/2)x – 1/5((1-1/2)-1/3(1-1/2))x$$ in the aesthetically preferable way: $$(1-1/2)(1-1/3)(1-1/5)x$$.

And if we keep going, canceling all of the multiples of 7 remaining, then the multiples of 11 remaining, then the multiples of 13 remaining, then the multiples of 17 remaining, …

$$= \frac{1}{1}$$ +1/7 + 1/11 + 1/13 + 1/17 + 1/19 $$+ \frac{1}{23} + …$$

What do we have left?

…1/23

Haha, just kidding, when we keep going we cancel everything but the 1 at the beginning:

$$[(1-\frac{1}{2}) (1-\frac{1}{3}) (1-\frac{1}{5}) (1-\frac{1}{7}) …]x$$
$$= \frac{1}{1}$$ +1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 + …

We only have to make a minor modification to have all of our shenanigans be above board, take the $$s$$th power of each denominator:

\$$[(1-\frac{1}{2^s}) (1-\frac{1}{3^s}) (1-\frac{1}{5^s}) (1-\frac{1}{7^s}) …]x = 1\$$

\$$x = \frac{1}{[(1-\frac{1}{2^s}) (1-\frac{1}{3^s}) (1-\frac{1}{5^s}) (1-\frac{1}{7^s}) …]}\$$

\$$x = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{5^s} + … = \Pi_p\frac{1}{1-\frac{1}{p^s}}\$$

This little guy has a name, his name is the zeta function:

\$$\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{5^s} + … =: \zeta(s) = \Pi_p\frac{1}{1-\frac{1}{p^s}}\$$

Terminology note: each step, $$\frac{1}{1-\frac{1}{p^s}}$$, is called an Euler factor (one for each prime), and the whole formula is called the Euler product.

The natural thing to ask is, well, why constrain ourselves to 1 in the numerator? Why not have a function:

$$\frac{\chi(1)}{1^s} + \frac{\chi(2)}{2^s} + \frac{\chi(3)}{3^s} + \frac{\chi(4)}{4^s} + … =: L(s)$$

For example, if I want my $$x$$ to look like this:

$$x = \frac{1}{1} – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + …$$

Then I need $$\chi(s)$$ that spits out $$\chi(1) = 1$$, $$\chi(2) = 0$$, $$\chi(3) = -1$$, etc.

This L(s) is called the L-function or Dirichlet series, and it is of great interest to people in general, particularly in the study of rational points on curves.

But, if you’ve been reading this blog for very long, you won’t be suprised to hear that I am interested because they come up in the dialogue between formal group laws and complex homology theories.

Specifically, I’ve been wondering about (5) in this excerpt of the paper “Dirchlet Series and Homology theory” by Smith and Stong:

Note that it is not ridiculous for (3) to be equivalent to (5): the $$b_i$$ are defined in terms of $$L_{\epsilon}$$, so secretly $$\epsilon$$ is present in the statement of (3).

We’ve been acquainted with the notion of a Dirichlet series and an Euler factor. At last, we can decipher the meaning of (5).

Recall that a genus is an operator that takes in a manifold (up to some equivalence relation) and spits out a number that characterizes it — of great interest to any who care about algebraic invariants. More specifically, an “$$R$$-genus” is a ring homomorphism \$$MU_* \xrightarrow{g} R\$$ from (complex-oriented) cobordism classes of manifolds to a commutative ring $$R$$.

Keep in mind that in the rational case, the generators of the ring $$MU_*$$ are the complex projective spaces of each dimension, $$P^n$$. \$$MU_* \otimes Q \simeq Z[P^n]\$$

Given a genus $$g$$, we can build a few different things to study its properties in a more manageable form. Today, we’re just going to mention two associated formal power series: the Dirichlet series, and the series we get out of Hirzebruch’s construction.

• \$$L_g(s) := \sum_{n \geq 1} \frac{g(P^{n-1})}{n^s}\$$
• \$$\hat{g}(z) := \sum_{n \geq 0} g(P^n)z^n\$$

For example, let’s say $$g(P^n) = 1$$ for all $$n$$, (that is, $$g$$ is the Todd genus) then $$L_g(x) = \sum_{n \geq 1} \frac{1}{n^s}$$. The classical fella, $$\zeta(s)$$, that we started this post with!

This isn’t just interesting, it also gives us a pretty useful theorem, which tells us that the p-primary divisiblility which takes place in $$MU_*$$ can be nicely described in terms of the L series associated to objects in $$Hom_{Ring}(MU_*, -)$$.

Keep in mind that Quillen proved that $$MU_*$$ is the universal formal group, so, for example, a ring homomorphism from $$g: MU_* \to Z_{(p)}$$ is nothing more than a formal group structure for $$Z_{(p)}$$.

This was a world-shaking result. It began to give us hope looking at the seemingly orderless, complicated, multi-layered signal sent by the stable homotopy groups of spheres. Quillen’s result led to first localizing this “signal” at a prime, then engineering various band-pass filters to isolate the individual messages, i.e., breaking this complicated multi-layered object into layers.

In this breathtaking paper, Morava eloquently states:
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To my dismay, I still don’t understand Quillen’s proof. But let’s get back to the Dirichlet story.

I’ll mention one more interesting thing about $$L$$-functions in this story.

As a subset of $$Hom_{Ring}(MU _*,Q)$$, let’s look at the set of maps that send $$Z[P^n] \to Z_{(p)}$$.

1. This set is in bijective correspondence with the set of { $$L$$-series with $$Z_{(p)}$$ coefficients, having leading term 1 }.
2. This set is endowed with an abelian group structure via the multiplication of $$L$$-series (series beginning with 1 are invertible).
3. Inside this group, we have a subgroup, $$G$$, given by the group of series:$$\sum \frac{g(P^n)}{n^s}$$ for which $$g(P^{p^r1-1}) \equiv 0 \mod p^r$$.
4. So the two series $$L_g$$ and $$L_f$$ with $$g, f$$ have the same Euler factor iff they lie in the same orbit of $$G$$.

Thanks to Laurens Gunnarsen for explaining the L-function, and to Peter Teichner for letting me borrow his copy of Landweber’s book “Elliptic curves and Modular forms in algebraic topology.”

## A First Look at an Equivariant Elliptic Cohomology

Usually, besides the information preserved by the formal group law of the elliptic curve, we can’t see any information about the elliptic curve when looking at the output of its associated cohomology theory. The formal group law only* remembers if the curve was singular/supersingular, and the characteristic of its field.

(*There’s also some crazy invocation of the Serre-Tate Theorem going on s.t. operations upstairs between elliptic curve formal group laws occur downstairs between elliptic cohomology theories but that’s for another post.)

A dream is to have a cohomology theory $$E^*(-)$$ that encodes all of the geometry of the elliptic curve $$C$$ used to construct it. One way to phrase this more precisely is by asking for a cohomology theory with a property along the lines of $$\textbf{E}^*(pt) = C$$.

This property is inspired by an analogue to K-theory, which satisfies something like $$\textbf{K}_G(pt) = \mathbb{G}_m$$, where $$\textbf{K}_G(-) := \text{Spec }K_G(-)$$.

Turns out that I’m not alone in this desire, such a cohomology theory has been constructed before!

Thanks to Eric Peterson for graciously answering all of my questions — all of them, and Minhyong Kim for carefully listening to me and clarifying my vague thoughts toward such an object. All errors in the following are mine and mine alone.

There are beasties called equivariant cohomology theories, for example, the singular beastie is defined like this: $$H_G(X) := H(X \times_G EG)$$

Where \$$X \times_G EG := X \times EG/(x \cdot g, e) \sim (x, g \cdot e)\$$, this is the Borel construction (“colimit”).

Note, $$X \times_G EG$$ is NOT the same shorthand for the below fiber product (“limit”):

If you’ve never encountered equivariant stuff before and want to get a feel for it beyond a beginner ranting at you, here’s an introductory lecture by Hopkins.

To the exciting part: Grojnowski constructed an equivariant $$E^*(-)$$ that trivially ensures that the elliptic curve is kept in the heart of the cohomology theory as its coefficient ring, $$E^*(pt) = C$$.

My rough understanding of Grojnowksi’s construction is:

• input:: $$X :=$$ space with an $$S^1$$-action
• look at an elliptic curve $$C$$
• over each point of the elliptic curve we have the stalk $$H^*_{S^1}(X) \otimes g$$, where $$g$$ is an element of the Lie algebra of the curve $$C$$
• output:: $$E^*_{S^1}(X) :=$$ these stalks are (magically*) glued together according to the Lie algebra of the elliptic curve $$C$$

*Here’s how I think the Lie algebra gluing together works:

Note that $$T_e\mathcal{E} : T\mathcal{E} = Lie \mathcal{E} \times \mathcal{E}$$ (tangent bundles of Lie groups are always trivial)
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Let’s look at the map $$T_e\mathcal{E} \to \mathbb{A}^1_\mathbb{C}$$:

For complex elliptic curve $$\mathcal{E} \simeq \mathbb{C}/L$$ , the Lie algebra $$\mathfrak{e}$$ is canonically $$\mathbb{C}$$ and the exponential map $$\mathfrak{e} to \mathcal{E}$$ is the reduction mod L.
For a point $$e$$ on the elliptic curve, we have a map to the tangent plane at that point:

$$\mathcal{E} \to T\mathcal{E}$$
$$e \mapsto T_e\mathcal{E} \simeq \mathbb{A}^1_\mathcal{C}$$

We also have an inclusion
$$e \hookrightarrow T_e\mathcal{E}$$ and we can identify an (open?) neighborhood of $$e \in E$$ with the open neighborhood of $$e \in T_e\mathcal{E}$$.
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Same picture with more labels:
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So, we can take something stalkwise defined over $$e \in \mathcal{E}$$ and glue it together based on transition maps between neighborhoods in the corresponding tangent spaces.

More specifically, we can take an object defined over $$e \in \mathcal{E}$$
and an object defined over $$e_1 \in \mathcal{E}$$,
and if their neighborhoods (which we’ll call $$N_e$$ and $$N_{e_1}$$) overlap, this allows us to define and test the sheaf gluing condition.

Since we identified the neighborhood of $$e$$ in the tangent space with the neighborhood of $$e$$ on the curve, this is all fine and good:
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But how does this connect to the cohomology theory as a sheaf over the elliptic curve?

I’m pretty sure that we can think of the map \$$\text{Spec }H^*_{S^1}(pt) \to \mathbb{A}^1_\mathbb{C}\$$ as mapping a copy of $$\text{Spec }H^*_{S^1}(pt)$$ to each tangent plane of $$\mathcal{E}$$.

Why? We’re assuming that:

1) If $$X$$ is a $$G$$-space $$\leadsto$$ $$H_G^*(X)$$ is a sheaf on $$\text{Spec }H^*_G(pt)$$

2) $$\text{Spec }H^*_{S^1}(pt) \simeq \mathbb{A}^1_\mathbb{C}$$

Expand for more detail

When $$G := S^1$$:

$$H_{S^1}^*(pt) \simeq (\text{Sym } \mathfrak{t}^*)^W \simeq \mathbb{C}[x]$$
$$\Rightarrow \text{Spec } H_{S^1}^*(pt) \simeq \mathbb{A}^1_{\mathbb{C}}$$

For points $$q$$ near the origin of $$\mathbb{A}^1_{\mathbb{C}}$$, there is a neighborhood $$U \in \mathbb{A}^1_{\mathbb{C}}$$ with $$\Gamma(U, H_{S^1}^*(X)) = H_{S^1}^*(X^{f(q)})$$

In this case, $$f$$ is just $$\text{\exp}: \mathbb{A}^1_{\mathbb{C}} to S^1$$ and the notation $$X^g$$ is the set of points in $$X$$ fixed by $$g$$ endowed with the subspace topology.

This $$E_{S^1}$$ beastie trivially satisfies the condition $$E_{S^1}^*(pt) = C$$, it just degenerates to (a mild modification of) the elliptic curve $$C$$ when X is a point.

We’re even supposed to expect that we have to mildly modify it!

It’s not possible for $$E_{S^1}^*(pt)$$ to give us the elliptic curve in the usual way (as a ring of global functions). I’ve been told that this is because elliptic curves (as examples of projective varieties) don’t have interesting global functions, but I don’t know why this is yet (first I must understand why there aren’t non-constant functions without poles on the Riemann sphere…).

I should probably tell you that when I say “mild modification” of our elliptic curve $$C$$, what I really mean is $$C \otimes Hom(S^1, T)$$, where we are idenifying $$Hom(S^1, T)$$ with the lattices of the Lie algebra of $$S^1$$; $$S^1$$ is the multiplicative circle group (i.e., chilling on the unit circle in $$\mathbb{C}$$, adding 2 angles to get a third angle), and $$T$$ is a compact torus.

Actually, the case where $$G = S^1$$ collapses Grojnowski’s construction quite a bit. Here, $$G = T, \mathfrak{g} = \mathfrak{t}$$, and $$Hom(T, S^1) = Hom(S^1, S^1) = \mathbb{Z}$$. So here, $$E_T = \mathbb{Z} \otimes_{\mathbb{Z}} C$$, and tensoring a ring with itself is a null operation, which gives us $$E_T = C$$.

But this isn’t the end of the story! Of course it isn’t.

Matt Ando (pg.29) hints at an extension of Grojnowski’s construction from complex elliptic curves to p-adic Tate curves!

That is, rather than looking at a complex elliptic curve $$C$$, which is represented by a map \$$\text{Spec }\mathbb{C} \xrightarrow{C} M_{\ell}\$$

We’re looking at the particular inclusion so that any further composite:

$$\text{Spec }R \xrightarrow{f} \text{Spf }Z[[u]] \hookrightarrow M_{\ell}$$ + a little compactification

selects the elliptic curve over $$R$$ with $$q$$-invariant specified by $$f(u)$$.

Something I really don’t get is if Grojnowski’s construction can be done in the non-cuspidal case, for some reason they say you can only do it in the cuspidal case but they don’t say WHY. Maybe it is too obvious to point out, but I know very little about curves so this continues to elude me.