Tolstoy: "Happy families are all alike; every unhappy family is unhappy in its own way."
Mathematics: "Real tori are all alike; every complex torus is complex in its own way."
To be precise, a 'n-dimensional real torus' is a real manifold of the form V/Λ where V is an n-dimensional real vector space and Λ ⊆ V is a lattice of rank n in this vector space. They are all isomorphic.
An 'n-dimensional complex torus' is a complex manifold of the form V/Λ where V is an n-dimensional complex vector space and Λ ⊆ V is a lattice of rank 2n in this vector space. These are not all isomorphic, because there are different ways the lattice can get along with multiplication by i. For example we might have iΛ = Λ or we might not.
And so, it's possible to write a whole book - and indeed a fascinating one - on complex tori. For example a 1-dimensional complex torus is an elliptic curve, and there are whole books just about those.
@johncarlosbaez this is another thing I thought would be somewhat useful in music theory, as Eisenstein series give us a moduli space for 2D lattices, which can also parameterize the subgroups of a 2D free abelian group. But is a similar representation possible for 3D, 4D, etc lattices?
@battaglia01 - sorry to take so long to reply. Are you still interested in this question?
"Eisenstein series give us a moduli space for 2D lattices, which can also parameterize the subgroups of a 2D free abelian group. But is a similar representation possible for 3D, 4D, etc lattices?"
@battaglia01 - Lattices in ℂⁿ give complex tori, and the nice ones give 'abelian varieties': complex tori that are projective varieties. Only the latter really act like generalizations of elliptic curves. The moduli space of elliptic curves, and the theory of modular forms, generalizes to higher dimensions this way. But people discovered you must work with abelian varieties equipped with an extra structure, a 'polarization', to get this to work. When we do, we're led to study 'Siegel modular forms'. But unfortunately it seems the Eisenstein series trick for getting modular forms doesn't work in higher dimensions, because this sum over points ℓ in a lattice Λ⊂ℂⁿ:
[ \sum_{\ell \in \Lambda} \frac{1}{(z-\ell)^n} ]
only makes sense when (n = 1). You might hope that some trick would save us, but I haven't seen any analogue of Eisenstein series in higher dimensions. I'm just learning this theory, so I might have missed something, but I've looked around.
@johncarlosbaez very interesting! Are Siegel modular forms are the way to parameterize arbitrary lattices? Why would we need Eisenstein series?
I had tried a different purely algebraic method, with partial results. My thinking was: the exterior algebra is uniquely the thing characterizing weighted subspaces given some very simple axioms:
Given independent ( a, b ), ({a, b}) and ({a, ra+b}) generate the same subspace. We want some binary operation such that ( a \wedge b = a \wedge (ra + b) ) for any scalar ( r ).
We want ( \wedge ) to be an algebra: it's bilinear, distributes over +, etc.
This uniquely specifies either the exterior algebra or the zero algebra: we have ( a \wedge b = a \wedge (ra + b) = a \wedge ra + a \wedge b = r(a \wedge a) + a \wedge b ), thus (a \wedge a = 0 ) for all vectors ( a ), and from there it's easy to see we also have ( a \wedge b = -b \wedge a ).
So, perhaps we could modify this to get a new operator (#) that represents "signed lattices." We'd want to replace that first axiom with the requirement that ( a # b = a # (ka + b) ) for any ( k \in \Bbb Z ) instead.
The main thing is, clearly this can't be an algebra: we don't want ((2a) # b = a # (2b)). But we already have something like anticommutativity:
[a # b = (-b) # a = b # (-a) = (-a) # (-b)]
The question is, what other properties should this have? My thought was to require that ( a # b ) be degree-2 homogeneous, rather than bilinear. This seemed to lead to some interesting stuff, although I don't have it all figured out.
It's also somewhat interesting we're parameterizing "signed lattices" here, or something like that - different from (G_4) and (G_6).
@battaglia01 wrote: "Are Siegel modular forms the way to parameterize arbitrary lattices? Why would we need Eisenstein series?"
As mentioned, Siegel modular form parametrize only certain special lattices L in ℂⁿ: those for which ℂⁿ/L is a projective variety. This phenomenon is invisible when n = 1 because every lattice in ℂ gives a projective variety (called an elliptic curve). Further, Siegel modular forms parametrize these certain special lattices only if we equip those lattices with some extra structure: at the very least, a polarization. Again this phenomenon is invisible for n = 1 because every elliptic curve has a god-given polarization!
In short, the whole subject seems complicated and technical if one is hoping for a straightforward generalization of elliptic curves, modular forms and Eisenstein series. But I think this is inevitable.
Like ordinary modular forms, Siegel modular forms are defined by some transformation properties, and you still have to work to actually find some. Eisenstein series are a nice way to get your hands on modular forms. I still don't know much about how we get Siegel modular forms.
@johncarlosbaez Thanks! A few things I thought of with regard to this:
In our situation, typically we're interested in lattices in (\Bbb R^n). The only reason we using the (\Bbb C) structure on (\Bbb R^2) is because it seems to give good results! We can do things like take a reciprocal of a vector and raise it to a power, and if we do that and sum on all nonzero vectors we seem to magically get a moduli space for the powers 4 and 6.
So perhaps we could put some algebra generalizing (\Bbb C ) on (Bbb R^n) and do something similar. The main thing is that, unless (n = 1, 2) or (4), then these have zero divisors. But certainly we could at least try with (\Bbb H), and see if it actually gives a correct moduli space.
The other thing is, all we really need is reciprocals and power-associativity! So we ought to be able to use octonions as well. If this actually gave the correct result it'd be great for music theory, giving us all primes up to 19 - particularly if there were some way to represent n-D sublattices embedded into (\Bbb R^8). I'm curious if there's something like Hurwitz's theorem which characterizes for what (\Bbb R^n) a "power-associative division algebra" structure can exist.
The other thought is: are zero divisors necessarily that much of an issue? We already skip the zero vector in Eisenstein series, so perhaps we could look at arbitrary power-associative algebras and just sum on non-zero divisors. Or perhaps we just don't care; there's simply a pole for any lattice which has a zero divisor in it. I'm not sure, but it seems like one ought to get something interesting.
I'm mostly thinking algebraically here - modular forms are still pretty new to me!
@battaglia01 wrote: "In our situation, typically we're interested in lattices in ℝⁿ."
Who is "us"? I can certainly imagine people getting interested in lattices in ℝⁿ. But all the work I mentioned on Siegel modular forms is about lattices in ℂⁿ, and it's trying to generalize work on lattices in ℂ which takes full advantage of complex analysis (or in modern terms, algebraic geometry). You can think of Eiseinstein series as giving functions of lattices in ℝ², but most people who work with them and their generalizations treat them as functions of lattices in ℂ and take full advantage of that fact: this is a prerequisite for treating them as 'modular forms', which unlocks much of their power.
But you're going in a different direction, so everything I said in my last post is basically irrelevant!
You can define something like Eisenstein series for lattices in 4 dimensions in the way you suggest, treating ℝ⁴ as the quaternions ℍ. I don't know anything about that except that the first publication mentioning octonions was the appendix of a completely unrelated paper by Cayley called "On Jacobi's Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions". This seems to be about Cayley's attempt to generalize elliptic functions from the complex numbers to the quaternions. Elliptic functions are closely connected to modular forms. So I wouldn't be shocked if Cayley had tried to study quaternionic analogues of Eisenstein series. Apparently everything in this particular paper was wrong except the appendix, so it was left out of his collected works. But Cayley wrote vast numbers of papers, and only god knows what's in all of them.
@battaglia01 - As you suggest, you can also write down a formula for Eisenstein series for lattices in 8 dimensions, treating ℝ⁸ as the octonions 𝕆. Maybe Cayley was even contemplating something like this! More likely he was trying to invent octonionic elliptic functions.
I've never seen anyone discuss quaternionic or octonionic Eisenstein series. I'll google those terms.
There are division algebras except in dimensions 1, 2, 4, and 8, but you don't need a division algebra to make sense of the usual formula for an Eisenstein series: you just need a power-associative algebra where every nonzero element has a two-sided inverse. (This is NOT the same thing as a power-associative division algebra: it's weaker!) There's an algebra of this sort for every dimension that's a power of 2:
These algebras have zero divisors when we reach dimension 16, but still every nonzero element has a left and right inverse, and that's good enough to write down the formula for Eisenstein series!
However, I think it's wise to see what (if anything) people have done with Eisenstein series in the quaternionic and octonionic cases, before tackling these still more exotic cases! I have never seen anyone do anything useful with sedenions.
@johncarlosbaez Thanks, very interesting! When I said I care mostly about real lattices, I meant in tuning theory, where the main reason I'm interested in this to begin with is in parameterizing the subgroups of a free abelian group (representing just intonation intervals, for instance).
This isn't a super pressing issue - typically, we don't even really need to parameterize arbitrary lattices in ( \Bbb R^n ), but just arbitrary subgroups of ( \Bbb Z^n ). And if we just want a bare-bones computational representation, we can always just use something like Hermite normal form to uniquely represent any such lattice. This gets the job done, but I've been curious if something better exists, particularly since the exterior algebra has such an elegant representation of subspaces and there are plenty of reasons you may want it instead of reduced row echelon form, for instance. And sometimes we do care about arbitrary lattices in ( \Bbb R^n ). So sometimes we just use the exterior algebra anyway, taking the exterior product of two basis vectors for the lattice, and just note that the result only corresponds to an equivalence class of lattices.
I see quite a bit of stuff involving quaternionic Eisenstein series, so it looks like they've at least been studied to some degree. I guess the main question is if they successfully manage to represent all lattices in ( \Bbb R^4 ) uniquely.
That is a good point regarding power-associative algebras which have inverses! So clearly such algebras exist in ( \Bbb R^{2^n} ). Do they only exist in those situations or are there others?
@johncarlosbaez that is not too surprising: if we take the analogous in the complex setting, we would have a (pseudo) "lattice" of rank n, a soubroup of V generated by n linearly independent vectors! these quotients would then all be isomorphic (right?)
but lattices in complex vector spaces are defined as being lattices with respect to viewing the complex vector space as a real vector space (therefore rank 2n). and then we look at the resulting spaces as a complex manifold.
@berber - Yes, everything you said is right. It's not surprising. Perhaps what's surprising is that there's such a rich and beautiful theory of complex tori. This especially true of the nicest ones, the 'abelian varieties', which are complex algebraic varieties.
@wnj - I have been enjoying 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝐴𝑏𝑒𝑙𝑖𝑎𝑛 𝑉𝑎𝑟𝑖𝑒𝑡𝑖𝑒𝑠 by Herbert Lange and Christina Birkenhake - not easy, but full of beautiful material. Complex abelian varieties are the 'best' complex tori. The same authors have an earlier book 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑇𝑜𝑟𝑖, but for some stupid reason I haven't looked at that yet.
@johncarlosbaez @wnj
I may not have learned much in my years on this planet, but I have learned this: if John Carlos Baez says "not easy", there is no point for me to even look at it... 🤷🏽♂️
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