This is a slight generalization of a [question](https://math.stackexchange.com/questions/3203167/which-root-lattices-have-a-theta-series-with-this-property) I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this site.


Suppose $\Lambda$ is an even lattice. Consider its theta series


$$\theta\_{\Lambda}(q) = \sum\_{a\in \Lambda} q^{(a,a)/2},$$


where $(\cdot,\cdot)$ denotes the Euclidean inner product.


My question is:



> 
> For which $\Lambda$ do we have
> 
> 
> $$\theta\_{\Lambda}(q) = 1+m\sum\_{n>0}\frac{f(n)\: q^n}{1-q^n}$$
> 
> 
> where $m$ is nonzero and $f$ is a [totally multiplicative](https://en.wikipedia.org/wiki/Completely_multiplicative_function) arithmetic function?
> 
> 
> 




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Examples
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I only know of two kinds of lattices with this property:


1. Maximal orders in rational division algebras with class number 1, scaled by $\sqrt{2}$:


	* Dimension 1: The integers, with $m=2$ and $f(n)=\lambda(n)$ is the Liouville function.
	* Dimension 2: The rings of integers of imaginary quadratic fields of discriminants $D = -3, -4, -7, -8, -11, -19, -43, -67, -163$. Here $m=\frac{2}{L(0,f)}$ and $f(n) = \left(\frac{D}{n}\right)$ is a Kronecker symbol, and $L(0,f)$ is given by $\sum\_{n=0}^{|D|} \frac{n}{D} \left(\frac{D}{n}\right)$.
	* Dimension 4: The maximal orders of totally definite quaternion algebras of discriminants $D = 4, 9, 25, 49, 169$. Here $m=\frac{24}{\sqrt{D}-1}$ and $f(n) = n \left(\frac{D}{n}\right)$.
	* Dimension 8: The Coxeter order in the rational octonions, with $m=240$ and $f(n)=n^3$.
2. The two 16-dimensional lattices of heterotic string theory, $E\_8\times E\_8$ and $D\_{16}^+$. Both lattices have the same theta series, with $m=480$ and $f(n)=n^7$.


These include in particular all the root lattices I mentioned in the original Math.SE post.




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Attempt
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(Feel free to skip this part)


I do not know much about modular forms so this may contain mistakes. Theorem 4 in [these notes](http://www.math.mcgill.ca/goren/GS/Thetas.pdf) implies that in even dimension there is a level $N$ and a character $\chi$ taking values in $\{-1,0,1\}$ for which $\theta\_{\Lambda}$ is a modular form of weight $k = (\mathrm{dim}\: \Lambda) /2$. The requested property, in turn, implies that the Epstein zeta function of the lattice has an Euler product


$$\zeta\_{\Lambda} (s) \propto \prod\_p \frac{1}{1-(1+f(p))p^{-s}+f(p)p^{-2s}} = \zeta(s) \prod\_p \frac{1}{1-f(p)p^{-s}},$$


which in even dimension means that $\theta\_{\Lambda}$ is a Hecke eigenform (noncuspidal, given the leading coefficient 1); we thus see that it must be an Eisenstein series of weight $k$, level $N$ and character $\chi$, by the decomposition of the space of modular forms into Eisenstein + cuspidal subspaces.


This Eisenstein series has the Fourier expansion $E\_{k,\chi}(q) = 1- (2k/B\_{k,\chi}) \sum (\cdots)$ where $B\_{k,\chi}$ is a [generalized Bernoulli number](https://en.wikipedia.org/wiki/Bernoulli_number#Generalized_Bernoulli_numbers) and the $(\cdots)$ part has integral coefficients. So one possible course of action would be to find those generalized Bernoulli numbers for which $2k/B\_{k,\chi} = -m$ is a negative even integer (since in $\Lambda$ there must be an even number of vectors of norm 2), and check case by case whether the associated Eisenstein series is the theta series of a lattice.


If this approach is correct, we can then use Tables 1-3 in [this paper](https://arxiv.org/abs/1110.6430), which shows that the only such cases with $\mathrm{dim}\: \Lambda \ge 4$ are the ones given in the Examples section, together with a certain Eisenstein series of weight 2 and level 42, which does not seem to correspond to a lattice.


On the other hand, I don't understand what happens in the odd-dimensional case (apart from dimension 1 which is trivial), where the modular forms involved are of half-integral weight. It seems that the concept of Hecke eigenform is defined a bit differently, so the above approach may not work here. I found [this answer](https://mathoverflow.net/a/118040) which says that zeta functions associated to modular forms of half-integer weight generally lack Euler products. Here are also some possibly relevant questions ([1](https://mathoverflow.net/questions/180203/ternary-quadratic-form-theta-series-as-hecke-eigenforms-and-class-number-one), [2](https://mathoverflow.net/questions/33905/odd-powers-of-the-theta-function-as-eigenforms)) dealing with special cases. Another particular case, namely products of powers of the Jacobi theta function and Dedekind eta function, was treated in [this paper](https://eudml.org/doc/160130) by Hecke himself.




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**Update:** following Prof. Kimball's suggestion, I used [this LMFDB online tool](http://www.lmfdb.org/Lattice/) to check the theta series of all even integral lattices with one class per genus. I found no further examples of lattices with the required property.


Given the Hecke eigenform condition and the [Siegel-Weil formula](https://en.wikipedia.org/wiki/Siegel%E2%80%93Weil_formula), I think this would suffice as a proof that the above list of 18 lattices is complete, *if* we can show that there is no case other than $(E\_8\times E\_8, D\_{16}^+)$ where all lattices in a single genus are isospectral (or if there is a finite, checkable number of such exceptional cases).



Which even lattices have a theta series with this property?
After revisiting my question, I think I have managed to find a proof that there are no other examples of lattices with the requested property. I'm posting it as a self-answer in case someone is interested. Here is the outline:


* The case of even dimension $\mathrm{dim}\: \Lambda \ge 4$ was already described$^\dagger$ in the "Attempt" section of my question; the most important part (bounding the generalized Bernoulli numbers) is done in Section 3 of [this 2011 paper](https://arxiv.org/abs/1110.6430) by M. Johnson which I cited in the question.
* We can also adapt the strategy in that paper to the odd-dimensional case. As in the even case, we start by observing that the theta series must be a certain Eisenstein series of half-integer weight $k$ and prescribed level $4N$ and character $\chi$, normalized such that the zeroth coefficient is $1$. One needs an expression for the Fourier coefficients of these series, which I found for $\mathrm{dim}\: \Lambda = 2k \ge 5$ in page 17 of [this 2015 thesis](https://escholarship.org/content/qt98k5g03d/qt98k5g03d.pdf) by M. Owen. We are only interested in the coefficient of $q^1$, which simplifies to


$$c\_1 = \left(-2\pi i/N\right)^k \:\Gamma(k)^{-1} A(1) X(1),$$


where


$$A(1) = \sum\_{r=1}^{4N} \epsilon\_r^{2k} \left(\frac{4N}{r}\right) \chi(r) e^{2\pi ir/N}, \quad X(1)= \prod\_{l\nmid 4N}(1+\chi(l) \epsilon\_l^{2k-1} l^{1/2-k})$$


(as in Owen's paper, here $l$ is prime, $\left(\frac{4N}{r}\right)$ is a Kronecker symbol, and $\epsilon\_n$ denotes the principal branch of $\sqrt{\left(\frac{-1}{n}\right)}$. Note also that I use $4N$ where the author uses $N$, since for half-integer weight forms the level is always a multiple of 4). This $c\_1$ is an analogue of the generalized Bernoulli numbers for half-integral weight.


The $A(1)$ factor is a sum of roots of unity whose modulus is obviously bounded by $4N$, and by the same argument as in page 9 of Johnson's paper, $|X(1)|$ is bounded by $\prod\_{l}(1+l^{1/2-k})=\frac{\zeta(k-1/2)}{\zeta(2k-1)}$. Putting everything together, we see that the bound on $|c\_1|$ will be lower than $2$ unless $N$ and $k$ are low enough (concretely we have either $N=1$ and $k \le 9/2$ or $1 < N \le 3$ and $k=5/2$), so recalling that $c\_1$ must be a nonzero even integer, we can restrict our analysis to these cases only. All of them are then ruled out by manual computation of the coefficients.
* The case $\mathrm{dim}\: \Lambda = 2$ is easily dealt with by using the relationship between 2D integer lattices and rings of integers of imaginary quadratic number fields, and applying the class number formula.
* This only leaves the case $\mathrm{dim}\: \Lambda = 3$. It is known that there are no isospectral lattices in dimension lower than $4$ (see e.g. the end of Section 2 [here](https://www.maths.ed.ac.uk/~v1ranick/papers/conwaysens.pdf)), so in this case it suffices to appeal to the Siegel-Weil formula and check the single-class examples in dimension $3$ with the LMFDB online tool, which I already did (this could also be an alternative way to treat the case $\mathrm{dim}\: \Lambda = 2$). This completes the proof.




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**Remark:** I admit that the proof above is somewhat "ugly", since it does not give any insight into the form of the classification. A more satisfying proof would explain why all of these lattices have the multiplicative structure of an order in a division algebra (and thus occur in dimensions 1, 2, 4, 8) except for the sporadic case $(E\_8 \times E\_8, D\_{16}^+)$.


There is an observation to be made regarding that last case. For any lattice $\Lambda$ it is possible to define [Siegel theta series](https://en.wikipedia.org/wiki/Siegel_theta_series) of genus $g$ generalizing the usual theta series, which essentially count the number of $g$-dimensional sublattices of $\Lambda$. Since there is a bijective correspondence between lattices and their theta series in genus $g\ge \mathrm{dim} \: \Lambda$, the issue cannot arise that two lattices have the same theta series; in fact, the sporadic example of $E\_8 \times E\_8 \leftrightarrow D\_{16}^+$ disappears already in genus 4 because of the [Schottky form](https://en.wikipedia.org/wiki/Schottky_form), and the corresponding Siegel-Weil formula for $g\ge 4$ takes the form $\mathrm{E}\_8^{(g)} = \frac{405}{691} \Theta\_{E\_8 \times E\_8}^{(g)} + \frac{286}{691} \Theta\_{D\_{16}^+}^{(g)}$ with $\Theta\_{E\_8 \times E\_8}^{(g)} \neq \Theta\_{D\_{16}^+}^{(g)}$. In contrast, the octonionic "$\mathrm{E} = \Theta$" identity between the Siegel theta series of the $E\_8$ lattice and the Siegel Eisenstein series of weight 4 does hold for all genera, and I believe the same happens for the other examples. This could perhaps be a starting point to explain why the sporadic case doesn't have a multiplicative structure.




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$\dagger$- The extra possible example of dimension $4$ and level $42$ that I mentioned does not correspond to any lattice; perhaps the easiest way to see this is to inspect the coefficients of the candidate theta series, which starts as


$$1+2q+2q^2+2q^3+2q^4+12q^5+2q^6+2q^7+2q^8+2q^9+12q^{10}+\ldots$$


(the general coefficient is twice the sum of the divisors of $n$ that don't divide $42$).


Since the coefficient of $q^n$ counts the number of vectors of norm $2n$, such a lattice would have a vector $\mathbf{a}$ of squared norm $2$ and another, linearly independent vector $\mathbf{b}$ of squared norm $10$. We can also see that all vectors of squared norm lower than $10$, as well as those with squared norms strictly between $10$ and $20$, must be integer multiples of $\mathbf{a}$, since there are only two vectors of each norm in these intervals.


The squared norm of the sum $\mathbf{a}+\mathbf{b}$ is a positive even integer bounded by $(\sqrt{2}+\sqrt{10})^2 \approx 20.944$; by the above observations, it can only be $20$. But then by the parallelogram law, $|\mathbf{a}-\mathbf{b}|^2 = 2|\mathbf{a}|^2+2|\mathbf{b}|^2-|\mathbf{a}+\mathbf{b}|^2 = 4+20-20=4$, which is a contradiction since $\mathbf{a}-\mathbf{b} \not\propto \mathbf{a}$.


