Which even lattices have a theta series with this property?

Question

This is a slight generalization of a question 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 arithmetic function?

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Examples

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

(Feel free to skip this part)

I do not know much about modular forms so this may contain mistakes. Theorem 4 in these notes 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 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, 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 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, 2) 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 by Hecke himself.

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Update: following Prof. Kimball's suggestion, I used this LMFDB online tool 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, 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).

Answer 1

I don't have a full answer to your question, and have not tried to work out details, but I can offer some thoughts, at least in even dimension:

• First, to satisfy that property, the theta series needs to be an Eisenstein series which is a Hecke eigenform (using non-trivial constant term and multiplicativity), say with weight $k$, level $N$ and character $\chi$. (Typically $N$ should divide the level $M$ of $\Lambda$ but I think there may be some edge cases to worry about with non-trivial character. At least you have $N | M^2$.)

• Presumably you can use your multiplicativity property to show that your Eisenstein series needs to be new of whatever level $N$ it is.

• Since your theta series must have positive coefficients, the character $\chi$ must be real (so quadratic) and the $p$-th coefficient in the $q$-series should be $p^{k-1} + \chi(p)$. Possibly your multiplicativity condition rules out nontrivial $\chi$.

• Then, as you suggest, you should be able to use facts about Bernoulli numbers, to rule out most weights. E.g., the von-Staudt-Clausen theorem which tells you the denominators should suffice for trivial character.

• If $\Lambda$ is an ideal in a definite quaternion algebra (over $\mathbb Q$ or a totally real field of narrow class number 1) and its (say right) order $O$ is Eichler, then one can show its theta series is an Eisenstein series if and only if we are in class number 1. Namely, the Eichler condition implies there is a unique Eisenstein in the space of theta series of the right $O$-ideals, but this Eisenstein series is a sum over all ideal classes.

Similarly it's reasonable that this (essentially? I'm not familiar with your 16-d example) can only happens for lattices with 1 class per genus.

For half-integral weight forms, I wouldn't expect multiplicativity either, but I don't know where this might be proved. For your case, I would expect you can just check this from details about half-integral weight Eisenstein series, but perhaps someone else can add something more?

Answer 2

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 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 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), 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 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, 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 $\mathbf{a}-\mathbf{b} \not\propto \mathbf{a}$.