# nt.number theory – Which peer networks have a theta series with this property?

This is a slight generalization of a question I asked in Math StackExchange and which remains unanswered after a month. So I decided to post it here. I am sorry in advance if this is inappropriate for this site.

assume $$Lambda$$ is an even trellis. Consider his theta series

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

or $$( cdot, cdot)$$ means the Euclidean domestic product.

My question is:

For who $$Lambda$$ have we

$$theta _ { Lambda} (q) = 1 + m sum_ {n> 0} frac {f (n) : q ^ n} {1-q ^ n}$$

or $$m$$ is not zero and $$f$$ is a fully multiplicative arithmetic function?

## Examples

I only know two types of networks with this property:

1. Maximum 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 function of Liouville.

• Dimension 2: The rings of imaginary quadratic fields of discriminants $$D = -3, -4, -7, -8, -11, -19, -43, -67, -163$$. Right 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 fully defined quaternion discriminant algebras $$D = 4, 9, 25, 49, 169$$. Right here $$m = frac {24} { sqrt {D} -1}$$ and $$f (n) = n left ( frac {D} {n} right)$$.

• Dimension 8: The order of Coxeter in rational octonions, with $$m = 240$$ and $$f (n) = n ^ 3$$.

2. The two 16-dimensional networks of Heterotic String Theory, $$E_8 times E_8$$ and $$D_ {16} ^ +$$. Both networks have the same theta series, with $$m = 480$$ and $$f (n) = n ^ 7$$.

These include in particular all the root networks I've mentioned in the original Math.SE article.

## Attempt

(Do not hesitate to skip this part)

I do not know much about modular forms, so it can contain errors. Theorem 4 in these notes implies that in an even dimension there is a level $$N$$ and a character $$chi$$ take values ​​in $${- 1,0,1 }$$ For who $$theta _ { Lambda}$$ is a modular form of weight $$k = ( mathrm {dim} : Lambda) / 2$$. The requested property in turn implies that the Epstein zeta function of the network has an Euler product.

$$zeta _ { Lambda} (s) propo prod_p frac {1} {1- (1 + f (p)) p ^ {- s} + f (p) p ^ {- 2s}} = zeta (s) prod_p frac {1} {1-f (p) p ^ {- s}},$$

which in the very dimension means that $$theta _ { Lambda}$$ is a proper form of Hecke (non cumulate, given the main coefficient 1); so we see that it has to act from a series of weights from Eisenstein $$k$$level $$N$$ and the character $$chi$$, by the decomposition of the space of modular forms into subspaces Eisenstein + cuspidal.

This series Eisenstein has Fourier expansion $$E_ {k, chi} (q) = 1- (2k / B_ {k, chi}) sum ( cdots)$$ or $$B_ {k, chi}$$ is a generalized number of Bernoulli and the $$( cdots)$$ the part has integral coefficients. So, one possible solution would be to find those generalized Bernoulli numbers for which $$2k / B_ {k, chi} = -m$$ is a negative integer (since in $$Lambda$$ there must be an even number of vectors of the standard 2) and check on a case-by-case basis if the associated Eisenstein series is the theta series of a network.

If this approach is correct, we can then use Tables 1 to 3 of this article, which show that the only cases of this type with $$mathrm {dim} : Lambda ge 4$$ are those given in the Examples section, as well as a certain series of Eisenstein weights 2 and 42, which do not appear to correspond to a network.

Moreover, I do not understand what happens in the case of odd dimensions, where the modular forms involved have a semi-integral weight. It seems that the concept of Hecke eigenform is defined a little differently, so the above approach may not work here. I found this answer which says that zeta functions associated with modular forms of an entire half-weight are usually devoid of Euler products. Here are also some potentially relevant questions (1, 2) about particular cases.

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