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 $ klevel $ 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.