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What does the theta series of the even, unimodular, positive definite lattice D16+ look like? Also, is there a way to look this information up for any lattice?

anon
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  • What did you try? Do you have any ideas? Where and why did this question arise? It's hard to see how much you know about modular forms and theta series. –  Jul 24 '17 at 16:22
  • I know that the theta series is supposed to be a modular form of weight 8, and that modular forms of weight 8 are polynomials in the eisenstein series $E_8.$ – anon Jul 24 '17 at 16:26
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    The space of modular forms of weight 8 is 1-dimensional. Thus the theta series is a multiple of $E_8$. –  Jul 24 '17 at 16:28
  • I also know what the D16+ lattice looks like, i.e it consists of all vectors $(a_1,...,a_{16})$ such that either all the $a_i$ are integers or they are all integers plus 1/2, and their sum is even. – anon Jul 24 '17 at 16:28
  • Alright, is there a way to find the multiplying factor? – anon Jul 24 '17 at 16:29
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    start with http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ – Will Jagy Jul 24 '17 at 16:30
  • Looking at the Fourier expansion is a possibility. –  Jul 24 '17 at 16:32
  • It's the square of the theta function for the $E_8$ lattice. – Angina Seng Jul 24 '17 at 16:45
  • You can try the LMFDB site. – Somos Jul 24 '17 at 18:02
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    This does not need any sites or hard calculations. We know that the space of weight $8$ modular forms has dimension one. Moreover, we know a non-zero element: $E_8$ with constant fourier coefficient $a(0) = 1$. Now, it is obvious that the constant fourier coefficient of the theta series is equal to $1$. As a consequence, we obtain that they are equal. Analogously one sees the equality @LordSharktheUnknown mentions. –  Jul 24 '17 at 18:53
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    Thank you Paul. I have one last comment, this means that both the even unimodular lattices of dimension 16 have same theta series? Is that usually the case? – anon Jul 24 '17 at 19:49
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    @anon The amount of even unimodular lattices of dimension $8n$ grows much faster then the dimension of the vector spaces of modular forms (for $n = 3$ we have $24$, for $n = 4$ there are more then a billion of them, while the corresponding dimensions are $1, 1, 2$). So in a sense, this should happen quite often. –  Jul 24 '17 at 21:52

1 Answers1

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  • $h(t) = e^{-\pi \|t\|^2}, t \in \mathbb{R}^k$ is its own Fourier transform. If $M \in \mathbb{R}^{k \times k}$ is an invertible matrix then the Fourier transform of $h(Mt) = e^{-\pi \|M t\|^2}$ is $\frac{1}{|\det M|} h(M^{-T} t)$.

  • If $\Lambda = M \mathbb{Z}^k$ is a lattice then we define its theta series $$\Theta_\Lambda(x) = \sum_{\lambda \in \Lambda} e^{- \pi x \|\lambda\|^2} =\sum_{n \in \mathbb{Z}^k} h(\sqrt{x} M n), \qquad \Re(x) > 0$$ and with $H(n)=h((\sqrt{x}I) M n)$ the Poisson summation formula gives $$\Theta_\Lambda(x) = \sum_{n \in \mathbb{Z}^k} H(n)=\sum_{n \in \mathbb{Z}^k} \widehat{H}(n)= \frac{x^{-k/2}}{|\det M|} \Theta_{\Lambda'}(1/x)$$ where $\Lambda' = M^{-T} \mathbb{Z}^k$ is the dual lattice.

  • Note $\Lambda$ and $P \Lambda$ have the same theta series for any orthogonal matrix $PP^T=P^TP = I$.

    Let $G = M^T M$. If $G \in \mathbb{Z}^{k \times k}$ then $\Lambda' = M^{-T} \mathbb{Z}^k \supseteq M^{-T} G \mathbb{Z}^k = M \mathbb{Z}^k = \Lambda$. If moreover $\det(G) = 1$ then $G^{-1} \in\mathbb{Z}^{k \times k}$ and $\Lambda= \Lambda'$. The lattice is then said unimodular. We also get $\|\lambda\|^2 \in \mathbb{Z}$ so that

    $$\Theta_\Lambda(2iz)=\Theta_\Lambda(2iz+1), \qquad \Theta_\Lambda(\frac{-2i}{4z}) = (i z)^{-k/2} \Theta_\Lambda(2iz)$$ and if $8 | k$ then $\Theta_\Lambda(2iz) \in M_{k/2}(\Gamma_0(4))$.

  • If also $\|\lambda\|^2 \in 2\mathbb{Z}$ then $\Lambda$ is said even unimodular, in that case $\Theta_\Lambda(iz) \in M_{k/2}(SL_2(\mathbb{Z}))$. This is the case here.

  • $M_{8}(SL_2(\mathbb{Z}))$ is one-dimensional (containing only the Eisenstein series) so you'll get $\Theta_\Lambda(iz) = E_8(z)$ the Eisenstein series.

reuns
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  • @PaulK I removed that part.. This is what we assume but it is not clear to me what are the properties of $M,A$ making a lattice unimodular/even. – reuns Jul 24 '17 at 22:09
  • @PaulK Thanks I edited accordingly – reuns Jul 24 '17 at 22:29