Lecture notes for
Math 272y: Rational Lattices and their Theta Functions (Fall 2019)

If you find a mistake, omission, etc., please let me know by e-mail. 4 September: Overview of Math 272
Corrected Sep.21-23 (A. Blumer): A few instances of $N_Q(m)$, and one of $N_m(E_8)$, should have been (and now are) $N_m(L)$; insert a phrase missing from the description of Siegel theta functions 9/9: Lattice basics
Corrected Sep.10 (A. Blumer): ${\rm Mat}_{n \times n}(R)$, not ${\rm Mat}_{n \times n}(A)$
[ring, not anneau; $A$ was already taken, so I changed it to $R$, but evidently didn’t catch every last instance of $A$]. September 11: Lattice basics II
The dual lattice; the discriminant group and form; elementary constructions of new lattices from known ones
Corrected Sep.12 (A. Blumer): generally no canonical isomorphism; trimodular, not “triimodular” In the context of Pontrjagin duality, you can think of dualizing the short exact sequence $0 \to L \to V \to V/L \to 0$ to get $0 \leftarrow V^*/L^* \leftarrow V^* \leftarrow L^* \leftarrow 0$. Quadratic forms $\sum_{j=1}^n c_j x_j^2$, corresponding to lattices $\oplus_{j=1}^n {\bf Z} \langle c \rangle$, are often called “diagonal” because they have diagonal Gram matrices. By a somewhat abusive analogy, degree-$d$ forms $\sum_{j=1}^n c_j x_j^d$, and their vanishing loci in ${\bf P}^{n-1}$, are sometimes called “diagonal forms” and “diagonal (hyper)surfaces” even for $d \geq 3$ when there is no diagonal matrix. Given a nondegenerate integral(*) quadratic form $Q$ on $L$ of some fixed discriminant, the signature gives the isomorphism class $Q$ on $L \otimes \bf R$; the $p$-part of the discriminant group likewise gives most of the isomorphism class $Q$ on $L \otimes {\bf Q}_p$ for $p$ odd, and the even/odd distinction and the discriminant form give additional information for ${\bf Q}_2$. (Already for $n=1$ it can be seen that the $2$-adic classification of quadratic forms is more delicate than the $p$-adic one for odd $p$.) This illustrates a common theme in number theory: studying an object over $\bf Q$ (or more generally over some global field) by describing it locally at each completion. For indefinite forms, it is known that the local information almost classifies $Q$ completely; the culminating result of the first part of Serre’s A Course in Arithmetic is the classification of unimodular quadratic form, for which the only local invariants are the signature and the even/odd distinction. (I wrote “almost” because rank $2$ is a special case, and there might be a rare further wrinkle involving the spinor genus.) We shall not pursue this very far in Math 272 because our main concern is the study of definite forms.
(*) If $Q$ is only rational, consider $d \cdot Q$ where $d$ is the least common denominator of the pairings $B(x,y)$. September 16: Theta functions; the functional equation relating $\theta_L$ with $\theta_{L^*}$ Note that the functional equation relating $\Theta_L$ with $\Theta_{L^*}$ passes the “sanity checks” of being consistent with $L^{**} = L$ (using $\mathop{\rm disc} L^* = (\mathop{\rm disc} L)^{-1}$) and $(L\langle c \rangle)^* = (L^*) \langle c^{-1} \rangle$ (together with $\mathop{\rm disc}(L\langle c \rangle) = c^n \mathop{\rm disc} L$). While the functional equation for $\Theta_{\bf Z}$ doesn't directly tell us anything about $\Theta_{\bf Z}(1) = \sum_{n \in \bf Z} \exp(-\pi n^2)$, there is a “closed form” for this number $1.08643481\ldots$: remarkably it equals $\pi^{1/4} \, / \, \Gamma(3/4)$ ! This is a nice “exercise” in the classical theory of elliptic functions, which alas would take us too far afield in Math 272. Theta functions probably use the letter $\theta$ because they generalize the value at zero of a Jacobi theta function, which is a theta function of a rank-1 lattice. I refrain from giving here a formula for a Jacobi theta function, because there is such a bewildering variety of notations that there is a Wikipedia page devoted just to “Jacobi theta functions (notational variations)”! September 23 and 25: The theta functions of unimodular lattices
Updated Sep.26: paragraph added at the end about A characterization of the ${\bf Z}^n$ lattice by its shortest characteristic vectors.
Corrected Oct.3 to remove stray text “Now define $M_k$ to be the vector space of modular forms f ” before (21) The fact that $\Delta$ has no finite zeros also follows from the product formula $\Delta = q \prod_{n=1}^\infty (1-q^n)^{24}$. Serre gives this as Theorem 6 in Chapter VII of A Course in Arithmetic (with an extra factor of $(2\pi)^{12}$ because he uses a different normalization of $\Delta$), with a “proof, which is ‘elementary’ but somewhat artificial”. There are several other approaches to this, some of which might eventually appear here. We shall at one point use the consequence that all coefficients of $1 / \Delta = q^{-1} + 24 + 324 q + 3200 q^2 + 25650 q^3 + 176256 q^4 + \cdots$ are positive (because this is true of the factor $(1-q)^{-24}$, and all other factors $(1-q^n)^{-24}$ for $n \geq 2$ have nonnegative coefficients). This Youtube video of Coxeter discussing the math behind Escher’s circle limit is not as mathematical as one might have liked, but still interesting because it was Coxeter who suggested to Escher that the Poincaré disc model of $\cal H$ was the natural setting for such a tiling. September 30 and October 2: Positive-definite integral lattices generated by vectors of norm at most 2
Corrected Oct.3 to fix several typos noted by A. Blumer Here are pictures of the ADE and affine ADE diagrams, with the multiplicities for $\tilde{D}_n$ and $\tilde{E}_n$ (adapted from an unfinished 2006 manuscript of notes on elliptic fibrations; NB: the coordinates for $D_n$ are numbered in reverse order from what I used in the lecture notes). October 7: Gluing No class October 9 (Yom Kippur) and October 14 (University holiday: Columbus/IP day). Next meeting is October 16. October 16 and 21: Harmonic polynomials The spaces of harmonic polynomials form one of those mathematical objects that connect several different parts of mathematics, each of which suggests a way to view and develop the theory. Notably, the decomposition ${\cal P}_d = \oplus_{k=0}^{\lfloor d/2 \rfloor} {\sf F}^k {\cal P}_{d-2k}^0$ is also the decomposition of the space ${\cal P}_d$ into irreducible representations of $O_n({\bf R})$; and the restriction of ${\cal P}_d^0$ to a sphere $\Sigma$ is an eigenspace for the Laplacian $\Delta_\Sigma$, so the fact that these spaces topologically span ${\cal C}(\Sigma)$ is a spectral decomposition for the action of $\Delta_\Sigma$. Either of these approaches lets us generalize our observation that the spherical average of a nonconstant harmonic polynomial is zero: if $P_1,P_2$ are harmonic of degrees $d_1,d_2$, and $d_1 \neq d_2$, then the average of $P_1 P_2$ over $\Sigma$ vanishes. In the $O_n({\bf R})$ viewpoint, this is a manifestation of Schur's lemma; in the $\Delta_\Sigma$ picture, it is a consequence of the orthogonality of eigenspaces of a self-adjoint operator for different eigenvalues. We proved the special case $d_2 = 0$ using our formula for the Fourier transform of $G_t P_1$; here’s a generalization that proves the orthogonality result for any distinct $d_1,d_2$. We may assume $d_1 \gt d_2$. It is enough to prove for some $t \gt 0$ that the integral over ${\bf R}^n$ of $G_t P_1 P_2$ is zero. By Theorem 1, the Fourier transform of $G_t P_1$ is a multiple of $G_{1/t} P_1$; in particular, it vanishes to order $d_1$ at the origin. But (as in Lemma 4, part (i)) the integral over ${\bf R}^n$ of $G_t P_1 P_2$ is some linear combination of partial derivatives of order $d_2$ of that Fourier transform. Since $d_1 \gt d_2$, these partials all vanish, and we are done. October 23: Some applications of the 7-design property for $E_8$ In these notes we use only the 7-design property for the first shell of the $E_8$ lattice. Similar counts can be made for the next few shells, though we get progressively less information because the number of possible inner products reaches the limits of the method more quickly. As Scott Kominers suggests, some of the counts for $\langle v_0, v_0 \rangle \gt 8$ can also be obtained from the “7½-design” property: while $\theta_{E_8,P}$ need not vanish when $P$ is a harmonic polynomial of degree 8, it does vanish for all harmonic polynomials $P$ of degree 10, because there are no nonzero cusp forms of weight 14 for $\Gamma$. This gives an additional linear relation on the $N_k$, namely $\sum_k N_k P_{10}(k/M) = 0$, where $M = (2 \langle v_0, v_0 \rangle)^{1/2}$, and $$P_{10}(x) = 7 - 560 x^2 + 6720 x^4 - 26880 x^6 + 42240 x^8 - 22528 x^{10},$$ a multiple of the Gegenbauer polynomial $C^{(3)}_{10}(x)$. Once $\langle v_0, v_0 \rangle > 8$, this condition is independent of the formulas for $\sum_k k^d N_k$ for $d=0,2,4,6$. (For $\langle v_0, v_0 \rangle = 8$, there is a linear dependence; if $\langle v_0, v_0 \rangle < 8$ there are too few $N_k$ for a consistent extra condition to be new.) Naturally we can push this a bit further yet by combining the “7½-design” property with the condition that each $N_k$ be a nonnegative integer; for example, all vectors with $\langle v_0, v_0 \rangle = 18$ that aren’t in $3E_8$ are equivalent, with $(N_0, N_1, N_2, N_3, N_4, N_5) = (42, 36, 33, 18, 10, 2)$. This notion of a “7½-design” was introduced by Venkov; likewise the shells of some other even self-dual lattices constitute 3½-, 7½-, or even 11½-designs. In most cases the the possible orthogonal complements $L = v_0^\perp \cap E_8$ can also be described by gluing back to an even unimodular lattice $L'$ of rank 16 (not 8, which would just bring us back where we started). For example, if $\langle v_0, v_0 \rangle = 10$ then $L$ is the orthogonal complement in $L'$ of some sublattice $A_9$; necessarily $L' \cong D_{16}^+$, and then all $A_9$'s are equivalent under ${\rm Aut}(D_{16}^+)$, so $L$ is identified uniquely (and indeed $R(L) \cong D_6$). For $\langle v_0, v_0 \rangle = 4, 6, 12, 18$ we can glue to $D_9$, $A_2 \oplus E_7$, $A_3 \oplus E_6$, $A_1 \oplus A_8$ respectively. In the first and last case $L'$ must be $D_{16}^+$ (since $D_9$ cannot fit in an $E_8$, nor can $A_8$ be a saturated sublattice of $E_8$); and in the other two cases, $L'$ must be $E_8^2$ (no $D_n$ lattice can contain an $E_{n'}$). We recover $L \cong D_7$, $A_1 \oplus E_6$, $A_2 \oplus D_5$ respectively. It still takes some more work to determine $N_k$ for $k \neq 0$. This alternative route is not available for studying analogous aspects of the Leech lattice, because the classification of even unimodular lattices of rank 32 is still well out of reach. Here’s the list of root numbers that determine a unique root lattice of rank at most $n$, for each $n \leq 8$ (and the gp code used to generate this list). The counts $N_k$ can also be interpreted as “quadrature rules” for the measure $(1-x^2)^{5/2}$ on $|x| \leq 1$; for example the counts for $\langle v_0, v_0 \rangle = 2$ yield $$\int_{-1}^1 f(x) \, (1-x^2)^{5/2} \, dx \doteq \frac\pi{768} \bigl[ 126 \, f(0) + 56( \,f(-1/2)+f(1/2)) + ( \,f(-1)+f(1)) \bigr],$$ with equality if $f$ is a polynomial of degree at most 7 (and we can't hope for more: consider $f(x) = (1-x^2) \prod_{k=-1}^1 (x-(k/2))^2$, which has degree 8). Similarly the familiar “Simpson’s Rule” $\int_{-1}^1 f(x) \, dx \doteq \frac16 [\, f(-1) + 4\, f(0) + f(1)]$ (exact for $\deg(\,f) \leq 3$) can be obtained by projecting to the $x$-axis the vertices of a regular octahedron, which constitute a 3-design of six points on the sphere in ${\bf R}^3$). For more along these lines, including higher-dimensional integration rules, see G. Kuperberg’s article Numerical cubature from Archimedes’ hat-box theorem, SIAM J. Numer. Anal. 44 (2006), 908-935 (arXiv:0405366). [“Quadrature” is an otherwise archaic synonym for “integration”, as in “squaring” the circle or other two-dimensional shape; “cubature” is sometimes used to mean the same thing for multiple integrals.] October 28 and 30: the Niemeier lattices These days Wikipedia even has a reasonably good summary and bibliography to start reading about the Niemeier lattices — and also a surprisingly detailed page on the mass formula for quadratic forms (note in particular the section with “Examples” for even self-dual lattices of ranks $8,16,24,32$). November 4 and 6: More about the Leech lattice; extremal theta functions and lattices, and spherical designs
Corrected Nov.13 to restore missing subscripts to several instances of ${\sf E}_4$ (noted by A. Blumer) If $L \subset {\bf R}^n$ is a unimodular lattice with minimal norm $N = N_{\min}(L)$ then the associated sphere packing uses balls of radius $\frac12\sqrt{N}$ and thus has density $(N/4)^{n/2} V_n$ where $V_n$ is the volume of a unit sphere in $n$-dimensional Euclidean space. It is well known that $V_n = \pi^{n/2} / \Gamma((n/2) + 1)$. [The late P.X.Gallagher was fond of the memorable equivalent formula $V_n = \pi^{n/2} / \Pi(n/2)$, where $\Pi(x) = \Gamma(x+1)$ is Euler’s original notation for the interpolation of $x!$.] So, for example, the packing densities of the sphere packings associated to the $E_8$ and Leech lattices are $\pi^4 / (2^4 4!) \doteq .2537$ and $\pi^{12} / 12! \doteq 1 / 518.249$. Now by Stirling’s approximation $V_n$ is within a factor $n^{\pm O(1)}$ of $(2 \pi e / n)^{n/2}$, whence the packing density is within a factor $N^{\pm O(1)}$ of $((\pi e N) / (2n))^{n/2}$. Now if $L$ is an extremal self-dual even lattice then $N = n/12 + O(1)$ so the packing density is $(\pi e / 24)^{n/2 + o(n)}$. Numerically, $\pi e / 24 \doteq 0.3558$ which is about $2^{-1.49}\!$, so the packing density is about $2^{-.745 n}$, safely below Rogers’s elementary upper bound of $2^{-.5 n + o(n)}$ and even the Kabatiansky-Levenshtein bound $2^{-.599\ldots\,n + o(n)}\!$, though well above the Minkowski(-Hlawka) lower bound $2^{-n+o(n)}$ for random lattices (which is still the best we have for large $n$ — the only improvements have been in the $2^{o(n)}$ factor!). But if $L$ is extremal among self-dual lattices that need not be even, then $N = n/8 + o(1)$ and the packing density is $(\pi e / 16)^{n/2 + o(1)} \gt 2^{-.453 n + o(n)}$, which is asymptotically above the Rogers bound, let alone Kabatiansky-Levenshtein; so it is not surprising that the last such lattice appears already at $n=24$. November 11: Statements of general modularity results for weighted theta functions; lattices of level N and their theta functions The happy cancellations going from (13) to (14) are not that miraculous: the special case $b=0$ is the consistency/sanity check between (11) and (12). I see now that it is not quite true that if $L$ has index $N$ then so does $L' = (L^*) \langle N \rangle$: the index might be strictly smaller; $L'$ might even have index $1$ (which means that $L = L'\langle N \rangle$ for some self-dual even $L'$). So we must require that $L$ is not $L_1\langle N_1 \rangle$ for some even lattice $L_1$ and integer $N_1 > 1$. If there do exist such $L_1$ and $N_1$ then we simply consider $L_1$. November 13, 18, and 20: Lattices of level 2 and their theta functions The “Arf” of “Arf invariant” is Cahit Arf (1910–1997); the lecture notes’ $Q_0$ and $Q_1$ are usually called quadratic forms of Arf invariant $0$ or $1$ respectively. Any $n$ and ${\rm disc}(L)$ allowed by Proposition 2 (with $1 \lt {\rm disc} \lt 2^n$) actually occurs for some lattice of level 2. For discriminant $2^{2k}$ with $1 \leq k \leq n/4$, use $D_4^{k-1} D_{n-4(k-1)}^{\phantom 0}$; for $k \gt n/4$, use $L'$ for $L$ of level 2 and discriminant $2^{(n/2)-2k}$. For extremal level 2 lattices $L \cong L'$, the asymptotic sphere-packing density is $(\pi e / 16\sqrt{2})^{n/2 + o(n)}$, a bit better than for even self-dual lattices (about $2^{-.703 n}$ compared with $2^{-.745n}$). Of course in both cases we have no evidence of the existence of such lattices past the first few candidate $n$, and we know that for large $n$ they cannot exist because the extremal theta functions have negative coefficients. According to Kenneth S. Williams’ “An arithmetic proof of Jacobi’s eight squares theorem” (Far East J. Math. Sci. 3(6), 1001–1005 (2001)), the formula $r_8(n) = 16 (-1)^n \sum_{d|n} (-1)^d d^3$ “first appeared implicitly in the work of Jacobi [...] and explicitly in the work of Eisenstein”, citing papers from 1829 and 1847 (sections 40–42 and page 501 of the respective collected works). As often happens in mathematical terminology, there are some alternative names and notations for Hadamard codes; if you see this term in the literature, be sure you know which convention the author(s) are using. The Nebe-Sloane catalogue of lattices lists three extremal “2-modular lattices” of rank 20, and cites the paper “Modular forms, lattices and spherical designs” by Bachoc and Venkov (in Reseaux euclidiens, “designs” spheriques et groupes (ed. J.Martinet), L’Enseignement Mathematique 37 (2001)) for a proof that there are no others. (I have now verified this result using my list of the $410$ level-2 lattices of rank 20 and discriminant $2^8$.) That paper also contains various results and tables of $N_k(v_0)$ along the lines of what we have been doing. [Warning: two of the three lattices are named for their symmetry groups, but “HS20” does not have an action of the sporadic Higman-Sims group: it is named for the initials of Boris Hemkemeier and Rudolf Scharlau, who found the lattice “by a random search in the [2-]neighborhood graph”. The Higman-Sims group does act on a 22-dimensional slice of $\Lambda_{24}$, but that’s a different lattice.] The Nebe-Sloane catalogue now has a table of enumerations of “extremal strongly modular lattices”, which links to a list of seven “2-modular lattices” of rank 32 and minimal norm 6. The main index lists five, the last of which has an automorphism of order 17 and “may be isomorphic” with the Mordell-Weil lattice of $y^2 + y = x^3 + t^{17}$ in characteristic $2$ or with its scaled dual. November 25: Lattices of level 3 and their theta functions We can also distinguish between the two classes of nondegenerate quadratic forms $Q$ on a vector space of dimension $d>0$ over a finite field $k$ of odd order $q$, as we did for quadratic forms over ${\bf Z} / 2{\bf Z}$, by the numbers of solutions of $Q(x) = c$ for each $c \in k$. When $d = 2r$, each nonzero $c$ appears $q^{2r-1} - q^{r-1}$ times for $Q_+$, and $q^{2r-1} + q^{r-1}$ times for $Q_-$; thus $Q(x) = 0$ has $(q^r-1)(q^{r-1}+1)$ nonzero solutions for $Q_+$, and $(q^r+1)(q^{r-1}-1)$ for $Q_-$. When $d$ is odd, there are $q^{d-1}$ solutions for $c=0$ (including $x=0$), and for $c\in k^*$ the count depends on $c \bmod {k^*}^2$. In the previous installment we noted that the vanishing of $\theta_{D_4}(z) = 1 + 24 q + 24 q^2 + \cdots$ at $z = (i-1)/2$ lets us estimate $e^\pi = -1/q(z)$: since $q$ is roughly $-1/24$, we can approximate $e^\pi$ by $24$, and the next term $24q^2$ in the $q$-expansion gives a closer estimate of $23$ (the numerical value is $23.14\ldots$). We can make similar use the vanishing of $\theta_{A_2}(z) = 1 + 6q + 6q^3 + \cdots$ at $z = (3^{-1/2}i - 1)/2$ to estimate $e^{\pi / \sqrt3}$: the first-order estimate is $6$, and the next estimate is $6 + \frac16$ because here the next term of the theta function involves $q^3$ rather than $q^2$. The numerical value of $e^{\pi/\sqrt3}$ is $6.1337\ldots$. For extremal level 3 lattices $L \in {\bf R}^n$ with ${\rm disc}(L) = 3^{n/2}$, the asymptotic sphere-packing density is $(\pi e / 12\sqrt{3})^{n/2 + o(n)}$, somewhat better than for level 2 (about $2^{-.642 n}$ compared with $2^{-.703n}$). Again we have no evidence of the existence of such lattices past the first few candidate $n$, and we know that for large $n$ they cannot exist because the extremal theta functions have negative coefficients. At least one such lattice exists in ${\bf R}^{24}$, but it is not as good as the Leech lattice: asymptotically $(\lfloor n/6 \rfloor + 2) / \sqrt 3 > \lfloor n/12 \rfloor + 2,$ but for $n=24$ (and even for $n=48$) the difference between the constant terms outweighs the difference between the leading terms. The one-dimensional space $S_7(\Gamma_1(3))$ is generated by the normalized cuspform $\frac16 \theta_{A_2}(z) \Delta_{(3)}(z)$, call it $f_7(z)$; since the first factor is $1 + 6 q + O(q^2)$, and the second is $(\eta(z)\eta(3z))^6 = q - 6q^2 + O(q^3)$, their product $f_7$ has a $q$-expansion that begins $q + O(q^3)$ with no $q^2$ term. This implies that the number of vectors of norm $4$ in a level-$3$ lattice $L \subset {\bf R}^{14}$ depends only on ${\rm disc}(L)$ (though the full theta function may differ among $L$ of the same discriminant — the only exception to this is the two lattices with ${\rm disc}(L) = 3$). Because $f_7$ is automatically a Hecke eigenform, it follows that its $q^m$ coefficients vanish for all $n \equiv 2 \bmod 4$. But in fact many more coefficients are zero: the $q$-expansion of $f_7$ begins $$q - 27 q^3 + 64 q^4 - 286 q^7 + 729 q^9 - 1728 q^{12} + 506 q^{13} + 4096 q^{16} - 10582 q^{19} + 7722 q^{21} + 15625 q^{25} - 19683 q^{27} + \cdots,$$ in which we recognize several coefficients as cubes and even sixth powers, and also recognize the exponents as the ones appearing in $\theta_{A_2}$. In fact $f_7$ is a weighted harmonic function $\theta_{A_2, P}$ with $\deg P = 6$, which we can write as $\frac16\sum_\alpha \alpha^6 q^{\alpha \bar\alpha}$, the sum running over all $\alpha \in {\bf Z} + {\bf Z} e^{2\pi i/3}$ (that is, over the Eisenstein integers; the coefficients are rational integers because the $\alpha$ and $\bar\alpha$ terms combine). The fact that $f_7$ is a “CM form” [CM = complex multiplication] lets us compute any $N_{2n}(L)$ in terms of the prime factorization of $n$, as we did for level-$3$ lattices of rank at most $10$ for which the theta functions could be expressed entirely in terms of Eisenstein series.
There is a similar phenomenon for $S_5(\Gamma_1(4))$, which is generated by a weighted theta function $\frac14 \sum_{a,b \in \bf Z} (a+bi)^4 q^{a^2+b^2}$ of the level-$4$ lattice $A_1^2$ — which makes it possible to give exact formulas for the number $r_{10}(n)$ of representations of an integer $n$ as a sum of $10$ squares. This was first observed by J.W.L. Glaisher in a 1907 paper “On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares” (Quart. J. Pure and Appl. Math. Oxford 38, 1-62), which led him to what we now call a “Ramanujan conjecture” for the coefficients of the (non-CM) modular form of weight $6$ that contributes to $r_{12}(n)$ for $n$ odd, several years before Ramanujan’s 1916 paper (Trans. Camb. Philos. Soc. XXII (9), 159-184).
P.S. The number of level-$3$ lattices of discriminant $3^r$ in ${\bf R}^{14}$ is 2, 8, 21, 29, 21, 8, 2 for $r=1,3,5,7,9,11,13$. We outline a Conway-style proof that the Coxeter-Todd lattice $K_{12}$ is the unique extremal level-$3$ lattice $L$ of rank $12$. The discriminant group ${\rm D}(L) = L^*/L$ is a $6$-dimensional vector space over ${\bf Z} / 3 {\bf Z}$ with a quadratic form $Q: [v] \mapsto (3\langle v,v \rangle) \bmod 3$ without isotropic spaces of dimension $3$; thus it has $(3^3+1) (3^2-1) = 224$ isotropic vectors, that is, $224$ nonzero cosets $c \in {\rm D}(L)$ for which $Q(c) = 0$. By our computation of $\theta_L$ (which here is the same as $\theta_{L'}$), the dual lattice $L^*$ has $4032 = 18 \cdot 224$ vectors $v$ with $\langle v,v \rangle = 2$, each of which maps to one of our $224$ isotropic $c$. Hence the average such $c$ has $18$ representatives of norm $2$. We claim that this requires each $c$ to have exactly $18$ short representatives, which form six pairwise orthogonal equilateral triangles. Indeed if $v,v'$ are distinct vectors of norm $2$ that map to the same $c$ then $v-v'$ is a nonzero vector in $L$, so $\langle v,v' \rangle$ is an integer that is either $0$ or $-1$, but not $-2$ because then $v' = -v$ which is impossible since $v \notin L$. Moreover, if $\langle v,v' \rangle = -1$ then $v'' := -(v+v')$ is also a vector of norm $2$ in the same coset, with $\langle v,v'' \rangle = \langle v',v'' \rangle = -1$. Thus $v,v',v''$ form an equilateral triangle, and any further norm-$2$ vector $w$ in the same coset must be orthogonal to $v,v',v''$ because $$\langle w,v \rangle + \langle w,v' \rangle + \langle w,v'' \rangle = \langle w,v+v'+v'' \rangle = \langle w,0 \rangle = 0$$ and each of $\langle w,v \rangle, \langle w,v' \rangle, \langle w,v'' \rangle$ is $\leq 0$. Therefore there are at most $18$ norm-$2$ vectors in the coset, with equality if and only if they form six pairwise orthogonal equilateral triangles. Now these triangles span a lattice $A_2^6 \subset L^*$, and their pairwise differences span a sublattice $(A_2^6)_0^{\phantom.}$ of $L$ of minimal norm 4 with $[A_2^6 : (A_2^6)_0^{\phantom.}] = 3$. Hence also $[L : (A_2^6)_0^{\phantom.}] = 3$ (by comparing discriminants), which soon identifies $L$ with $K_{12}$.

These days it takes only a few seconds to compute the $2$-neighbor graph of level-$3$ lattices of rank $12$ and discriminant $3^6$; this finds a total of $10$ such lattices, and confirms that $K_{12}$ is the only extremal one. As with the Barnes-Wall lattice $BW_{16} = \Lambda_{16}$, those $10$ lattices were already determined by Scharlau and Venkov (1995) by adapting Venkov’s proof of the Niemeier classification. My analytic number theory notes include some further information about Dirichlet characters (see pages 4-5, and also page 10, exercises 7-8 for the notion of a primitive character) and Gauss sums (see pages 3-4 and 6). Of course there are many other sources. For example, if you’ve read Serre’s A Course in Arithmetic then you’ve already encountered Dirichlet characters in Chapter VI. December 2: Shifted theta functions $\theta_{L+v_0,P}$ with $v_0 \in L^*$, and their modularity for $\mathop{\rm disc} L \leq 5$
Corrected Dec.3: twice (before (12)) $\bigl({1\;N\atop 0\;\,1}\bigr)$ appeared as $\bigl({1\;\,1\atop 0\;N}\bigr)$ (noted by A. Blumer) For some discriminant groups (such as ${\bf Z} / p {\bf Z}$ with $p$ prime), the matrices $\rho(g)$ for $g = \bigl({1\;1\atop0\;1}\bigr)$ and $g = \bigl({0\;-1\atop1\;\phantom-0}\bigr)$ will also appear in Wednesday’s lecture as linear transformations that preserve the complete weight numerator of a self-dual linear code. The finite Heisenberg group also arises naturally in that setting when the code contains the all-$1$'s word. Using the 4-design property of the $27$ minimal vectors in a nontrivial coset of $E_6$ in $E_6^*$, we can for instance compute that if $v$ is one such vector then as $v'$ ranges over those $27$ vectors the possible values $4/3, 1/3, -2/3$ of the inner product $\langle v,v' \rangle$ occur with multiplicities $1, 16, 10$. This reflects the configuration of 27 lines on a smooth cubic surface over an algebraically closed field: each line meets $10$ of the others, and is skew to the remaining $16$. The evaluation of the Gauss sum for the quadratic character $\chi_5$ is equivalent to the familiar(?) evaluation of $\cos(2\pi / 5)$ (a.k.a. “$\sin 18^\circ$”) as $(\sqrt{5} - 1)/4$, which in turn yields the Euclidean construction of the regular pentagon. [Alternatively one can use $\sin 54^\circ = (\sqrt{5}+1)/4$, which is also equivalent with $\tau_1(\chi_5) = \sqrt{5}$.] December 4: Linear codes and their weight enumerators, and some connections with lattices and their theta functions, roughly following “Lattices, Linear Codes, and Invariants, Part II” (Notices of the American Math. Society 47 (2000), 1382-1391).

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