--- title: Lattice Theory aliases: - Lattice Theory flashcard: Research::Lattices created: 2023-01-28T21:01 updated: 2024-01-29T22:53 --- # Lattice Theory ## Basic Definitions - [x] What is the signature of a lattice? ✅ 2023-01-17 - [ ] The signature of the associated bilinear form $\beta$. - [ ] Change variables to write $\beta(x) = \sum \lambda_i x_i^2$ where $\lambda_i \in \ts{0, \pm 1}$, then the signature is $(n_+, n_-)$, the counts of how many times $+1$ and $-1$ appear respectively. - [x] What is a positive/negative definite form? A nondegenerate form? ✅ 2023-01-17 - [ ] Positive: $\lambda_i = +1$ for all $i$. - [ ] Negative: $\lambda_i = -1$ for all $i$. - [ ] Nondegenerate: $\lambda_i \neq 0$ for all $i$. - [x] What is an integral lattice? ✅ 2023-01-17 - [ ] $\inp{x}{y}\in \ZZ$ for all $x,y\in L$. - [x] What is a unimodular lattice? ✅ 2023-01-17 - [ ] Integral and generated by columns of a matrix $M$ with $\det M = \pm 1$. - [ ] Volume of fundamental domain is 1. - [x] Discuss dimensions of unimodular lattices. ✅ 2023-01-17 - [ ] Only possible in dimensions divisible by 8 - [ ] $\dim = 8\implies E_8$ - [ ] $\dim = 16 \implies \Gamma_8\sumpower 2, \Gamma_{16}$. - [ ] $\dim = 32 \implies N_i, i\leq 24$ the Niemeir lattices, including the Leech lattices. - [x] What is an even lattice? ✅ 2023-01-17 - [ ] $\norm{x}_L \in 2\ZZ$ for any $x\in L$. - [x] What are type I/type II lattices? - [ ] Type II $\iff$ even - [ ] Type I $\iff$ not even (odd). - [x] What is the $E_8$ lattice? ✅ 2023-01-17 - [ ] The unique positive-definite, even, unimodular lattice of rank 8. - [ ] Points in $\ZZ^8$ whose coordinates are either all integers or all half-integers and the sum of coordinates is even. - [ ] Root lattice of $E_8$ ![](attachments/2023-01-17-E8root.png) $$\left(\begin{array}{cccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2\end{array}\right)$$ - [ ] - [x] What is a lattice? ✅ 2023-01-07 - [ ] A lattice is a free abelian group with an $\ZZ\dash$valued symmetric bilinear form. - [x] What is a root in a lattice? ✅ 2023-01-07 - [ ] A vector $v\in \Lambda$ with $v^2 = -2$. - [x] What is the Weyl group of a lattice? ✅ 2023-01-07 - [ ] $W(\Lambda)\normal \Orth(\Lambda)$ generated by reflections $v\mapsto v + (v, \delta)\delta$ for $\delta$ the roots of $\Lambda$. - [x] What is the hyperbolic lattice $H$? What is its bilinear form? ✅ 2023-02-07 - [ ] The matrix of the quadratic form is $\matt 0 1 1 0$; so a basis with two isotropic elements. - [ ] Nondegenerate since $\disc H = -1 \neq 0$. - [ ] $Q(x,y)= xy$, equivalently to $Q(x, y) = x^2-y^2$. - [x] What is the definition of a lattice in $V \in \mods{\RR}$? ✅ 2023-01-28 - [ ] $\Lambda \subseteq V$ a discrete cocompact subgroup, or a free $\ZZ\dash$submodule of $\GG_a(V)$. - [ ] Usually equipped with $\beta: \Sym^2 V\to R$. - [x] What is the dual of a lattice? ✅ 2023-02-07 - [ ] $L\dual \da \zmod(L\to \ZZ) \cong \ts{q\in L_\QQ \st (q, \Lambda) \subseteq \ZZ}$. - [x] What is $\len \Lambda$? ✅ 2023-02-07 - [ ] The minimal number of generators of $D_\Lambda$. - [x] What is a $p\dash$elementary lattice? ✅ 2023-02-07 - [ ] $D_\Lambda \cong C_p\sumpower{k}$ for some $k$; $\implies \len \Lambda = k$. - [x] What is a quadratic form on $V\in \rmod$? ✅ 2023-02-07 - [ ] $Q: V\to R$ where $Q(rx) = r^2 Q(x)$ and $\beta(x,y) \da Q(x+y) - Q(x) - Q(y)$ defines $\beta:\Sym^2(V)\to R$. - [ ] Generalizes the polarization identity $$2\inp{x}{y} = \inp{x+y}{x+y} - \inp{x}{x} -\inp{y}{y}$$. - [x] What is the Gram matrix of a bilinear form? ✅ 2023-01-28 - [ ] $M_{ij} = \beta(e_i, e_j)$ where $\gens{e_i}_\ZZ = \Lambda$ for a choice of $\ZZ\dash$basis. - [x] What is the hyperbolic module? ✅ 2023-02-07 - [ ] $H(M) \da M \oplus M\dual$ with quadratic form $Q(v+v\dual) \da v\dual(v)$. - [ ] Yields $Q(\sum x_i e_i + \sum y_i e_i\dual) = \sum x_i y_i$, so for $n=1$ this yields $Q(x e + ye\dual) = xy$. - [x] What is the reflection formula for a quadratic module? ✅ 2023-02-07 - [ ] $$\tau_z(x) \da x - {\beta(x,z) \over Q(x)}\cdot z$$ - [x] What is Witt's theorem on quadratic modules? ✅ 2023-02-07 - [ ] If $(V, Q)\in \rmod$ is a quadratic space, then $\Orth(V) = \gens{\tau_z \st z\in V}_R$ and in fact every element is a product of at most $\dim V$ reflections. - [x] What is a nondegenerate pairing on $\Lambda$? ✅ 2023-01-28 - [ ] The linear extension of $\beta$ to $V = \Lambda\tensor_\ZZ \RR$ induces an isomorphism $V\iso V\dual$. - [ ] Equivalently $\disc \Lambda \neq 0$. - [x] What is $\disc(\Lambda)$ for a lattice $(\Lambda, \beta)$? ✅ 2023-01-28 - [ ] $\disc(\Lambda) \da \det G(\beta)$ where $G(\beta)$ is the Gram matrix. - [ ] Interpret as $\disc(\Lambda) = \covol(\Lambda)^2 \da \vol(V/\Lambda)^2$. - [ ] When integral, $\disc(\Lambda ) = \size(\Lambda\dual/\Lambda)$. - [ ] Can compute by taking $M$ a generator matrix for $\Lambda$, then $\disc \Lambda = M^tM = (\det M)^2$. - [x] What is a unimodular lattice? ✅ 2023-01-28 - [ ] Integral with $\disc \Lambda = +1$. - [ ] Equivalently, $\covol(\Lambda) \da \vol(V/\Lambda) = 1$. - [ ] Equivalently, $\Lambda = \Lambda\dual$. - [ ] Equivalently $D_\Lambda = \ts{0}$. - [ ] More generally, $I\dash$modular if $\Lambda = I\Lambda\dual$. - [x] Give a consequence of having a unimodular lattice. ✅ 2023-02-07 - [ ] If $M\leq \Lambda$ is unimodular then there is an isometry $\Lambda \iso M\oplus M^\perp$. - [x] What is an overlattice? Give an example. ✅ 2023-02-07 - [ ] $\Lambda$ is an overlattice of $M$ iff $M\leq \Lambda$ is a sublattice and $[\Lambda: M] < \infty$, so $\rank_\ZZ \Lambda = \rank_\ZZ M$. - [ ] $\Lambda\dual$ is always an overlattice of $\Lambda$, - [ ] $\Lambda$ is always an overlattice of $M\oplus M^\perp$ for any $M\leq \Lambda$. - [x] What is an even unimodular lattices? ✅ 2023-02-07 - [ ] $\Lambda = \Lambda\dual$ and $Q(\Lambda) \subseteq R$, or equivalently $\beta(x,x)\in 2R$ for all $x\in \Lambda$. - [ ] Equivalently the Gram matrix is in $\GL_n(\ZZ)$ with even diagonal entries. - [x] Give example computations of Witt groups. ✅ 2023-02-07 - [ ] $W(\CC) = C_2$. - [ ] $W(\RR) = \ZZ$ - [x] What are Type I and Type II lattices? ✅ 2023-01-28 - [ ] Type II: even unimodular. - [ ] Type I: other ("odd") unimodular. - [x] What are the rank, signature, and discriminant of $U$? ✅ 2023-01-28 - [ ] $\rank U = 2, \sgn U = (1,1), \disc U = 1$. - [x] What are the rank, signature, and discriminant of $E_8$? ✅ 2023-01-28 - [ ] $\rank E_8 = 8, \sgn E_8 = (8, 0), \disc E_8 = 1$. - [x] What is the twist $\Lambda(n)$ of a lattice $\Lambda$? ✅ 2023-01-28 - [ ] $\beta_{\Lambda(n)}(x,y) = n\beta_\Lambda(x, y)$. - [x] What is the signature of a discriminant form $q_\Lambda$? ✅ 2023-01-28 - [ ] If $\sgn \Lambda = (\ell_+, \ell_-)$ then $\sgn q_\Lambda = \ell_+ - \ell_-$. - [x] What is the theta function of a lattice? ✅ 2023-01-28 - [ ] $$\theta_{\Lambda}(z)=\sum_{x \in \Lambda} q^{x^2/2}, \qquad q = \exp(2\pi i z), \,\,z\in \HH$$ - [ ] Coefficients count the number of vectors of a fixed norm. - [x] What is a 2-elementary lattice? ✅ 2023-01-28 - [ ] $\Lambda$ with discriminant group $D_\Lambda\cong C_2^r$ for some $r$. - [x] What is $\ell(D_\Lambda)$? ✅ 2023-02-01 - [ ] The minimal number of generators. - [x] What is a primitive sublattice? ✅ 2023-01-28 - [ ] $R \leq \Lambda$ a co-torsionfree sublattice, ie $\Lambda/R$ is torsionfree. - [ ] Yields a split SES $R\injects \Lambda \surjects \Lambda/R$ so $\Lambda \cong R \oplus \Lambda/R$, not necessarily orthogonal wrt $\beta$. - [ ] Examples: any $R \leq \Lambda$ of the form $R = S^\perp$ for some $S\leq \Lambda$ is always primitive. - [ ] What is the importance of primitive sublattices? - [ ] If $H \leq \Lambda$ is primitive, then $H$ admits a complementary sublattice so $H \oplus H^\perp = \Lambda$. - [x] What is a primitive element of a lattice? ✅ 2023-02-07 - [ ] $x\in \Lambda$ where $\gens{x}_\ZZ$ is a primitive sublattice. - [ ] Corresponds to $x$ being the closest element in $\gens{x}_\QQ \intersect \Lambda$ to the origin. - [ ] Equivalently, $v$ is primitive iff ${1\over n} v\not \in \Lambda$ for any $n$. - [x] What is a root lattice? ✅ 2023-02-07 - [ ] A sublattice $\Lambda' \leq \Lambda$ generated by all $-2$ vectors. - [x] What is a lattice isometry? ✅ 2023-01-30 - [ ] A map $f: (\Lambda_1, \beta_1) \to (\Lambda_2, \beta_2)$ where $\beta_2(f(x), f(y)) = \beta_1(x,y)$. - [ ] Forms a group $\Orth(\Lambda)$. - [ ] What is a primitive element of a lattice? - [ ] What is the quadratic form associated to a discriminant form? - [x] What is a root in a lattice? ✅ 2023-02-07 - [ ] A nonzero primitive $x\in \Lambda$ where the reflection $s_x: \Lambda_\QQ \selfmap$ satisfies $s_x(\Lambda) = \Lambda$. - [ ] Reflection formula: $$s_x(y) = y - {\beta(x,y) \over \beta(x,x)}x$$ - [ ] If $\Lambda$ is even, it suffices to check $\beta(x,x) = -2$ for all $x$. - [x] Discuss roots in an even unimodular lattice. ✅ 2023-02-07 - [ ] Could generally have $\beta(x,x) = -n$, but even and unimodular forces $n=2$ for every $x$. - [x] What is the Weyl group $W(\Lambda)$ of a lattice? What is $W^{(2)}(\Lambda)$? ✅ 2023-03-27 - [ ] $W(\Lambda) \da \gens{s_\alpha\st \alpha\in \Lambda} \leq \Orth(\Lambda)$ the subgroup generated by reflections? - [ ] $W^{(2)}(\Lambda) = \gens{s_\alpha \st \alpha\in \Lambda, \alpha^2 = -2}$. - [x] What is a fundamental domain for $W(\Lambda)$ in $\Lambda_\RR$? ✅ 2023-03-27 - [ ] $$\Lambda_\RR \sm\Union_{\delta \in W^{(2)}(\Lambda) }\delta^\perp$$ - [x] What is the discriminant group? Why care? ✅ 2023-02-01 - [ ] $D_\Lambda \da \Lambda\dual/\Lambda$. - [ ] $\Lambda$ not unimodular $\implies D_\Lambda \neq 0$. - [x] What is the discriminant form? ✅ 2023-02-01 - [ ] For an even lattice, $$\begin{align*}q_\Lambda: D_\Lambda &\to \QQ/2\ZZ \\ x + \Lambda &\mapsto \beta(x,x) +2\ZZ\end{align*}$$ - [ ] Obtained by extending $\beta$ to $\Lambda\dual$ and then mapping to the quotient. - [x] What is the bilinear form associated to the discriminant form $q_\Lambda$? ✅ 2023-02-07 - [ ] $$\begin{align*}\beta_\Lambda: \Sym^2 D_\Lambda &\to \QQ/\ZZ \\x\tensor y&\mapsto {1\over 2}\qty{ q_{\Lambda}(x+y,x+y) - q_{\Lambda}(x,x)-q_{\Lambda}(y,y)} \\ \\ \implies q_\Lambda(x) &= \beta_\Lambda(x,x)\end{align*}$$ - [x] What is an isotropic element of a lattice $V$? An isotropic subspace? ✅ 2023-02-07 - [ ] $x\in V$ with $Q(x) = 0$. - [ ] Subspaces $U$: $U \subseteq U^\perp \iff\ro{Q}{U} = 0$. - [x] What is the radical of a bilinear form? ✅ 2023-02-07 - [ ] For $(V, \beta)$, set $\Rad V \da V^\perp \da\ts{x\in V \st \beta(x,v) = 0\, \forall v\in V} = \Intersect_{x\in V}$. - [x] What is a nondegenerate quadratic module? ✅ 2023-02-07 - [ ] $(V, \beta)$ with $\Rad \beta = 0$ - [ ] Equivalently, $\disc Q \neq 0 \in k\units/\squares{k\units}$. - [ ] Equivalently, $q_V: V\iso V\dual$ where $q_V(x) \da \beta(x, \wait)$. - [x] What is an integral lattice? What is a common mitake here? ✅ 2023-02-07 - [ ] $\im \beta \subseteq \ZZ$. - [ ] Equivalently $\Lambda \subseteq \Lambda\dual$. - [ ] Not sufficient for $\im Q \subseteq \ZZ$, e.g. $Q = \matt{2}{-1}{-1}{2}$ has $Q(\vector x) = x_1^2 -x_1 x_2 + x_2^2$ so $Q(\ZZ^2) \subseteq \ZZ$ but $\beta((1,0), (0, 1)) = -1/2\not\in \ZZ$. - [x] What are $W^{(2)}$ and $W^r$? ✅ 2023-02-07 - [ ] $\subseteq \Orth(\Lambda)$ generated by reflections about $(-2)\dash$vectors, and by all vectors respectively. - [x] What are 2-reflective and reflective lattices? ✅ 2023-02-07 - [ ] $[\Orth(\Lambda): W_2]<\infty$ and $[\Orth(\Lambda): W_r]<\infty$ - [ ] Equivalently, the fundamental polyhedra have finite (usually hyperbolic) volume. - [x] What is the coparity invariant $\delta$? ✅ 2023-02-07 - [ ] For $\Lambda$ a 2-elementary lattice, $\delta = 0 \iff q_M(D_\Lambda) \subseteq \ZZ/2\ZZ \subset \QQ/2\ZZ$ and 1 otherwise. - [x] What is a hyperbolic lattice? ✅ 2023-02-07 - [ ] $\sgn \Lambda = (1, m)$ for any $m$. - [ ] Describe the $A_n, D_n, E_n$ lattices. - [x] When are two lattices in the same genus? ✅ 2023-02-07 - [ ] $\sgn L_1 = \sgn L_2$ and their $p\dash$adic completions $(L_i)_p \da L_i \tensor_\ZZ \ZZpadic$ are isomorphic for all primes as $\ZZpadic\dash$bilinear forms. - [ ] Genus determines $L_\QQ$ uniquely, and each genus has only finitely many iso classes. - [x] What is the index of a lattice? ✅ 2023-02-07 - [ ] $\ind \Lambda = n_+ - n_- \in C_8$. - [x] What is the exponent of a lattice? ✅ 2023-03-06 - [ ] The exponent of $D_\Lambda$: the minimal $n$ such that $n \Lambda\dual \subseteq \Lambda$. - [x] What is the level of a lattice? ✅ 2023-03-06 - [ ] For even lattices: the smallest $n$ such that $\Lambda\dual(n)$ is again even. - [ ] Always have level = $c\cdot \exp\Lambda$ where $c= 1,2$. - [ ] The smallest $n$ such that $n\cdot \inp{v}{v}\in 2\ZZ$ for all $v\in \Lambda$. - [x] When are lattice similar? ✅ 2023-03-06 - [ ] $\sigma: \Lambda_1 \to \Lambda_2$ with $\beta_{\Lambda_2}(\sigma v_1, \sigma v_2) = c \beta_{\Lambda_1}(v_1, v_2)$ for some $c$. - [x] When do two lattices have the same genus? ✅ 2023-03-06 - [ ] If $L_p \cong M_p$ for all primes $p$ and $\infty$. - [ ] Equivalently: equivalence classes under "isometry under all localizations". - [x] What is the theta series of an even lattice of dimension $n=2k$ and level $\ell$? ✅ 2023-03-06 - [ ] $$\theta_\Lambda(q)\da \sum_{m\in \ZZ_{\geq 0}} f_\Lambda(m) q^m,\qquad f_\Lambda(m) \da \size\ts{x\in \Lambda \st \beta(x,x) = 2m}$$ - [ ] Modular form of weight $k$ for $\Gamma_0(\ell)$ and a quadratic character $\varepsilon: \Gamma_0(\ell) \to \mu_2$ - [ ] Alternatively, $\theta_\Lambda(q) = \sum_{x\in \Lambda} 1\cdot q^{{1\over 2}\beta(x,x)}$. - [x] What is the genus of a lattice? ✅ 2023-03-27 - [ ] All lattices with the same signature and discriminant form up to isomorphism. - [ ] Notation: $g(M) \da \rm{II}_{p, q}(D_M)$. - [x] What is a reflective modular form? ✅ 2023-03-27 - [ ] Modular form: $f\in \Hol(\Aff\Omega_{\Lambda}, \CC)$ where $f(tz) = t^{-k}f(z)$ for $t\in \CC\units$ and $f(gz) = \chi(g) f(z)$ for $g\in \Gamma \leq \Orth^+(\Lambda)$ where $\Omega_{\Lambda}\da \ts{z\in \PP(\Lambda\tensor \CC)\st z^2 = 0, \norm{z} < 0}$ and $\Aff$ is its affine cone. - [ ] $f$ is reflective iff zeros of $f$ are contained in $\union \gamma^\perp$ for $\gamma$ roots of $\Lambda$. - [x] What is an overlattice? Why care? ✅ 2024-01-29 - [ ] $M$ is an overlattice of $L$ iff $M\contains L$ and $[M: L] < \infty \in \zmod$. - [ ] $\sgn M = \sgn L$ for any $M$ over $L$ - [ ] Overlattices of $L$ biject with subgroups of $D_L$. - [x] What is $\operatorname{Iso}\left(L_1, L_2\right)$ ✅ 2023-03-27 - [ ] The set of isomorphisms between lattices $L_1$ and $L_2$ compatible with the quadratic forms of $L_1$ and $L_2$. - [x] What is the genus of a lattice? ✅ 2023-03-27 - [ ] A lattice $L^{\prime}$ is said to belong to the same genus as $L$ if the quadratic forms $L^{\prime} \otimes \mathbb{Z}_p \cong L \otimes \mathbb{Z}_p$ for all primes $p$ and $L^{\prime} \otimes \mathbb{R} \cong L \otimes \mathbb{R}$. - [ ] Equivalently, $D_L\iso D_{L'}$ as quadratic spaces. - [x] What is the class of a lattice? ✅ 2023-03-27 - [ ] If Iso $\left(L, L^{\prime}\right)$ is non-trivial then $L$ and $L^{\prime}$ are said to belong to the same class. We let $\operatorname{gen}(L)$ denote the set of classes in the same genus as $L$. - [x] What is the Witt index of a lattice? ✅ 2023-03-27 - [ ] The (real) Witt index of $L$ is defined as the maximal dimension of a totally isotropic subspace of $L \otimes \mathbb{R}$ - [x] What is the $p\dash$rank of a lattice? ✅ 2023-03-27 - [ ] For prime $p$, the $p$-rank $\operatorname{rank}_p(L)$ of $L$ is the maximal rank of all sublattices $S \subset L$ such that $\operatorname{det}(S)$ is coprime to $p$. - [x] What is the discriminant of a lattice? ✅ 2023-03-27 - [ ] $\size D_L$. - [x] What is the saturation of an embedding $\iota: L\injects M$? ✅ 2023-03-27 - [ ] The smallest primitive $M' \leq M$ containing $\iota(L)$. - [ ] $\operatorname{Sat}(L):=\left\{y \in L^{\prime} \mid n y \in L \text { for some positive integer } n\right\}$ - [x] What is an isotropic lattice? ✅ 2023-03-27 - [ ] Contains at least one isotropic vector. - [x] What is a spacelike vector? Lightlike? ✅ 2023-03-27 - [ ] Spacelike: $\norm{v} > 0$ or $\norm{v} \lt 0$. - [ ] Lightlike: $v^2 = 0$ (isotropic) with $v\neq 0$; two connected components of such (future/past). ## Specialized Definitions - [x] What is the divisibility of a vector? ✅ 2023-03-28 - [ ] For $v\in T$, $\div(v) = n$ where $v.T = n\ZZ$. - [ ] If $0 \neq x \in L$, we let $\operatorname{div}(x)$ denote the positive generator of the ideal $(x, L)$ - [x] What is $x^*$? How is it related to $\div(x)$? ✅ 2023-03-27 - [ ] Define $x^*:=x / \operatorname{div}(x) \in L^{\vee}$. - [ ] In our case: $v\in T\implies v^* \in A_T$. - [ ] $v^* = 0\iff \div(v) = 1$. - [x] What is a characteristic vector? ✅ 2023-03-28 - [ ] For $H$ 2-elementary, $v^* \in A_H$ where $q(x) = \inp{x}{v^*} \mod \ZZ$ for all $x\in A_H$. - [x] What is an ordinary vector? ✅ 2023-03-28 - [ ] Non-characteristic. - [x] What is an odd/simple primitive isotropic vector $e\in H$? ✅ 2023-03-28 - [ ] $\div(e) = 1$ - [x] What is an even ordinary primitive isotropic vector $e\in H$? ✅ 2023-03-28 - [ ] $\div(e) = 2, e^*$ ordinary. - [x] What is an even characteristic primitive isotropic vector $e\in H$? ✅ 2023-03-28 - [ ] $\div(e) = 2, e^*$ characteristic. - [x] What is the scale of a lattice? ✅ 2023-03-28 - [ ] $s(L) = \inp{L}{L}$. - [x] What is the norm of a lattice? ✅ 2023-03-28 - [ ] $\mfn(L)$ is the fractional ideal generated by $\ts{\inp x x \st x\in L}$. - [x] What is the spinor norm of a reflection $\tau_v$? ✅ 2023-03-28 - [ ] $\tau_v(x) = x - 2xv/v^2$ has spinor norm $\norm{v}_\Spin \da Q(v) = v^2/2\in k\units/(k\units)^\square$. - [x] What is an $R\dash$lattice? ✅ 2023-03-28 - [ ] $L\in \rmod^{\fg, \proj}$ with a nondegenerate symmetric bilinear form $\inp{\wait}\wait\in \Sym^2(L\dual)$. - [x] What is a Hermitian space? ✅ 2023-03-28 - [ ] A pair $(V, h)$ where.. - [ ] $E$ is an etale $\QQ\dash$algebra with a $\QQ\dash$linear involution $\bar{(\wait)}: E\selfmap$, - [ ] $V\in \mods{E}$, - [ ] $h: V\times V\to E$ is a sesquilinear form with $h(a,b) = \bar{h(b, a)}$. - [x] What is a $E\dash$bilinear form on $V\in \mods{E}$? ✅ 2023-03-28 - [ ] A $\QQ\dash$bilinear form $b$ with $b(ev, w) = b(v, \bar e w)$. - [ ] Example: trace forms of Hermitian forms. - [x] What is the trace form of $h$ a Hermitian form? ✅ 2023-03-28 - [ ] $b \da \Trace_{E/\QQ} \circ h$. - [x] What is the minimum of a lattice? ✅ 2023-03-28 - [ ] $m(L) \da \min_{x\in L\smz} x^2$. - [x] What is the Hermite constant of a lattice? Why care? ✅ 2023-03-28 - [ ] For $n\da \rank_\ZZ L$, $\gamma(L) \da m(L) / \det L^{1\over n}$. - [ ] Measures density of sphere packing associated to $L$. - [x] What is the kissing number? ✅ 2023-03-28 - [ ] $2s = \size S(L)$ where $S(L) \da \ts{x\in L \st x^2 = m(L)}$, the set of minimal vectors. - [x] What is a modular lattice of level $\ell$? Why care? ✅ 2024-01-29 - [ ] $\rank_\ZZ L = n$ with $\sigma: L\to L\dual$ satisfying $\sigma(v^2) = \ell v^2$. - [ ] $\theta_L(q)$ is a modular form of weight $n$ wrt a Fricke group of level $\ell$: $H(\ell)\leq \SL_2(\RR)$ with $[H(\ell): \Gamma_0(\ell)] = 2$. ## Classification - [x] Discuss classification of lattices. ✅ 2023-03-27 - [ ] Over algebraic number fields: classified by Hasse-Minkowski - [ ] Unimodular, indefinite, integral: classified by Milnor. - [ ] Odd: $\Lambda \cong \gens{1}^{n_+} \oplus \gens{-1}^{n_-}$. - [ ] Even: Exist $\iff \sgn \Lambda \equiv 0 \mod 8$; if indefinite, classified by $\sgn \Lambda$. - [ ] Indefinite, integral: Nikulin, using discriminant forms. - [ ] Even: - [ ] Indefinite, unimodular, nondegenerate: determined by $\sgn \Lambda = (n_+, n_-)$ - [ ] (Negative) definite ($n_+ = 0$), unimodular, nondegenerate: Only in ranks $r\equiv 0 \mod 8$, not unique for $r\neq 8$. - [ ] Indefinite, $\len \Lambda \leq\rank \Lambda - 2$: determined by $(\sgn \Lambda, q_\Lambda)$. - [x] Discuss classification of 2-elementary lattices. ✅ 2023-02-07 - [ ] Indefinite 2-elementary lattices classified by $(r, a,\delta)$: - [ ] Rank, $\ZZ/2\ZZ\dash$rank, and coparity. - [ ] Alternatively: $(g,k,\delta)$ where $g= 11 - {r+a\over 2}$ and $k={r-a\over 2}$; $\Fix(\iota)$ has $k+1$ connected components of genera summing to $g$. - [ ] Nikulin's classifications: 75 K3 lattices - [ ] K3 = hyperbolic, admitting a primitive embedding into $\Lambda_{\K3}$. - [x] What is Nikulin's theorem on lattices? ✅ 2023-01-28 - [ ] For $\Lambda$ even, negative definite, $\rank_\ZZ(\Lambda) \geq \ell(D_\Lambda) + 2$: - [ ] $\Lambda$ is determined up to isometry by its rank, signature, and discriminant form. - [x] Discuss classification of 2-modular lattices? ✅ 2023-01-28 - [ ] If 2-modular and indefinite, uniquely determined by rank, signature, $\ell(A_\Lambda)$, and $\Delta$ where $\Delta = 0 \iff q_\Lambda(\Lambda) \subseteq \ts{0, 1}$ and 1 otherwise. - [x] Give a classification of indefinite unimodular integral lattices. ✅ 2023-01-30 - [ ] ![](attachments/2023-01-30-lattice.png) - [x] Discuss classification of **even**, **unimodular** , **indefinite** integral lattices: ✅ 2023-01-31 - [ ] For signature $(p, q)$, exist if $p\equiv q \mod 8$, uniquely determined by rank and signature. - [ ] Given by $$\Pi_{p, q} \da U\sumpower{p} \oplus E_8 \sumpower{m}, \qquad p-q=-8m$$ or $$\Pi_{p, q} = U\sumpower{q}\oplus E_8(-1)\sumpower{m}, \qquad p-q = +8m$$ - [x] Discuss classification of **even**, **unimodular**, **definite** integral lattices ✅ 2023-02-01 - [ ] Definite means losing uniqueness, compared to indefinite. - [ ] $\dim = 8: E_8$ - [ ] $\dim = 16: E_8\sumpower{2}, \Lambda_{16}$. - [ ] $\dim = 24$: 24 Neimeier lattices, - [ ] Including the Leech lattice: uniquely has no roots. - [ ] $\dim = 32$: billions. - [ ] $\dim = 40: > 10^{80}$?? ## Results - [x] Discuss consequences of having a nondegenerate pairing. ✅ 2023-01-28 - [ ] Induces $V\iso V\dual$, so $\Lambda\dual, \Lambda \subset V$ can be identified as lattices in the same space $V\da \Lambda\tensor_\ZZ \RR$. - [ ] If $\Lambda$ is integral then $\Lambda\dual \contains \Lambda$ and $[\Lambda\dual : \Lambda] = \abs{\disc \Lambda}$. - [x] Describe ADE dynkin diagrams. ✅ 2023-01-28 - [ ] ![](attachments/2023-01-28-ade-dynkin.png) - [x] Discuss the lattice structure on $\Pic(F_4)$ ✅ 2023-01-28 - [ ] Rank 20, Signature $(1, 19)$, Discriminant -64. - [ ] Generated by the 48 lines it contains. - [x] Describe the Leech lattice. ✅ 2023-01-28 - [ ] There exists a unique even unimodular lattice **without roots** in $\mathbf{R}^{24}$. - [ ] It has 196560 vectors of weight 4. - [ ] $\Lambda = \Lambda\dual$ and $\det\Lambda = 1$ so $\Theta_\Lambda(z)$ is a modular form of weight 12. - [ ] The unique even unimodular negative definite lattice with no elements of square $-2$ and $\operatorname{Aut}(\Lambda) / \pm 1=$ $C o_1$ - [ ] Conway's group $\Orth(N)$ the isometries of the Leech lattice, contains the Mathieu group $M_{23}$ - [x] Discuss Neimeier lattices. ✅ 2023-01-28 - [ ] Neimeir classifies definite even unimodular lattices of rank 24. - [ ] All obtained as primitive sublattices of $\Pi_{1, 25} =U \oplus E_8(-1)\sumpower 3$ by specifying a primitive isotropic vector. - [x] What are Conway's groups $C_{o_0}$ and $C_{o_1}$? ✅ 2023-01-28 - [ ] $C_{o_0} = \Aut(\Lambda_{\mathrm{Leech}})$ and $C_{o_1} = \PP C_{o_0} = C_{o_0}/ \pm \id$ the quotient by the center. - [ ] $C_{o_1}$ is Conway's "first sporadic group", known to be simple. - [ ] How can you reconstruct a lattice from a Dynkin diagram? - [ ] What is the Conway group? - [ ] What is the Matheiu group? - [ ] What is the Leech lattice? - [ ] What is the Barnes-Wall lattice? - [ ] What is the Coxeter-Todd lattice? - [x] Why are 8 and 24 special? ✅ 2023-01-30 - [ ] 8 is the smallest dimension of an even unimodular attice, namely $E_8$ - [ ] 24 is the smallest dimension of an even unimodular lattice with no roots (check $\theta$ functions) - [ ] $0^2 + 1^2 + \cdot + 24^2 = 70^2$ and this is the only $n$ for which the sum of the first $n$ squares is a square. - [x] What is a sporadic simple group? ✅ 2023-01-30 - [ ] One of 26 exceptional groups: not cyclic, alternating, or of Lie type. - [x] What are the generators of $\SL_2(\ZZ)$? ✅ 2023-01-30 - [ ] ![](attachments/2023-02-01-s-t.png) - [x] What is Rohklin's theorem? ✅ 2023-01-31 - [ ] If $X$ is a smooth closed spin 4-manifold, $H^2(X)$ is an even lattice and $\sgn H^2(X) \equiv 0 \mod 16$. - [ ] Converse: if $H^2(X)$ is even and $\pi_1 X = 1$, then $X$ is spin. - [ ] Counterexample if $\pi_1 X\neq 1$: the Enriques surface has $\pi_1 X = C_2$ and $H^2 = E_8 \oplus H$ which has signature 8. - [x] What is the $E_8$ matrix? ✅ 2023-01-31 - [ ] ![](attachments/2023-01-31-E8.png) # Modular forms - [x] What are the Eisenstein series $E_4$ and $E_6$? ✅ 2024-01-29 - [ ] $$E_4(z)=1+240 \sum_{m=1}^{\infty} \sigma_3(m) q^m \quad \text { and } \quad E_6(z)=1-504 \sum_{m=1}^{\infty} \sigma_5(m) q^m$$ - [x] What is the Eisenstein series $G_k$? ✅ 2024-01-29 - [ ] $$G_k(z)=\sum_{\substack{(m, n) \in \mathbb{Z}^2 \\(m, n) \neq(0,0)}} \frac{1}{(m z+n)^k}$$ - [ ] Modular form of weight $k$ for any even $k\geq 4$. - [x] What is the normalized Eisenstein series $E_k$? ✅ 2024-01-29 - [ ] $$\frac{1}{2 \zeta(k)} G_k$$ - [x] Give an explicit formula for the normalized Eistenstein series $E_k$. ✅ 2024-01-29 - [ ] $$E_k(z)=1-\frac{2 k}{B_k} \sum_{m=1}^{\infty} \sigma_{k-1}(m) q^m$$.