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Math 596+726: Topics in Number Theory

Quadratic forms, orthogonal groups, and modular forms




Questions to think about.



1. Let $V$ be a non-degenerate quadratic space. Show that any two maximal isotropic spaces of $V$ have the same dimension, $t$. This integer is called the Witt index of $V$. Show that $V$ is isomorphic to an orthogonal direct sum of $t$ hyperbolic spaces and an anisotropic space $W$ of dimension $n-2t$ (where $n=\dim(V)$).
(The solution of this problem was presented by Hazem and Arihant at the end of lecture 3 on September 13.)


2. Describe the connected component of the identity in the symmetry group of $4$-dimensional Minkowski space $\mathbb R^{3,1}$. How many connected components does the full orthogonal group have? Same questions for $\mathbb R^{4,0}$ and $\mathbb R^{2,2}$.
The solution was presented by Marcel and Jad at the end of the recording of Wednesday the 22nd.


3. Show that the Hilbert symbol $(a,b)$, for $a,b\in \mathbb Q_p^\times$, is equal to $-1$ if and only the idoneous central simple algebra over $\mathbb Q_p$ defined by $$ B = \mathbb Q_p + \mathbb Q_p i + \mathbb Q_p j + \mathbb Q_p k, \quad i^2 = a, \quad j^2 = b, \quad ij=-ji = k, $$ is a division algebra, and that it is isomorphic to the matrix algebra $M_2(\mathbb Q_p)$ if $(a,b)=1$.
The solution was presented by Sun Kai and Paul-Antoine at the end of the lecture of September 29


4. Show that the quadratic form $ax^2 +by^2 + cz^2$ with coefficients in ${\mathbb Q}_p$, in which $a$, $b$ and $c$ belong to $\mathbb Z_p^\times$, has a non-trivial zero, i.e., the associated quadratic space over $\mathbb Q_p$ has a non-zero isotropic vector. (The idea here is to give a brief survey of Hensel's lemma and how it is used to solve equations like this on over the $p$-adics).
Davide and Niccolo discussed this question at the end of the lecture of Monday, October 18.


5. Show that all even unimodular lattices of a given signature $(r,s)$ are in the same genus. Show that all odd unimodular lattices are in the same genus. Give an example of two quadratic forms of the same discriminant that lie in different genera. (Marti is the designated victim for this one.)


6. In class we talked a lot about even unimodular lattices, because their theta series are modular forms on all of ${\mathbf{SL}}_2(\mathbb Z)$. But this leaves out the most natural quadratic forms like $x^2+y^2$ and $x^2+y^2+z^2+w^2$ related to the problem of expressing integers as sums of $2$ and $4$ squares, corresponding to the celebrated theorems of Fermat and Lagrange, which were touched on in the very first (motivational) lecture of the course.

The present exercise aims to dot the i's and cross the t's.

(a) Let $L$ be an odd unimodular lattice of rank $2k$. Show that the theta series $\theta_L(z)$ is invariant under the group $\Gamma(2)$ consisting of matrices in $\mathbf{SL}_2(\mathbb Z)$ which are congruent to $1$ modulo $2$.

(b) To avoid having theta series with fractional powers of $q$, it is useful to redefine $\theta_L(q)$ by the rules $$ \theta_L(z) = \sum_{v\in L} e^{2\pi i (v\cdot v)z}, \qquad \theta_L(q) = \sum_{v\in L} q^{v\cdot v} = \sum_{n=0}^\infty r_L(n) q^n,$$ where $r_L(n)$ now denotes the number of vectors $v\in L$ with $v \cdot v = n$. Show that $\theta_L(q)$ is a modular form of weight $k$ on the subgroup $\Gamma_0(4)$ consisting of matrices in ${\mathbf SL}_2(\mathbb Z)$ that are upper triangular mod $4$.

(c) Although the Eisenstein series $E_2$ defined by the series $$ E_2(q) = 1-24 \sum_{n=1}^\infty \sigma_{1}(n) q^n, \qquad \sigma_1(n) := \sum_{d|n} d, $$ fails to be invariants under the action of $\mathbf{SL}_2(\mathbb Z)$, the modification $$ E_2^*(z) := 1- 24\sum_{n=1}^\infty \sigma_1(n) q^n - \frac{3}{\pi y}, \qquad q:= e^{2\pi i z}, \quad y:= {\rm im}(z)$$ is invariant, after subtraction of the non-holomorphic term $\frac{3}{\pi y}$. Use this fact to show that the $q$-series $$ E_2^{(2)}:= E_2(z) - 2 E_2(2z) = E_2(q) - 2 E_2(q^2), \qquad E_2^{(4)}:= E_2(z) - 4 E_2(4z) = E_2(q) - 4 E_2(q^4) $$ are (holomorphic) modular forms of weight two on $\Gamma_0(4)$.

(d) Show that any modular form of weight $k$ on $\Gamma_0(4)$ has exactly $k/2$ zeroes on any fundamental region. Use this to concluce that $M_2(\Gamma_0(4))$ is two dimensional, and thus spanned by the two Eisenstein series $E_2^{(2)}$ and $E_2^{(4)}$.

(e) Use the result of (d) to calculate the number of vectors of odd length $n$ in any odd unimodular quaternary quadratic form.

(f) Write the theta series attached to the standard quaternary lattice $\mathbb Z^4$ with the standard dot product, and the theta series attached to the lattice $D_4$ (consisting of vectors in $\mathbb Q^4$ whose entries are either integers summing to an even integer, or halves of odd integers), as linear combinations of $E_2^{(2)}$ and $E_2^{(4)}$. Deduce a closed form expression for the number of vectors of a given length in each of these two lattices. Conclude that these lattices are not isometric, although they have the same number of vectors of any given odd length.


7. Let $G$ be a group acting transitively on a set $X$, let $x_0\in X$, and let $S$ be a subset of $G$ which satisfies, for all $x\in X$, there exists $g\in S$ with $g x_0 = x$. Show that $G$ is generated by $S$ together with the stabiliser of $x_0$.


8. Let $K$ be a field. Show that $\mathbf{SL}_2(K)$ is generated by matrices of the form $$ \left(\begin{array}{cc} 1 & t \\ 0 & 1 \end{array}\right), \quad t\in K, \qquad \qquad \left(\begin{array}{cc} a & 0 \\ 0 & a^{-1} \end{array}\right), \quad a \in K^\times, \qquad \qquad w := \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right). $$ More precisely, show that $$ \mathbf{SL}_2(K) = B \sqcup B w B,$$ where $B$ is the Borel subgroup of upper triangular matrices. This is known as the Bruhat decomposition of ${\mathbf SL}_2(K)$. (Hint: use question 7.)


9. Show that every element of ${\mathbf SL}_2({\mathbb R})$ can be written uniquely in the form $$ \left(\begin{array}{cc} 1 & x \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} y^{1/2} & 0 \\ 0 & y^{-1/2} \end{array}\right) \left(\begin{array}{cc} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{array}\right), \qquad x\in \mathbb R, \quad y\in \mathbb R^{>0}, \quad \theta \in [0,2\pi).$$ This is known as the Iwasawa decomposition of ${\mathbf SL}_2({\mathbb R})$. (Hint: use a suitable refinement of your answer to question 7.)


10. Show that the characteristic function on ${\mathbb Z}_p$ is equal to its fourier transform. More generally, let $V$ be a quadratic space over ${\mathbb Q}_p$ and let $L$ be a ${\mathbb Z}_p$-sublattice in $V$. Show that the fourier transform of the characteristic function of $L$ is $\mu(L)$ times the characteristic function of the ${\mathbb Z}_p$-dual lattice: $$ {\widehat{ {\bf 1}_L}} = \mu(L) \cdot {\bf 1}_{L^\vee}.$$


11. Let $L$ be a lattice of co-volume 1 in a quadratic space $V$.

The Poisson summation formula on $V_{\mathbb R}$ asserts that $$ \sum_{v\in L} \phi(v) = \sum_{v\in L^\vee} \hat{\phi}(v),$$ for all Schwartz functions $\phi$ on $V_{\mathbb R}$, where $L^\vee $ is the dual lattice of $V$.

The adèlic Poisson summation formula asserts that $$ \sum_{v\in V} \phi(v) = \sum_{v\in V} \hat{\phi}(v),$$ for all adèlic Schwartz functions $\phi$ on $V_{{\mathbb A}_{\mathbb Q}}$.

Show that the adèlic Poisson summation formula implies its more familiar analogue on $V_{\mathbb R}$.


12. Let $G = {\mathbb GL}_n(\mathbb Q_p)$ and let $X = {\mathbb GL}_n(\mathbb Q_p)/ {\mathbb GL}_n(\mathbb Z_p)$. Show that the action of $G$ on $X$ satisfies the finitness assumption that was made in class when we discussed Hecke operators, i.e., that the stabiliser of any $x\in X$ acts on $X$ with finite orbits.


13. Let $L$ be a unimodular $\mathbb Z_p$-lattice in a quadratic space $V$ over $\mathbb Q_p$, and let $G$ be the orthogonal group over $\mathbb Z_p$ attached to $L$. Show that $G(\mathbb Q_p)$ acts transitively on the set of pairs $(L_1,L_2)$ of unimodular lattices satisfying $L_1/(L_1\cap L_2) \simeq \mathbb Z/p\mathbb Z$. (More generally, given a finite abelian $p$-group $A$, it acts transitively on the set of pairs $(L_1,L_2)$ for which $L_1/(L_1\cap L_2\simeq A$, when this set of pairs is non-empty.)


14. Write down the character table of the Heisenberg group $H(W)$ where $W$ is the two-dimensional symplectic space over the field with $p$ elements.