User emiliocba - MathOverflow

comprehensive presentation of the unitary dual of $SO_0(n,1)$

2012-08-24T13:32:20Z

The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(n,1)$. I know the paper of Baldoni Silva-Barbasch ("The unitary spectrum for real rank one", Invent. math. 72) and the reference therein, but I would like to have a more comprehensive presentation, for example in a book. Thanks in advance.-.

Local densities of hermitian forms

2011-12-30T14:42:31Z

I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see 1). I would have liked to create the new tag [local-densities], but I don't have enough reputation.

Let $\mathcal O$ be a ring of integers of an imaginary quadratic number field, like $\mathbb Z(\sqrt{-1})$. Let $H\in \mathrm{GL}(n,\mathcal O)$ be a Hermitian matrix and $\ell$ an integer number. For $p$ an integer prime number, I need to compute the $p$-local density given by $$ \delta_p(H,\ell) = \lim_{j\to\infty} \frac{A_j(H,\ell)}{p^{j(2n-1)}} $$ where $$ A_j(H,\ell) = \# \{ x\in ({{\mathcal O}/p^j{\mathcal O}})^n:x^*Hx\equiv \ell\pmod {p^j} \}. $$

Now I introduce the setting on Hironaka's paper: let $k$ be a nonarchimedian local field of characteristic $0$, $\mathcal O_k$ the ring of integers in $k$ (note that $\mathcal O$ and $\mathcal O_k$ are different rings), $*$ an involution on $k$ and denote $k_0$ the fixed field by $*$. Assume that $k$ is unramified over $k_0$. Let $q$ be the residue class field of $k_0$, $\pi\in k_0$ be a prime element of $k$ and $\mathfrak{p}=\pi\mathcal O_k$. She gives a formula for $$ \mu_p(H,\ell) = \lim_{j\to\infty} \frac{N_j(H,\ell)}{q^{j(2n-1)}} $$ where $$ N_j(H,\ell)= \# \{ x\in ({{\mathcal O_k}/\mathfrak{p}})^n:x^*Hx\equiv \ell\pmod {\mathfrak{p}^j} \}. $$

Let $D_{\mathcal O}(<0)$ be the discriminant of $\mathcal O$.

If $(\frac{D_{\mathcal O}}{p})=-1$, we have that $p\mathcal O$ is a prime ideal in $\mathcal O$, and we can apply Hironaka's formula to the nonarquimedian local field $k=\mathbb Q[\sqrt{D_\mathcal O}]\otimes_\mathbb Q \mathbb Q_p$, where the involution $*$ is given by $\alpha\otimes x\mapsto \bar\alpha\otimes x$, and $k_0=\mathbb Q_p$. This can be done because $\mathcal O_k/\mathfrak{p} \simeq \mathcal O/p\mathcal O$, since $\mathfrak{p}=p\mathcal O_k$.

Now, if $(\frac{D_{\mathcal O}}{p})=+1$, we have that $p\mathcal O$ decomposes as a product of two different prime ideals (are conjugated). Also, $\mathbb Q[\sqrt{D_\mathcal O}]\otimes_\mathbb Q \mathbb Q_p$ is isomorphic to $\mathbb Q_p\times \mathbb Q_p$. Finally, my question is:

How do I apply Hironaka's formula to this case? Who is the local field $k$? Who is the conjungation on $k$ and the fixed field $k_0$? Who is the prime element $\pi$?

I'm sorry for this long question. I hope that somebody helps me with an understandable answer since I usually don't work in this area. Thanks.-.

1 Y. Hironaka. "Local zeta functions on hermitian forms and its application to local densities". Journal of Number Theory 71, 40--64 (1998). Link.

Answer by emiliocba for Local densities of hermitian forms

2012-10-21T10:34:18Z

Several months later, I have the answer to my question. I would like to share here.

Such as Maurice Mischler (Local densities of hermitian forms, in Contemporary Mathematics vol 272 (2000), 201--208) mentioned in his abstract, Hironaka compute the local density only in the inert prime case ($p\mathcal O$ is a prime ideal).

However, we can compute $\delta_p(H,\ell)$ as follows: let $1,\omega$ a $\mathbb Z$-basis of $\mathcal O$. Writing $z_j=x_{2j-1}+x_{2j}\omega$, we have that $H[z]$ induces a quadratic forms $Q_H\in\mathrm{GL}(2n,\mathbb Z)$. Then $$ \delta_p^\mathbb C(H,\ell) = \delta_p^\mathbb R(Q_H,\ell). $$ Now, we can compute the right-hand side by applying Yang's paper (An explicit formula for local densities of quadratic forms, J. Number Theory, vol 72 (1998), 309--356).

Answer by emiliocba for The Gauss circle problem on a hexagonal lattice

2012-10-20T11:19:42Z

Lax and Phillips (J. Funct. Anal. vol 46 (1982), 280--350) showed, for any crystallographic group $\Gamma$ in the Euclidean plane, that $$ N(r;x,x_0)= \frac{\pi r^2}{|F|} + O(r^{2/3} (\log r)^{1/2}), $$ as $r\to+\infty$, where $|F|$ denotes the volume of the fundamental domain of $\Gamma$, $x,x_0\in\mathbb R^2$ and $N(r;x,x_0)$ is the number of elements $\gamma\in\Gamma$ such that $$ |x-\gamma (x_0)|\leq r. $$

Later, Levitan (Russian Math. Surveys vol 42:3 (1987), 13--42) improved the error term to $O(r^{2/3})$.

In your particular case, $\Gamma$ is the subgroup of the isometries of the plane generated by the translations for $(1,0)$ and $(1/2,\sqrt{3}/2)$, thus $|F|=\sqrt{3}/2$.

Both papers works in higher dimensions and in hyperbolic spaces.

finitely many orbits of integer solutions module unimodular groups

2011-12-21T00:45:38Z

Let $Q$ be the Lorentzian matrix, that is $Q=\mathrm{diag}(I_n,-1)$, where $I_n$ denotes de $n\times n$ identity matrix. Let $\mathcal R$ the set of integer solutions $x\in\mathbb Z^{n+1}$ of $$ Q[x]:=x^tQx=x_1^2+\dots+x_{n}^2-x_{n+1}^2=-1. $$ The set $\Gamma=\mathrm{O}(n,1)\cap \mathrm{M}(n+1,\mathbb Z)$ acts by left multiplication on $\mathcal R$. It's known that there are finitely many $\Gamma$-orbits.

I think (but I'm not completely sure) that $\Gamma\backslash \mathcal R$ has only one class if $2\leq n\leq 7$, and two if $n=8$ (since $(0,0,0,0,0,0,0,0,1)$ and $(1,1,1,1,1,1,1,1,3)$ leaves in different $\Gamma$-orbits).

What happens in the hermitian case?

More precisely, let $\Gamma=\mathrm{U}(n,1)\cap\mathrm M(n+1,\mathcal O)$ with $\mathcal O$ the ring of integer of a quadratic imaginary extension of $\mathbb Q$, and let $\mathcal R$ be the set of "integer" solutions $x\in\mathcal O^{n+1}$ of $$ Q[x]=x^*Qx=|x_1|^2+\dots+|x_{n}|^2-|x_{n+1}|^2=-1. $$

How many $\Gamma$-orbits are there in the set $\mathcal R$?

[Edit: It's enough for $n=2,3$ and some particular ring $\mathcal O$, like $\mathbb Z[\sqrt{-1}]$.]

Thank you in advance.-.

volume form in a symmetric space of real rank one

2012-02-20T13:19:26Z

I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one. The first one is the volume form induced by the Riemannian structure given by the Killing form restricted on $\mathfrak p$. The second one is $d\bar g$ induced by $dg =\gamma(a_t)\ dk_1 \ da \ dk_2$. It is known that $$ dx=c \ d\bar g,\qquad \text{with $c\in\mathbb R$.} $$ What is $c$?

Now, a more detailed exposition. Let $G$ be a connected semisimple Lie group of real rank one and finite center. Let:

$\mathfrak g=\mathfrak k \oplus \mathfrak p$ a Cartan decomposition;
$G=NAK$ be an Iwasawa decomposition of $G$;
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak n$ the corresponding decomposition of $\mathfrak g$;
$B$ Killing form on $\mathfrak g$,
$\alpha$ the simple root ($G$ has real rank one);
$p=\dim \mathfrak n_\alpha$, $q=\dim \mathfrak n_{2\alpha}$;
$H_\alpha\in\mathfrak a$ with $\alpha(H_\alpha)=1$;
$A^+=\{ a_t:=\exp(t H_\alpha) : t>0 \}$,
$M$ the centralizer of $A$ in $K$.

Let $X=G/K$ be the symmetric space with the Riemannian structure induced by $B$ over $\mathfrak p$. Let $dx$ the volume form induced by this Riemannian structure.

Let $dg$ (resp. $d\bar g$) be the Haar measure on $G$ (resp. $G/K$) such that $$ dg = \gamma(a_t)\ dk_1 \ da \ dk_2 $$ on $KA^+K$, where $\gamma(a_t) = (e^t-e^{-t})^p (e^{2t}-e^{-2t})^q=2^{p+q}(\sinh t)^p(\sinh 2t)^q$, $da=dt$ and $dk$ is the Haar measure on $K$ normalized so that $K$ has volume 1.

Thanks.-.

Hurwitz integers and $F_4$

2012-02-12T23:53:51Z

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number of elements $\alpha\in\mathcal H$ such that $|\alpha|^2=m$.

This is equivalent to known the number of vectors $v$ in the lattice $F_4$ such that $\|v\|^2=m$.

I think this formula already exist but I can find it. An appropiate reference would be appreciated. Thanks.-.

Siegel's Mass Formula for ternary indefinite quadratic forms

2012-02-03T16:15:18Z

In his paper "On the theory of indefnite quadratic forms", Siegel gives the formula (Thm. 1) $$ \mu(S,T)=\prod_p \alpha_p(S,T), $$ where

$S$ is an $m\times m$ non singular integral symmetric matrix of signature $(r,m-r)$,
$T$ is an $n\times n$ integral symmetric matrix,
$\mu(S,T)$ is the measure of the representation of $T$ by $S$,
$\alpha_p(S,T)$ is the $p$-adic density of the representation of $T$ by $S$ ($p$ over the rational primes),

and $$ n\leq r,\qquad n\leq m-r,\qquad 2(n+1) < m. $$

In my case, $n=1$ thus $T=t$ is just a (non zero) integer number, and $S$ has signature $(m-1,1)$ (or $(1,m-1)$). Hence, the theorem holds for $$ m>4. $$ However, in the fourth page, he added that the formula also holds for $m=4$.

I known that when $m=3$ ($S$ is a ternary quadratic form), the formula doesn't holds in general and strange things happen (see BOROVOI ).

My question is: Does the formula works for the following particular case?:

$$ S= \begin{pmatrix} A & \\ &-a \end{pmatrix} $$

where

$A$ is a $2\times2$ integral positive definite symmetric matrix,
$a$ is a positive integer, and
$t$ is a negative integer.

I known that this is true for $A=I_2$ and $a=1$ (see this paper).

Thank you in advance.-.

How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?

2012-01-23T15:29:47Z

Let $k$ be a positive integer. Let $$Q= \begin{pmatrix} 1 &1/2& & & & \\ 1/2& 1 & & & & \\ & & 1 &1/2& & \\ & &1/2& 1 & & \\ & & & & 1 &1/2\\ & & & &1/2& 1 \end{pmatrix}. $$

How many solution $x\in\mathbb Z^6$ are there to $\quad x^tQx=k$?

This is equivalent to:

How many solution $x\in\mathbb Z^6$ are there to $$x_1^2+x_1x_2+x_2^2+ x_3^2+x_3x_4+x_4^2+ x_5^2+x_5x_6+x_6^2=k?$$

or to

How many solution $x\in \mathbb Z\left[\omega \right]^3$ are there to $\quad x^* I_3 x=k$?

where $I_3$ is the $3\times3$-identity matrix and $\omega=\frac{1+\sqrt{-3}}{2}$.

I know that there is a formula for this number (there is only one class in its genus), but I don't know it.

This question is related to

but they don't answer my question.

Answer by emiliocba for Shortest distance along the surface of the hyperboloid

2012-01-18T10:17:19Z

Using the following formula $$ \cosh(d(A,B))=\frac{|q(A,B)|}{|q(A)|^{-1/2}|q(B)|^{-1/2}}, $$ where $$ q(A,B)=A^t I_{2,1} B = a_1b_1+a_2b_2-a_3b_3 $$ with $I_{2,1}=diag(1,1,-1)$.

Representations by positive definite binary quadratic forms

2012-01-15T14:44:16Z

It's known that the number of representations of an integer $k$ by sum of two squares is $$ 4\;\sum_{d|k}\left(\frac{-4}{d}\right) $$ or $$ 4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= 4(d_1(k)-d_3(k)) $$ where $d_1(k)$ and $d_3(k)$ are the numbers of the divisors of $k$ of the forms $4m+1$ or $4m+3$ respectively.

I would like to have references about similar formulas for the number of representations of an integer by the quadratic form $$ x^2+N\;y^2 $$ with $N$ a positive integer, or more generally, any positive definite binary quadratic form. Thanks.-.

Values of Dirichlet L-funcions at natural numbers

2012-01-03T17:14:23Z

I want to know about reference of formulas for $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s} $$ for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have the Dirichlet class number formula.

I would like to have some reference for $s\geq2$. Thanks.-.

Answer by emiliocba for Books you would like to see translated into English.

2012-01-03T04:53:03Z

"Quadratische Formen" by Martin Kneser.

Answer by emiliocba for Books you would like to see translated into English.

2012-01-03T04:52:07Z

"Introduction aux groupes arithmétiques" by Armand Borel.

Norm map and units in local rings

2011-12-29T22:33:38Z

Let $$ L=\mathbb{Q}(\sqrt{-1})\otimes_\mathbb{Q} \mathbb{Q}_3 $$ where $\mathbb{Q}_3$ denotes de $3$-adic rational numbers. Then $L$ is a quadratic extension of the local field $\mathbb{Q}_3$. Furthermore, the valuation ring of $L$ is $B:=\mathbb{Z}[\sqrt{-1}] \otimes \mathbb{Z}_3$.

It is known that the norm map $N$ maps the set $U_L$ of units in $B$ onto the set $U_{\mathbb{Q}_3}$ of units in $\mathbb{Z}_3$ (see Serre's book "Local Fields", Chapter V, Prop. 3). This implies that there is an element $x\in U_L$ such that $N(x)=-1$ since $-1$ is a unit in $\mathbb{Q}_3$. Could someone specify this element $x$?

volume of complex hyperbolic manifolds

2011-12-27T12:00:51Z

I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds.

More precisely, let $\mathcal O$ be an imaginary quadratic number field, and let $\Gamma$ be the set of "integer" elements in $U(n,1)$, that is $$ \Gamma = U(n,1) \cap M(n+1,\mathcal O). $$

How much is the volume of the complex hyperbolic manifold $M_\Gamma = H_{\mathbb C}^n / \Gamma$?

The answer for $\mathcal O=\mathbb Z[\sqrt{-1}]$ is enough for me.

It's known that $$ \mathrm{vol}(M_\Gamma) = \frac{(-\pi)^n 2^{2n}}{(n+1)!}\; \chi(M_\Gamma), $$ where $\chi(M_\Gamma)$ denotes the Euler characteristic of $M_\Gamma$, but I couldn't find computed the term $\chi(M_\Gamma)$ in the literature.

Thank you in advance.-.

Answer by emiliocba for volume of complex hyperbolic manifolds

2011-12-27T21:53:00Z

This answer belongs to the second author of this paper (=:[ES]).

First, let $\Gamma_0 = SU(n, 1) \cap M(n + 1, \mathcal O)$ and $M_0 = H_\mathbb{C}^n / \Gamma_0$. By (1) and (28) in [ES], we obtain that $$ \mathrm{vol}(M_0) = \frac{(4 \pi)^n}{(n + 1)!} |d_{\mathcal O}|^s \left( \prod_{j = 1}^n \frac{j!}{(2 \pi)^{j + 1}} \right) \zeta(2) L_{\mathcal O}(3) \zeta(4) L_{\mathcal O}(5) \cdots F(n + 1), $$ where $d_{\mathcal O}$ is the discriminant of $\mathcal O$, $s$ is $\frac{n (n + 3)}{4}$ when $n$ is even and $\frac{(n - 1)(n - 2)}{4}$ when $n$ is odd, and $F(n + 1)$ is $\zeta(n + 1)$ when $n$ is odd and $L_{\mathcal O}(n + 1)$ when $n$ is even.

Now, we have a finite-sheeted covering $M_0 \to M:= H_\mathbb{C}^n / \Gamma$, so $$ \mathrm{vol}(M)=[PU(n, 1; \mathcal O) : PSU(n, 1; \mathcal O)]\; \mathrm{vol}(M_0). $$

An easy computation shows that the index $[PU(n, 1; \mathcal O) : PSU(n, 1; \mathcal O)]$ is equal to $$ 1 \quad\text{if $n$ is even and $d_{\mathcal O}<-3$}, $$ $$ 1 \quad\text{if $n$ is even, $d_{\mathcal O}=-3$ and $n\not\equiv 2\pmod6$}, $$ $$ 3 \quad\text{if $n$ is even, $d_{\mathcal O}=-3$ and $n\equiv 2\pmod6$}, $$ $$ 2 \quad\text{if $n$ is odd and $d_{\mathcal O}<-3$}, $$ $$ 2 \quad\text{if $n$ is odd, $d_{\mathcal O}=-3$ and $n\equiv1\pmod6$,} $$ $$ 1 \quad\text{if $n$ is odd, $d_{\mathcal O}=-3$ and $n\equiv3\pmod6$}, $$ $$ 6 \quad\text{if $n$ is odd, $d_{\mathcal O}=-3$ and $n\equiv5\pmod6$}. $$

Answer by emiliocba for The number of cusps of higher-dimensional hyperbolic manifolds

2011-12-27T11:47:40Z

Dear Roberto, to add information of Agol's comment, in Theorem 1.3 of this paper it is proved that there aren't one-cusped arithmetic hyperbolic $n$-orbifolds for $n\geq 30$. Moreover, Stover shows one-cusped arithmetic hyperbolic orbifolds in dimensions 10 and 11.

Comment by emiliocba

2012-10-20T13:54:49Z

@unknown: Now, I understand what you are looking for, but I don't known the answer. Good luck!

Comment by emiliocba

2012-10-20T13:21:13Z

@unknown: I'm sorry, but I can't understand when you say "an exact solution". The circle problem is to find the best error term of $E(r):=N(r)-\pi r^2$ as $r\to+\infty$, which is still open as far as I known. The conjecture is $N(r)=O(r^{1/2+\varepsilon})$.

Comment by emiliocba

2012-10-20T11:42:32Z

Surely, the exact counting solution in your case is as difficult as the Gauss circle problem.

Comment by emiliocba

2012-08-28T17:30:04Z

Dear Professor, thank you for your suggestions. I can see that this area is unavoidably complicated to arrive... I will try to undestand Thm 3 in Thieleker (page 362) and to read the survey in Collingwood.

Comment by emiliocba

2012-02-13T01:38:00Z

perfect answer! nice proof. Thank you.-.

Comment by emiliocba

2012-02-04T17:34:21Z

Thank you again. I want to known if these particular case holds generalizing the fact that it holds for $S=diag(1,1,-1)$ and $t<0$.

Comment by emiliocba

2012-02-04T00:16:46Z

Are you saying that if the ternary form $S=diag(A,-a)$ (with $A$ pos def and $a\in\mathbb N$) is in a genus with only one integer equivalence class, then Siegel's formula holds?

Comment by emiliocba

2012-02-04T00:13:25Z

Will, it's a nice answer! Thank you very much. However, I want to check my conclusion from your answer in the next comment.

Comment by emiliocba

2012-01-24T20:35:18Z

Your formula works for $k\leq 10^4$. This is a great answer! Thank you for your time.

Comment by emiliocba

2012-01-23T18:15:56Z

Thank Noam, I had not been able to explain it. Do you know some reference in where this formula "could be"?

Comment by emiliocba

2012-01-03T17:27:48Z

yes! Thank you Robert.-.

Comment by emiliocba

2012-01-03T13:37:59Z

Isn't the mass a genus invariant?

Comment by emiliocba

2011-12-30T14:51:42Z

Chandan, thank you for your comments. I changed the label as you suggested.

Comment by emiliocba

2011-12-29T00:22:05Z

With respect to (1), I think that there are topological arguments to prove that there is no embedding from $S^1\times S^1$ to an sphere $S^n$ with $n\geq2$.

Comment by emiliocba

2011-12-28T18:23:52Z

Dear Agol, I'm just interested in the number of elements of $\Gamma\backslash \mathcal R$ for $n=2$ or $3$.