User emiliocba - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:14:05Z http://mathoverflow.net/feeds/user/20052 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105387/comprehensive-presentation-of-the-unitary-dual-of-so-0n-1 comprehensive presentation of the unitary dual of $SO_0(n,1)$ emiliocba 2012-08-24T13:32:20Z 2013-01-13T12:39:39Z <p>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.-.</p> http://mathoverflow.net/questions/84592/local-densities-of-hermitian-forms Local densities of hermitian forms emiliocba 2011-12-30T14:42:31Z 2012-10-21T10:36:01Z <p>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 <a href="http://www.sciencedirect.com/science/article/pii/S0022314X98922375" rel="nofollow">1</a>). I would have liked to create the new tag [local-densities], but I don't have enough reputation. </p> <p>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 $$A_j(H,\ell) = \# \{ x\in ({{\mathcal O}/p^j{\mathcal O}})^n:x^*Hx\equiv \ell\pmod {p^j} \}.$$</p> <p>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 $$N_j(H,\ell)= \# \{ x\in ({{\mathcal O_k}/\mathfrak{p}})^n:x^*Hx\equiv \ell\pmod {\mathfrak{p}^j} \}.$$</p> <p>Let $D_{\mathcal O}(&lt;0)$ be the discriminant of $\mathcal O$. </p> <p>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 $\mathfrak{p}=p\mathcal O_k$.</p> <p>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: </p> <p>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 $\pi$?</p> <p>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.-.</p> <p><a href="http://www.sciencedirect.com/science/article/pii/S0022314X98922375" rel="nofollow">1</a> Y. Hironaka. "Local zeta functions on hermitian forms and its application to local densities". Journal of Number Theory 71, 40--64 (1998). <a href="http://www.sciencedirect.com/science/article/pii/S0022314X98922375" rel="nofollow">Link</a>. </p> http://mathoverflow.net/questions/84592/local-densities-of-hermitian-forms/110226#110226 Answer by emiliocba for Local densities of hermitian forms emiliocba 2012-10-21T10:34:18Z 2012-10-21T10:34:18Z <p>Several months later, I have the answer to my question. I would like to share here.</p> <p>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).</p> <p>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).</p> http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice/110143#110143 Answer by emiliocba for The Gauss circle problem on a hexagonal lattice emiliocba 2012-10-20T11:19:42Z 2012-10-20T11:37:02Z <p>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 $$|x-\gamma (x_0)|\leq r.$$</p> <p>Later, Levitan (Russian Math. Surveys vol 42:3 (1987), 13--42) improved the error term to $O(r^{2/3})$.</p> <p>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 $|F|=\sqrt{3}/2$.</p> <p>Both papers works in higher dimensions and in hyperbolic spaces.</p> http://mathoverflow.net/questions/83982/finitely-many-orbits-of-integer-solutions-module-unimodular-groups finitely many orbits of integer solutions module unimodular groups emiliocba 2011-12-21T00:45:38Z 2012-05-02T22:22:01Z <p>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.</p> <p>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).</p> <p>What happens in the hermitian case?</p> <p>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 $$Q[x]=x^*Qx=|x_1|^2+\dots+|x_{n}|^2-|x_{n+1}|^2=-1.$$</p> <p>How many $\Gamma$-orbits are there in the set $\mathcal R$? </p> <p>[Edit: It's enough for $n=2,3$ and some particular ring $\mathcal O$, like $\mathbb Z[\sqrt{-1}]$.]</p> <p>Thank you in advance.-.</p> http://mathoverflow.net/questions/89013/volume-form-in-a-symmetric-space-of-real-rank-one volume form in a symmetric space of real rank one emiliocba 2012-02-20T13:19:26Z 2012-02-20T14:42:49Z <p>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 $c$?</p> <p>Now, a more detailed exposition. Let $G$ be a connected semisimple Lie group of real rank one and finite center. Let:</p> <ul> <li>$\mathfrak g=\mathfrak k \oplus \mathfrak p$ a Cartan decomposition;</li> <li>$G=NAK$ be an Iwasawa decomposition of $G$;</li> <li>$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak n$ the corresponding decomposition of $\mathfrak g$;</li> <li>$B$ Killing form on $\mathfrak g$,</li> <li>$\alpha$ the simple root ($G$ has real rank one);</li> <li>$p=\dim \mathfrak n_\alpha$, $q=\dim \mathfrak n_{2\alpha}$;</li> <li>$H_\alpha\in\mathfrak a$ with $\alpha(H_\alpha)=1$; </li> <li>$A^+=\{ a_t:=\exp(t H_\alpha) : t>0 \}$, </li> <li>$M$ the centralizer of $A$ in $K$.</li> </ul> <p>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.</p> <p>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.</p> <p>Thanks.-.</p> http://mathoverflow.net/questions/88308/hurwitz-integers-and-f-4 Hurwitz integers and $F_4$ emiliocba 2012-02-12T23:53:51Z 2012-02-13T00:07:13Z <p>The Hurwitz integers are <code>$$\mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.$$</code> 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 $|\alpha|^2=m$.</p> <p>This is equivalent to known the number of vectors $v$ in the lattice $F_4$ such that $\|v\|^2=m$.</p> <p>I think this formula already exist but I can find it. An appropiate reference would be appreciated. Thanks.-.</p> http://mathoverflow.net/questions/87445/siegels-mass-formula-for-ternary-indefinite-quadratic-forms Siegel's Mass Formula for ternary indefinite quadratic forms emiliocba 2012-02-03T16:15:18Z 2012-02-05T03:27:34Z <p>In his paper "<a href="http://www.jstor.org/pss/1969191" rel="nofollow">On the theory of indefnite quadratic forms</a>", Siegel gives the formula (Thm. 1) $$\mu(S,T)=\prod_p \alpha_p(S,T),$$ where </p> <ul> <li>$S$ is an $m\times m$ non singular integral symmetric matrix of signature $(r,m-r)$,</li> <li>$T$ is an $n\times n$ integral symmetric matrix, </li> <li>$\mu(S,T)$ is the measure of the representation of $T$ by $S$,</li> <li>$\alpha_p(S,T)$ is the $p$-adic density of the representation of $T$ by $S$ ($p$ over the rational primes), </li> </ul> <p>and $$n\leq r,\qquad n\leq m-r,\qquad 2(n+1) &lt; m.$$</p> <p>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 $m=4$.</p> <p>I known that when $m=3$ ($S$ is a ternary quadratic form), the formula doesn't holds in general and strange things happen (see <a href="http://www.math.tau.ac.il/~borovoi/papers/reprfin.pdf" rel="nofollow">BOROVOI</a> ). </p> <p>My question is: Does the formula works for the following particular case?:</p> <p><code>$$S= \begin{pmatrix} A &amp; \\ &amp;-a \end{pmatrix}$$</code></p> <p>where</p> <ul> <li>$A$ is a $2\times2$ integral positive definite symmetric matrix,</li> <li>$a$ is a positive integer, and</li> <li>$t$ is a negative integer.</li> </ul> <p>I known that this is true for $A=I_2$ and $a=1$ (see <a href="http://www.sciencedirect.com/science/article/pii/S0022123697931293" rel="nofollow">this paper</a>).</p> <p>Thank you in advance.-.</p> http://mathoverflow.net/questions/86456/how-many-solutions-are-there-to-sum-i13-x-i2x-iy-iy-i2k How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$? emiliocba 2012-01-23T15:29:47Z 2012-01-25T08:23:17Z <p>Let $k$ be a positive integer. Let <code>$$Q= \begin{pmatrix} 1 &amp;1/2&amp; &amp; &amp; &amp; \\ 1/2&amp; 1 &amp; &amp; &amp; &amp; \\ &amp; &amp; 1 &amp;1/2&amp; &amp; \\ &amp; &amp;1/2&amp; 1 &amp; &amp; \\ &amp; &amp; &amp; &amp; 1 &amp;1/2\\ &amp; &amp; &amp; &amp;1/2&amp; 1 \end{pmatrix}.$$</code></p> <blockquote> <p>How many solution $x\in\mathbb Z^6$ are there to $\quad x^tQx=k$?</p> </blockquote> <p>This is equivalent to: </p> <blockquote> <p>How many solution $x\in\mathbb Z^6$ are there to $$x_1^2+x_1x_2+x_2^2+ x_3^2+x_3x_4+x_4^2+ x_5^2+x_5x_6+x_6^2=k?$$</p> </blockquote> <p>or to</p> <blockquote> <p>How many solution $x\in \mathbb Z\left[\omega \right]^3$ are there to $\quad x^* I_3 x=k$?</p> </blockquote> <p>where $I_3$ is the $3\times3$-identity matrix and $\omega=\frac{1+\sqrt{-3}}{2}$.</p> <p>I know that there is a formula for this number (there is only one class in its genus), but I don't know it.</p> <p>This question is related to</p> <ul> <li><a href="http://mathoverflow.net/questions/78361/which-integers-take-the-form-x2-xy-y2" rel="nofollow">http://mathoverflow.net/questions/78361/which-integers-take-the-form-x2-xy-y2</a></li> <li><a href="http://math.stackexchange.com/questions/44139/how-many-solutions-are-there-to-fn-m-n2nmm2-q/" rel="nofollow">http://math.stackexchange.com/questions/44139/how-many-solutions-are-there-to-fn-m-n2nmm2-q/</a></li> </ul> <p>but they don't answer my question. </p> http://mathoverflow.net/questions/85977/shortest-distance-along-the-surface-of-the-hyperboloid/85978#85978 Answer by emiliocba for Shortest distance along the surface of the hyperboloid emiliocba 2012-01-18T10:17:19Z 2012-01-18T10:17:19Z <p>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 $I_{2,1}=diag(1,1,-1)$.</p> http://mathoverflow.net/questions/85741/representations-by-positive-definite-binary-quadratic-forms Representations by positive definite binary quadratic forms emiliocba 2012-01-15T14:44:16Z 2012-01-15T23:04:51Z <p>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.</p> <p>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.-.</p> http://mathoverflow.net/questions/84812/values-of-dirichlet-l-funcions-at-natural-numbers Values of Dirichlet L-funcions at natural numbers emiliocba 2012-01-03T17:14:23Z 2012-01-04T09:26:34Z <p>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 <em>Dirichlet class number formula</em>. </p> <p>I would like to have some reference for $s\geq2$. Thanks.-.</p> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/84787#84787 Answer by emiliocba for Books you would like to see translated into English. emiliocba 2012-01-03T04:53:03Z 2012-01-03T04:53:03Z <p>"Quadratische Formen" by Martin Kneser.</p> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/84786#84786 Answer by emiliocba for Books you would like to see translated into English. emiliocba 2012-01-03T04:52:07Z 2012-01-03T04:52:07Z <p>"Introduction aux groupes arithmétiques" by Armand Borel.</p> http://mathoverflow.net/questions/84553/norm-map-and-units-in-local-rings Norm map and units in local rings emiliocba 2011-12-29T22:33:38Z 2011-12-30T14:49:55Z <p>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 $B:=\mathbb{Z}[\sqrt{-1}] \otimes \mathbb{Z}_3$.</p> <p>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 $x$?</p> http://mathoverflow.net/questions/84379/volume-of-complex-hyperbolic-manifolds volume of complex hyperbolic manifolds emiliocba 2011-12-27T12:00:51Z 2011-12-27T21:53:00Z <p>I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds. </p> <p>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 $$\Gamma = U(n,1) \cap M(n+1,\mathcal O).$$</p> <p>How much is the volume of the complex hyperbolic manifold $M_\Gamma = H_{\mathbb C}^n / \Gamma$?</p> <p>The answer for $\mathcal O=\mathbb Z[\sqrt{-1}]$ is enough for me.</p> <p>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.</p> <p>Thank you in advance.-.</p> http://mathoverflow.net/questions/84379/volume-of-complex-hyperbolic-manifolds/84416#84416 Answer by emiliocba for volume of complex hyperbolic manifolds emiliocba 2011-12-27T21:53:00Z 2011-12-27T21:53:00Z <p>This answer belongs to the second author of this <a href="http://arxiv.org/abs/1107.5281" rel="nofollow">paper</a> (=:[ES]). </p> <p>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.</p> <p>Now, we have a finite-sheeted covering $M_0 \to M:= H_\mathbb{C}^n / \Gamma$, so $$\mathrm{vol}(M)=[PU(n, 1; \mathcal O) : PSU(n, 1; \mathcal O)]\; \mathrm{vol}(M_0).$$</p> <p>An easy computation shows that the index $[PU(n, 1; \mathcal O) : PSU(n, 1; \mathcal O)]$ is equal to $$1 \quad\text{if n is even and d_{\mathcal O}&lt;-3},$$ $$1 \quad\text{if n is even, d_{\mathcal O}=-3 and n\not\equiv 2\pmod6},$$ $$3 \quad\text{if n is even, d_{\mathcal O}=-3 and n\equiv 2\pmod6},$$ $$2 \quad\text{if n is odd and d_{\mathcal O}&lt;-3},$$ $$2 \quad\text{if n is odd, d_{\mathcal O}=-3 and n\equiv1\pmod6,}$$ $$1 \quad\text{if n is odd, d_{\mathcal O}=-3 and n\equiv3\pmod6},$$ $$6 \quad\text{if n is odd, d_{\mathcal O}=-3 and n\equiv5\pmod6}.$$</p> http://mathoverflow.net/questions/37982/the-number-of-cusps-of-higher-dimensional-hyperbolic-manifolds/84377#84377 Answer by emiliocba for The number of cusps of higher-dimensional hyperbolic manifolds emiliocba 2011-12-27T11:47:40Z 2011-12-27T11:47:40Z <p>Dear Roberto, to add information of Agol's comment, in Theorem 1.3 of this <a href="http://arxiv.org/abs/1112.4495" rel="nofollow">paper</a> 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. </p> http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice/110143#110143 Comment by emiliocba emiliocba 2012-10-20T13:54:49Z 2012-10-20T13:54:49Z @unknown: Now, I understand what you are looking for, but I don't known the answer. Good luck! http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice/110143#110143 Comment by emiliocba emiliocba 2012-10-20T13:21:13Z 2012-10-20T13:21:13Z @unknown: I'm sorry, but I can't understand when you say &quot;an exact solution&quot;. 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 $N(r)=O(r^{1/2+\varepsilon})$. http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice/110143#110143 Comment by emiliocba emiliocba 2012-10-20T11:42:32Z 2012-10-20T11:42:32Z Surely, the exact counting solution in your case is as difficult as the Gauss circle problem. http://mathoverflow.net/questions/105387/comprehensive-presentation-of-the-unitary-dual-of-so-0n-1/105666#105666 Comment by emiliocba emiliocba 2012-08-28T17:30:04Z 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. http://mathoverflow.net/questions/88308/hurwitz-integers-and-f-4 Comment by emiliocba emiliocba 2012-02-13T01:38:00Z 2012-02-13T01:38:00Z perfect answer! nice proof. Thank you.-. http://mathoverflow.net/questions/87445/siegels-mass-formula-for-ternary-indefinite-quadratic-forms/87490#87490 Comment by emiliocba emiliocba 2012-02-04T17:34:21Z 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 $t&lt;0$. http://mathoverflow.net/questions/87445/siegels-mass-formula-for-ternary-indefinite-quadratic-forms/87490#87490 Comment by emiliocba emiliocba 2012-02-04T00:16:46Z 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? http://mathoverflow.net/questions/87445/siegels-mass-formula-for-ternary-indefinite-quadratic-forms/87490#87490 Comment by emiliocba emiliocba 2012-02-04T00:13:25Z 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. http://mathoverflow.net/questions/86456/how-many-solutions-are-there-to-sum-i13-x-i2x-iy-iy-i2k/86543#86543 Comment by emiliocba emiliocba 2012-01-24T20:35:18Z 2012-01-24T20:35:18Z Your formula works for $k\leq 10^4$. This is a great answer! Thank you for your time. http://mathoverflow.net/questions/86456/how-many-solutions-are-there-to-sum-i13-x-i2x-iy-iy-i2k Comment by emiliocba emiliocba 2012-01-23T18:15:56Z 2012-01-23T18:15:56Z Thank Noam, I had not been able to explain it. Do you know some reference in where this formula &quot;could be&quot;? http://mathoverflow.net/questions/84812/values-of-dirichlet-l-funcions-at-natural-numbers Comment by emiliocba emiliocba 2012-01-03T17:27:48Z 2012-01-03T17:27:48Z yes! Thank you Robert.-. http://mathoverflow.net/questions/54027/mass-of-spinor-genus-positive-integral-quadratic-forms Comment by emiliocba emiliocba 2012-01-03T13:37:59Z 2012-01-03T13:37:59Z Isn't the mass a genus invariant? http://mathoverflow.net/questions/84553/norm-map-and-units-in-local-rings Comment by emiliocba emiliocba 2011-12-30T14:51:42Z 2011-12-30T14:51:42Z Chandan, thank you for your comments. I changed the label as you suggested. http://mathoverflow.net/questions/84482/maximal-tori-and-group-structures-on-spheres Comment by emiliocba emiliocba 2011-12-29T00:22:05Z 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 $n\geq2$. http://mathoverflow.net/questions/83982/finitely-many-orbits-of-integer-solutions-module-unimodular-groups/84014#84014 Comment by emiliocba emiliocba 2011-12-28T18:23:52Z 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 $3$.