Cohomology of quaternions on an abelian variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:01:23Z http://mathoverflow.net/feeds/question/11777 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11777/cohomology-of-quaternions-on-an-abelian-variety Cohomology of quaternions on an abelian variety TonyS 2010-01-14T20:43:52Z 2010-02-01T11:39:18Z <p>Given two non-isogenous elliptic curves $E_1$ and $E_2$ over $\mathbb{C}$. Set $A:=E_1 \times E_2$. Given a nontrivial sheaf of quaternion algebras $D$ over $A$, what is the dimension of the vector space $H^1(A,D)$?</p> <p>If one thinks of $D$ as an element in the Brauer group $Br(A)$, then it is $2$-torsion, hence belongs to $Br(A)[2]$. Since the curves are non-isogenous there is an isomorphism $Pic(E_1)[2] \otimes Pic(E_2)[2] \to Br(A)[2]$. So there should be a connection between such quaternions and $2$-torsion line bundles on the curves, but i cannot find an explicit description for this isomorphism. If there is one, i thought one could use the Künneth formula to compute $H^1(A,D)$ in terms of the cohomology of the line bundles on the curves.</p> <p>For now i could only work out the bound $d=dim(H^1(A,D)) \geq 2$: using Hirzebruch-Riemann-Roch and simplifying terms one gets $d=c_2(D)+2$. After a result of M.Lieblich one has $c_2(D)\geq 0$. </p> <p>Does anyone see/have an explicit description of the isomorphism mentioned above? Is the idea using Künneth a promising approach to this problem at all? Or does anyone have another approach? Are there some calculations regarding this in the literature (i couldn't find one)?</p> <p>Another question in this context is: what is the image of such an algebra under the map $Br(A) \rightarrow Br(\mathbb{C}(A))$. This should be nontrivial $\mathbb{C}(A)$-quaternions, since the map "looking at the genric point $\eta$" is injective, i.e. $D_{\eta}$ is generated by elements $i,j$ with $i^2=a, j^2=b and ij=-ji$. But what are a resp. b? I think they should have something to do with functions h such that 2*Y=div(h), where Y defines one of the line bundles. Is this true?</p> http://mathoverflow.net/questions/11777/cohomology-of-quaternions-on-an-abelian-variety/13158#13158 Answer by Oren Ben-Bassat for Cohomology of quaternions on an abelian variety Oren Ben-Bassat 2010-01-27T18:27:48Z 2010-01-27T18:27:48Z <p>To compute the cohomology of $D$, you could consider its pullback via some finite to one map $A\to A$ to for which $D$ trivializes. Then there is a Lyndon-Serre spectral sequence using the group cohomology of $G$ the group you quotient. The $E_{2}$ term looks like $H^{i}(G,H^{j}(A,p^{-1}D))$ and it converges to $H^{i+j}(A,D)$. See the book by Mumford Abelian Varieties. There are 3 other helpful references that you can find in <a href="http://arxiv.org/abs/0811.2746" rel="nofollow">http://arxiv.org/abs/0811.2746</a>. See the articles cited in that paper by Hoobler, Elencwajg and Narasimhan, and Berkovich. </p> <p>The map from the tensor product of the Picard Groups to the Brauer group can also probably be made explicit using group or Cech Etale cohomology. </p> <p>--Oren</p> http://mathoverflow.net/questions/11777/cohomology-of-quaternions-on-an-abelian-variety/13160#13160 Answer by Tyler Lawson for Cohomology of quaternions on an abelian variety Tyler Lawson 2010-01-27T18:47:05Z 2010-01-28T16:24:43Z <p>For the description of the quaternion algebra associated to a pair of torsion line bundles, try the following. Take line bundles <code>${\cal L}_i$</code> on <code>$E_i$</code> equipped with isomorphisms <code>${\cal L}_i^{\otimes 2} \to {\cal O}$</code>, and pull these back to $A$. Define <code> $$D = {\cal O} \oplus {\cal L}_1 \oplus {\cal L}_2 \oplus {\cal L}_1 \otimes {\cal L}_2$$ </code> with multiplication induced by the maps <code>${\cal L}_i^{\otimes 2} \to {\cal O}$</code>, ${\cal O}$ being the unit, and the elements of <code>${\cal L}_1$</code> and <code>${\cal L}_2$</code> anticommuting.</p> <p>ADDED: I wish I had a more conceptual explanation for why this represents the cup-product in the Brauer group, but here is a cocycle description along the lines of what Oren suggested.</p> <p>Suppose $X$ is given with 2-torsion line bundles $\cal L$ and $\cal M$. Choose cocycles representing these, in the form of a cover (either open in the analytic case, or etale in the algebraic case) <code>$U_\alpha$</code> of $X$ together with sections <code>$s_\alpha$</code> of $\cal L$ and $t_\alpha$ of $\cal M$ on <code>$U_\alpha$</code> such that <code>$s_\beta / s_\alpha = u_{\alpha \beta} \in \{\pm 1\}$</code> and similarly <code>$t_\beta / t_\alpha = v_{\alpha \beta}$</code>; these latter two are the representing cocycles.</p> <p>Then D has basis <code>$\{1,s_\alpha, t_\alpha, s_\alpha t_\alpha\}$</code> on <code>$U_\alpha$</code>, where <code>$s_\alpha^2 = t_\alpha^2 = 1$</code>, and you can explicitly make this isomorphic to a matrix algebra. The change-of-basis sends <code>$s_\alpha$</code> to <code>$s_\beta = u_{\alpha \beta} s_\alpha$</code> and similarly for $t$. This can be achieved by conjugation by the element <code>$t_\alpha^{(1 - u_{\alpha \beta})/2} s_\alpha^{(1 - v_{\alpha \beta})/2} = g_{\alpha\beta} \in D \cong M_2(\mathbb{C})$</code>. These change-of-basis matrices reduce to a cocycle in <code>$PGL_2(\mathbb{C})$</code> representing the algebra, and so the image in the Brauer group is represented by the coboundary <code>$(\delta g)_{\alpha \beta \gamma} \in \{\pm 1\}$</code>.</p> <p>EDIT: fixed up following description of the coboundary so that it correctly described where the cup product lands.</p> <p>Explicit computation finds <code>$(\delta g)_{\alpha \beta \gamma}$</code> is equivalent to the cocycle <code>$v_{\alpha \beta} \otimes u_{\beta \gamma}$</code> with coefficients in <code>$\{\pm 1\} \otimes \{\pm 1\} \cong \{\pm 1\}$</code> (I may have mixed the indices, if I did please let me know and I'll correct it), which is precisely the formula for the cup product of the cocycles $u$ and $v$.</p> <p>So based on your description of the cup product inducing an isomorphism between 2-torsion in the Brauer group and cup products of 2-torsion elements in the Picard group, this genuinely should provide you with the bundles you're looking for.</p>