Tamagawa numbers of abelian varieties and torsion. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:07:34Z http://mathoverflow.net/feeds/question/58363 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58363/tamagawa-numbers-of-abelian-varieties-and-torsion Tamagawa numbers of abelian varieties and torsion. jvo 2011-03-13T21:21:23Z 2011-03-14T08:57:46Z <p>Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there be between the local Tamagawa number $c_v(A) = [A(K_v): A_0(K_v)]$ and the cardinality of the $p$-primary subgroup $A(K_v)(p)$ of $A(K_v)$? </p> <p>I am aware of the recent work of Lorenzini, which considers possible cancellations of the ratio \begin{align*} \frac{\prod_{v \subset \mathcal{O}_K} c_v(A)}{\vert A(K)_{\operatorname{tors}}\vert}, \end{align*} and shows for instance this ratio for $A$ an elliptic curve defined over $K ={\bf{Q}}$ is always greater than or equal to $\frac{1}{5}$. </p> <p>The question comes up naturally in a certain Euler characteristic computations (e.g. for the <code>$p^{\infty}$</code>-Selmer group of <code>$A$</code> defined over some profinite Galois extension <code>$K_{\infty}$</code> of <code>$K$</code>, where certain primes of <code>$K$</code> are known to split completely, and hence where the torsion subgroup <code>$A(K_{\infty})_{\operatorname{tors}}$</code> is known to be finite). In particular, granted the refined conjecture of Birch and Swinnerton-Dyer, it is apparent from these computations that <code>$c_v(A)$</code> for a prime of bad (multiplicative) reduction <code>$v$</code> is given essentially by the quotient <code>\begin{align*}\frac{\vert H^1(G_w, A(K_{\infty, w}))(p)\vert }{\vert H^2(G_w, A(K_{\infty, w}))(p)\vert }. \end{align*}</code> Here, <code>$K_{\infty, w}$</code> is the union of all completions at primes above <code>$v$</code> in <code>$K_{\infty}$</code>, and <code>$G_w$</code> denotes the Galois group <code>$\operatorname{Gal}(K_{\infty, w}/K_v)$</code>. If <code>$A$</code> has good ordinary reduction at all primes above <code>$p$</code> in <code>$K$</code>, then it is possible to use the Coates-Greenberg theory of deeply ramified extensions to characterize the local factor at <code>$v$</code> in the Euler characteristic formula coming from this quotient as <code>$\vert \widetilde{A}(\kappa_v)(p)\vert^2$</code>, where <code>$\widetilde{A}$</code> is the reduction of <code>$A$</code> mod <code>$v$</code>, and <code>$\kappa_v$</code> the residue field at <code>$v$</code> (which is consistent with B+S-D). But, this characterization seems to break down when <code>$A$</code> does not have good ordinary reduction at <code>$v$</code>, and I have not found any calculations in the literature for this case of bad reduction. Hence why I ask about any possible known relation to torsion. Sorry if the question is silly!</p> http://mathoverflow.net/questions/58363/tamagawa-numbers-of-abelian-varieties-and-torsion/58376#58376 Answer by Chris Wuthrich for Tamagawa numbers of abelian varieties and torsion. Chris Wuthrich 2011-03-14T00:10:02Z 2011-03-14T00:10:02Z <p>There is no general relation between the local $p$-primary torsion and the Tamagawa numbers. I believe one can have $p$-torsion points that map to non-trivial or to the trivial element in the group of components of the Neron model.</p> <p>This should indicate you that, in your Euler characteristic formula, you can not hope to replace the square of the local $p$-primary part with just the Tamagawa numbers when the reduction is not good ordinary. </p> <p>If I understand correctly what you want to do is to compute the cokernel of the universal norms on the local points at a place above $p$. I would start by looking at papers about the case of a $\mathbb{Z}_p$-extension. If the reduction is multiplicative, then the computations were carried out by John W. Jones in Compositio 73 (1990). He has also a paper about the additive case. If your reduction is good but not ordinary, then it is a completely different story since the cokernel is not finite. I am sure there are many other references</p>