MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$ be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture of Beilinson and Bloch predicts that the dimension of $\mathrm{Ch}^j(V)_{0} \otimes \mathbf{Q}$ as a $\mathbf{Q}$-vector space is given by the order of vanishing of the L-function $L(s,H^{2j-1}_{\mathrm{et}}(V \times_{K} \overline{K},\mathbf{Q}_{\ell}))$ at its central critical point. On the other hand, the Bloch-Kato conjecture predicts that this order of vanishing is equal to the dimension of the Bloch-Kato Selmer group $H^1_f(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell})(j))$. So it seems reasonable to ask: is there a natural map

$\phi_j : \mathrm{Ch}^j(V)_{0} \to H^{1}_{f}(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell})(j))$ `

which is "close" to being an isomorphism, which explains the compatibility of these conjectures?

share|cite|improve this question
I deleted my earlier comment, since it seems unnecessary now. – Donu Arapura Mar 2 '11 at 21:23
You might want to make explicit that $K$ is a number field. – Emerton Mar 2 '11 at 21:44
up vote 9 down vote accepted

The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map

$$\phi : Ch^j(V)_0 \to H^1(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell})(j))$$

The map $\phi$ is also called the cycle class map and is defined for any field $K$, say of characteristic $0$.

Now if $K$ is a number field, the conjecture of Bloch-Kato predicts that

(1) $\phi$ takes values in $H^{1}_f(G_K,\cdot)$.

(2) $\phi \otimes \mathbf{Q}_\ell$ is an isomorphism.

In fact (1) is a purely local question : this is really a question about the Abel-Jacobi map associated to a variety over $\mathbf{Q}_p$ (where $p$ can be equal to $\ell$ or not). The conjecture (2) together with the Beilinson-Bloch conjecture implies the statement you mention about the order of vanishing.

You can find a good survey on the $\ell$-adic Abel-Jacobi map and more details about what is known here :

J. Nekovar, p-adic Abel-Jacobi maps and p-adic heights, In: The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), 367 - 379, CRM Proc. and Lect. Notes 24, Amer. Math. Soc., Providence, RI, 2000.

See also the "Conjecture $\mathrm{Mot}_\ell$" in the following article :

M. Flach, The Equivariant Tamagawa Number Conjecture : A Survey.

share|cite|improve this answer
Sorry, I definitely do not manage to parse the formula for $\phi$... Anyway, this is the same as in David's question but with $H^1_f$ replaced by $H^1$. – François Brunault Mar 2 '11 at 18:51
Thanks, this is very clear. – Donu Arapura Mar 2 '11 at 21:38
@Charles Rezk : Thanks for fixing the formula! – François Brunault Mar 2 '11 at 22:33
Thank you, Francois - this is exactly what I was looking for. – David Hansen Mar 2 '11 at 23:58

There is the etale Abel-Jacobi map as defined by Nekovar. (See page 28 of the paper below)

I have no idea - I don't know if it's conjectured, known, or known to be false - if this map provides a compatibility between the conjectures.

share|cite|improve this answer
Yes, the $\ell$-adic Abel-Jacobi map is the right object to consider here. Note that this map lands a priori in $H^1(G_K,\cdot)$. It is conjectured that the image is in fact contained in the subgroup $H^1_f(G_K,\cdot)$. I am not sure this has been proved in full generality (especially in the bad reduction case). – François Brunault Mar 2 '11 at 17:19

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.