Question

The parity conjecture for elliptic curves predicts that the rank of an elliptic curve defined over the rationals has the same parity as the p-Selmer rank for a prime number p. Could anyone familiar with the recent development sketch what has happened in the last few years, and what the state of the art is?

Answer 1

For convenience, restrict to elliptic curves over $\mathbf{Q}$ (there are more general results/conjectures over number fields). There are three possible parities one could consider:

(i) The parity of the rank of $E(\mathbf{Q})$.

(ii) The parity of the $p$-Selmer rank of $E$ for a prime $p$.

(iii) The parity of the order of vanishing of the $L$-function of $E$ (determined by the root number).

Conjecturally these parities are all the same, and this follows from the BSD conjecture. The parity conjecture usually refers to the claim that (i) and (iii) are the same. The claim that (i) and (ii) are the same (your question) follows from the conjecture that Sha(E) is finite (it is almost equivalent to this conjecture), and this is only known in the cases where one can (essentially) prove the entire BSD for $E$ (for example, by work of Kolyvagin). The conjecture that (ii) and (iii) are the same is also sometimes called the parity conjecture. There has been progress on this question in recent years by Nekovar, and most recently by the brothers Dokchister. The current state of the art is that (ii) and (iii) are now known to be the same for all elliptic curves over $\mathbf{Q}$. A survey article by Tim Dokchitser can be found here:

http://arxiv.org/pdf/1009.5389v1

Answer 2

Let me add a few remarks to the very nice CW answer already given.

• The parity conjecture (i.e. algebraic rank equals analytic rank modulo 2) is known for all elliptic curves over number fields (not just over $\mathbb{Q}$) under the assumption that Tate-Shafarevich groups of elliptic curves over number fields are finite. The survey by Tim Dokchitser that has already been linked to describes the proof.

• There are refined parity conjectures for twists by Artin representations. I will also take this opportunity to explain the content of the $p$-parity conjectures in a little more detail. Let $A/K$ be an Abelian variety and let $\tau$ be an Artin representation of $G_K$. Let $p$ be a prime number. Consider the Pontryagin dual of the $p^{\infty}$-Selmer group of $A/K$ and take the tensor product with $\mathbb{Q}_p$, call this $\chi_p(A/K)$. This is a $\mathbb{Q}_p$-vector space. If we believe that the $p$-primary part of the Tate-Shafarevich group of $A/K$ is finite, then the $\mathbb{Q}_p$-dimension of $\chi_p(A/K)$ is exactly the rank of $A(K)$. If we don't assume this, then we have to allow for the possibility of some copies of $\mathbb{Q}_p/\mathbb{Z}_p$ inside the Tate-Shafarevich group increasing the dimension. In any case, $\chi_p(A/K)$ is a $G_K$-representation, and we can consider the number of copies of $\tau$ inside it: $\langle\tau,\chi_p(A/K)\rangle$. On the analytic side, we have the twisted $L$-function $L(A/K,\tau,s)$ and its root number $w(A/K,\tau)$. The $p$-parity conjecture for twists now predicts that $$ (-1)^{\langle\tau,\chi_p(A/K)\rangle} = w(A/K,\tau). $$ If we believe in the finiteness of Tate-Shafarevich groups, we could instead work with the $\tau$-isotypical component of the Mordell-Weil group $A(K)$. Anyway, what I wanted to say is that we now know the $p$-parity conjecture for various different twists, here are some examples: DD1, Theorems 1.3, 1.11, 1.12, DD2, Theorem 1.11.