# What is the status of the equidistribution root numbers of elliptic curves' L-functions

In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated

The Bhargava Conjecture: For each $n > > 1$ the average size of $S_{n}(E/\mathbb{Q})$ is $\displaystyle\sum\limits_{d|n} d$.

Here $S_{n}(E/\mathbb{Q})$ is the $n$-Selmer group of $E/\mathbb{Q}$. Silverberg remarks that assuming the Bhargava conjecture for infinitely many $n$, the parity conjecture, and the equidistribution of root numbers of $L$-functions of elliptic curves over $\mathbb{Q}$, it follows that $50$% of elliptic curves have rank $0$ and $50$% of elliptic curves have rank $1$ (this is called the Rank Distribution Conjecture).

Does anyone have a conjectural strategy for proving the equidistribution of root numbers?

Silverberg mentions the Poonen-Rains conjecture together with the parity conjecture implying the Rank Distribution Conjecture (which in turn implies equidistribution of root number), so it could be that trying to prove the Poonen-Rains conjecture offers a possible approach, but it seems to me that one in fact needs the equidistribution conjecture as a hypothesis in the latter statement of Silverberg...

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This was the topic of Harald Helfgott's thesis. arxiv.org/abs/math.NT/0305435 See his website for some later papers on the subject, where he proves equiparity in some families. I think a big obstruction is proving something like Chowla's conjecture that the parity of the number of prime divisors of a random integer is equidistributed. – Jamie Weigandt Oct 14 '12 at 1:15
Also, Poonen-Rains for just one p implies that the parity of the p-Selmer rank is equidistributed. By a theorem of the Dokchitser brothers, the parity of the p-Selmer rank is exactly the parity of the analytic rank, except sometimes when there is p-torsion, which happens with density 0, so equidistribution of the root numbers follows. – Jamie Weigandt Oct 14 '12 at 1:22