[Paper Review] The $p$-parity conjecture for elliptic curves with a $p$-isogeny

Kęstutis Česnavičius|arXiv (Cornell University)|Jul 2, 2012

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper proves the $p$-parity conjecture for elliptic curves over number fields that possess a $p$-isogeny, completing the conjecture for all primes $p > 3$ by removing the prior semistable reduction restriction. The proof relies on establishing a conjectural formula for local root numbers in non-semistable cases and applies to complex multiplication elliptic curves, showing that if the $p$-primary Shafarevich-Tate group is infinite, it must contain $\mathbb{Q}_p/\mathbb{Z}_p$ as a direct summand twice.

ABSTRACT

For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\\mathrm{Sha}(E/K)[p^\\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\\mathbb{Z}_p$-corank of the $p^\\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \\le 3$ were known). T. and V. Dokchitser have showed this in the case when $E$ has semistable reduction at all places above $p$ by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the $p$-parity conjecture holds for every $E$ with complex multiplication defined over $K$. Consequently, if for such an elliptic curve $\\mathrm{Sha}(E/K)[p^\\infty]$ is infinite, it must contain $(\\mathbb{Q}_p/\\mathbb{Z}_p)^2$.

Motivation & Objective

• To complete the proof of the $p$-parity conjecture for elliptic curves with a $p$-isogeny over number fields, removing the prior restriction to semistable reduction at primes above $p$.
• To establish the conjectural formula for local root numbers in non-semistable reduction cases, thereby generalizing results by Dokchitser and Dokchitser.
• To show that the $p$-parity conjecture holds for all elliptic curves with complex multiplication defined over a number field $K$, regardless of $p$.
• To deduce that if $\Sha(E/K)[p^\infty]$ is infinite for such curves, it must contain $\mathbb{Q}_p/\mathbb{Z}_p$ as a direct summand with multiplicity at least two.

Proposed method

• Proves the conjectural formula for local root numbers in non-semistable reduction cases using Galois representations and local class field theory.
• Applies the formula to extend the $p$-parity conjecture to all $p$-isogeny cases by verifying that the global root number matches the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group.
• Uses the fact that the $p$-parity conjecture is invariant under odd-degree Galois extensions to reduce the problem to curves acquiring a $p$-isogeny over such extensions.
• Applies the theory of Hecke characters and complex multiplication to show that the global root number is $1$ for CM elliptic curves over $K$.
• Leverages isogeny invariance of both the $p$-Selmer rank and the root number to reduce to the case where the curve has maximal $\mathcal{O}_F$-action.
• Analyzes the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group via the action of the imaginary quadratic field $F$ on the $p$-adic Tate module.

Experimental results

Research questions

• RQ1 Does the $p$-parity conjecture hold for elliptic curves with a $p$-isogeny over a number field when the reduction at $p$-adic places is not semistable?
• RQ2 Can the conjectural formula for local root numbers in non-semistable cases be proven to establish the $p$-parity conjecture?
• RQ3 Does the $p$-parity conjecture hold for all elliptic curves with complex multiplication defined over a number field $K$?
• RQ4 What is the structure of $\Sha(E/K)[p^\infty]$ if it is infinite for a CM elliptic curve over $K$?

Key findings

• The $p$-parity conjecture is proven for all elliptic curves over a number field $K$ that possess a $p$-isogeny, for all primes $p > 3$, without requiring semistable reduction at places above $p$.
• The conjectural formula for local root numbers in non-semistable cases is established, completing the work of Dokchitser and Dokchitser for $p > 3$.
• For every elliptic curve with complex multiplication defined over $K$, the $p$-parity conjecture holds for all primes $p$, due to the global root number being $1$ and the $p$-Selmer rank being even in the $p$-split or inert case.
• If $\Sha(E/K)[p^\infty]$ is infinite for a CM elliptic curve over $K$, then it must contain $\mathbb{Q}_p/\mathbb{Z}_p$ as a direct summand with multiplicity at least two.
• The $p$-parity conjecture is invariant under odd-degree Galois extensions, so it holds for curves that acquire a $p$-isogeny over such extensions.
• The $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group is even when $p$ is inert or ramified in the CM field, and matches the root number when $p$ splits or ramifies, due to the existence of a $p$-isogeny.