$p$-twisted Selmer near-companion curves

Technical Summary of "p-TWISTED SELMER NEAR-COMPANION CURVES"

Problem Statement and Context
The paper addresses the relationship between the arithmetic properties of elliptic curves and their Selmer groups under twisting. Building on the work of Mazur and Rubin [7], who defined n-Selmer near-companion curves, the paper investigates the relationship between the boundedness of Selmer rank differences and the isomorphism of torsion subgroups. Mazur and Rubin conjectured that if two elliptic curves $E_1$ and $E_2$ over a number field $K$ are $n$-Selmer near-companions (meaning the difference in the dimensions of their $n$-Selmer groups under quadratic twists is bounded), then their $n$-torsion subgroups $E_1[n]$ and $E_2[n]$ must be isomorphic as $G_K$-modules.

While Yu [11] proved this conjecture for the case $n=2$, the general case for odd primes $p$ remained open. A key difficulty noted in the introduction is that the equality of the fields generated by the torsion points, $K(E_1[p]) = K(E_2[p])$, does not automatically imply a $G_K$-module isomorphism $E_1[p] \cong E_2[p]$. To bridge this gap and extend the framework, the author introduces the concept of $p$-twisted Selmer near-companions ($p$-TSNC). Unlike the original definition which considers only quadratic twists, $p$-TSNC considers twists by all characters $\chi \in \text{Hom}(G_K, \mu_p)$.

Methodology
The core strategy involves analyzing the behavior of Selmer ranks under $p$-twists. The author defines the $p$-twisted Selmer group $S_p(E/K, \chi)$ and investigates the difference in dimensions $r_p(E_1, \chi) - r_p(E_2, \chi)$.

The proof relies on a "density argument" utilizing Chebotarev's Density Theorem. The methodology proceeds as follows:

1. Field Extension Analysis: The author assumes $K(E_1[p]) \neq K(E_2[p])$ and analyzes the Galois representations associated with $E_1$ and $E_2$. The proof is divided into cases based on the structure of the fields $M_i = K(E_i[p])$ and the $p$-torsion points $E_i(K)[p]$.
2. Local-Global Principles: Using tools from global class field theory and the properties of local Selmer conditions (specifically the images of Kummer maps $\gamma_v(\chi_v)$), the author constructs global characters $\chi$ that satisfy specific local conditions at carefully chosen primes $q$.
3. Rank Manipulation: By selecting primes $q$ with specific Frobenius properties (e.g., $q \in P_2(E)$ where the Frobenius acts trivially on $E[p]$), the author constructs characters that increase the Selmer rank of one curve by 2 while leaving the other unchanged, or increasing them by different amounts.
4. Inductive Construction: Through a series of lemmas (Sections 3–8), the paper demonstrates that if $K(E_1[p]) \neq K(E_2[p])$, one can find infinitely many characters $\chi$ such that the difference $|r_p(E_1, \chi) - r_p(E_2, \chi)|$ grows arbitrarily large. This contradicts the definition of $p$-TSNC, which requires this difference to be bounded by a constant $C$.

Key Contributions and Results
The paper introduces the definition of $p$-twisted Selmer near-companions and establishes the following main theorems:

• Theorem 1.6 (Non-existence of $p$-TSNC with distinct fields): If $E_1$ and $E_2$ are $p$-TSNC over $K$, then $K(E_1[p]) = K(E_2[p])$, provided at least one of the following conditions holds:

  • $E_i[p] \subset E_i(K)$ for some $i$.
  • $E_i(K)[p] \cong \mathbb{Z}/p\mathbb{Z}$ for $i=1,2$.
  • $E_1(K)[p] \cong \mathbb{Z}/p\mathbb{Z}$ and $K(\mu_p) \subsetneq M_2$.
  • The degree $[M_i : K(\mu_p)]$ does not divide $p$ for $i=1,2$.
  • $\mu_p \subset K$.
    Essentially, under these conditions, the property of being $p$-TSNC forces the fields generated by the $p$-torsion points to be identical.

• Theorem 1.7 (Sufficiency of Isomorphism): If there exists a $G_K$-module isomorphism $E_1[p] \cong E_2[p]$ (for $p \ge 3$), then $E_1$ and $E_2$ are $p$-TSNC over $K$. This confirms the converse direction of the main conjecture in the context of the new definition.

• Corollaries: The paper derives that if $\mu_p \subset K$ and $E_i(K)[p] \cong \mathbb{Z}/p\mathbb{Z}$, then $p$-TSNC implies $E_1[p] \cong E_2[p]$ as $G_K$-modules (Corollary 1.9). Similarly, if $E_1[p] \subset E_1(K)$, $p$-TSNC implies isomorphism (Corollary 1.10).

Significance
The paper extends the framework of Mazur and Rubin by generalizing the twisting mechanism from quadratic characters to all characters into $\mu_p$. The primary significance lies in proving that for a wide range of elliptic curves (covering cases where the $p$-torsion is rational, partially rational, or where the Galois image is large), the condition of being $p$-twisted Selmer near-companions is sufficient to force the equality of the fields $K(E_1[p])$ and $K(E_2[p])$.

The author notes that for elliptic curves without complex multiplication, Faltings' theorem implies that $K(E_1[p]) \neq K(E_2[p])$ for infinitely many primes if the curves are not isogenous. Consequently, the paper suggests that if two non-isogenous curves are $p$-TSNC for all but finitely many primes, they must be isogenous. The work provides a rigorous partial resolution to the Mazur-Rubin conjecture in the context of $p$-twisted Selmer groups, specifically establishing the link between bounded Selmer rank differences and the equality of torsion fields under specific arithmetic constraints.