[Paper Review] Euler Characteristics and their Congruences for Multi-signed Selmer Groups

Anwesh Ray, R. Sujatha|arXiv (Cornell University)|Nov 10, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes congruence relations for truncated Euler characteristics of multi-signed Selmer groups in the context of $p$-congruent elliptic curves over number fields with mixed semistable reduction at $p$. By analyzing imprimitive Selmer groups and leveraging Iwasawa invariants, the authors prove that when the $\mu$-invariants vanish, the $\lambda$-invariants and truncated Euler characteristics of $p$-congruent curves are congruent modulo $p$, extending prior results to the mixed-reduction setting with optimal sets of primes.

ABSTRACT

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ be elliptic curves over a number field $F$ with semistable reduction at all primes $v|p$ such that the $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ and $E_2[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_1$ and $E_2$. Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb{Z}_p^m$-extensions of $F$ are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$.

Motivation & Objective

• To generalize Greenberg-Vatsal and B.D. Kim's results on $p$-congruent elliptic curves to the case of mixed semistable reduction at $p$.
• To study the relationship between Iwasawa invariants of imprimitive multi-signed Selmer groups for $p$-congruent elliptic curves $E_1$ and $E_2$ over a number field $F$.
• To establish congruence relations for truncated Euler characteristics of these Selmer groups over $\mathbb{Z}_p^m$-extensions.
• To identify an optimal set of primes $\Sigma_1$ smaller than the full bad reduction set $\Sigma_0$, such that imprimitive invariants match under $p$-congruence.
• To verify these congruences via explicit examples over $\mathbb{Q}$ and $\mathbb{Q}(i)$, demonstrating divisibility of $\Phi_{E,\Sigma_1}$ and matching Euler characteristics.

Proposed method

• Define multi-signed Selmer groups $\operatorname{Sel}^{\ddagger}(E/F^{\operatorname{cyc}})$ indexed by $\ddagger \in \{+,-\}^d$, where $d$ is the number of primes above $p$ with good supersingular reduction.
• Use imprimitive Selmer groups with respect to a set $\Sigma_1$ of primes $v \nmid p$ where $E_1$ or $E_2$ has bad reduction, to compare Iwasawa invariants.
• Apply techniques from Iwasawa theory over $\mathbb{Z}_p^m$-extensions, assuming the Selmer groups lie in the category $\mathfrak{M}_H(G)$.
• Leverage the structure of the Iwasawa algebra and the theory of characteristic ideals to compare $\mu$- and $\lambda$-invariants of $E_1$ and $E_2$.
• Define the truncated Euler characteristic $\chi_t(G,E)$ as the Euler characteristic of the $p$-primary Selmer group when the rank is positive, and relate it to $L$-values via $\Phi_{E,\Sigma_1}$.
• Use explicit computation of $L$-factors and Tamagawa products to verify congruences in examples, particularly showing $5$-divisibility of $\Phi_{E_1,\Sigma_1}$.

Experimental results

Research questions

• RQ1 Under what conditions do $p$-congruent elliptic curves with mixed semistable reduction at $p$ have isomorphic $\mu$-invariants for their imprimitive multi-signed Selmer groups?
• RQ2 Can the $\lambda$-invariants of imprimitive multi-signed Selmer groups of $p$-congruent elliptic curves be shown to match when their $\mu$-invariants vanish?
• RQ3 How do truncated Euler characteristics of multi-signed Selmer groups behave under $p$-congruence in the mixed-reduction case?
• RQ4 Can an optimal set of primes $\Sigma_1 \subset \Sigma_0$ be constructed such that the imprimitive Iwasawa invariants of $E_1$ and $E_2$ match, and $\Sigma_1$ is strictly smaller than $\Sigma_0$?
• RQ5 What is the relationship between the truncated Euler characteristic $\chi_t(G,E)$ and the product $\Phi_{E,\Sigma_1}$, and how does this relate to congruences modulo $p$?

Key findings

• If the $\mu$-invariant of the imprimitive multi-signed Selmer group of $E_1$ is zero, then the $\mu$-invariant of $E_2$ is also zero, under $p$-congruence.
• When both $\mu$-invariants vanish, the $\lambda$-invariants of the imprimitive multi-signed Selmer groups of $E_1$ and $E_2$ are equal.
• The truncated Euler characteristics $\chi_t(\Gamma, E_1)$ and $\chi_t(\Gamma, E_2)$ are congruent modulo $p$ when $r_{E_1}^\ddagger < r_{E_2}^\ddagger$, as shown in Example 2.
• In Example 1 over $\mathbb{Q}$, both $\chi^\pm(\Gamma, E_1)$ and $\chi^\pm(\Gamma, E_2)$ are equal to 1, demonstrating matching Euler characteristics.
• In Example 2 over $\mathbb{Q}(i)$, $\Phi_{E_1,\Sigma_1}$ is divisible by 5, and since $r_{E_1}^\ddagger = 0 < r_{E_2}^\ddagger = 1$, Theorem 5.5 implies $5$ divides $\Phi_{E_1,\Sigma_1} \times \chi(\Gamma, E_1)$, which is verified by explicit computation.
• Over the $\mathbb{Z}_p^2$-extension $\mathcal{F}_\infty$, Corollary 6.6 confirms that $\chi_t(G, E_1) = \chi_t(G, E_2)$, establishing congruence of truncated Euler characteristics.