arXiv subjects

Mulun Yin

Publications and source records attributed to Mulun Yin.

7 recordsLinked to original sources

arXiv2026

On the Iwasawa invariants of admissible quadratic twists of elliptic curves at ordinary Eisenstein primes

We study the Iwasawa $μ$- and $λ$-invariants of quadratic twists of rational elliptic curves at Eisenstein primes $p\geq 3$. We verify Greenberg's $μ=0$ conjecture for all such twists, which gives an Euler system free proof of Mazur's cotorsion conjecture, and study the $λ$-invariants when the twist is admissible in the sense of [GV00]. As a consequence, we prove the cyclotomic Iwasawa Main Conjecture and obtain the $p$-part BSD formulae in rank $0$ for admissible twists under mild conditions, covering many elliptic curves of potentially ordinary reduction.

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arXiv2026

On Greenberg's conjecture for rational elliptic curves at Eisenstein primes

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $μ$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}_p$-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic $μ$-invariants.

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arXiv2026

On Conjectures of $μ$-Invariant for Selmer Groups at $p=2$

Let $E$ be an elliptic curve define over $\mathbb{Q}$. Further assume $E$ has good ordinary reduction at $p=2$. In this article, we prove Greenberg's conjecture on the value of the algebraic Iwasawa $μ$-invariant when $E$ has a rational isogeny of degree 2, and obtain a sharp absolute upper bound of the $2$-adic $μ$-invariants for all such curves. We also prove the vanishing of the algebraic $μ$-invariant of the fine Selmer group introduced by Coates and Sujatha at $p=2$.

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arXiv2026

Computing the Iwasawa $μ$-Invariants for Elliptic Curves over $\mathbb{Z}_2$-Extensions

Let $E$ be an elliptic curve defined over rational numbers. Further assume $E$ has good ordinary reduction at $p=2$ and $E[4]$ is reducible as a $G_\mathbb{Q}$-representation. In this paper, we offer sufficient and necessary computational criteria for the algebraic Iwasawa $μ$-invariant over the cyclotomic $\mathbb{Z}_2$-extension of rational number to be $0$, $1$ or $\geq 2$. We also show that elliptic curves with isomorphic mod-$4$ representations have algebraic $μ$-invariant equal to $0$ if one of the curves does.

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arXiv2025

On the Tamagawa number conjecture for newforms at Eisenstein primes

We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods. Under standard hypotheses (i.e. the injectivity of the $p$-adic Abel-Jabobi map and the non-degeneracy of the Gillet-Soulé height pairing), we also discuss some partial results towards a rank $1$ result. A conditional higher weight $p$-converse theorem to Gross-Zagier-Zhang-Kolyvagin-Nekovář is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures.

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arXiv2024

On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Let $f$ be a newform of weight $k=2r$ and level $N$ with trivial nebentypus. Let $\mathfrak{p}\nmid 2N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ such that the self-dual twist of the mod-$\mathfrak{p}$ Galois representation of $f$ is reducible with constituents $ϕ,ψ$. Denote a decomposition group over the rational prime $p$ below $\mathfrak{p}$ by $G_p$. We remove the condition $ϕ|_{G_p} \neq \mathbf{1}, ω$ from [CGLS22], and generalize their results to newforms of higher weights $2r$ with $r$ being odd. As a consequence, we prove some Iwasawa Main Conjectures and get the $p$-part of the strong BSD Conjecture for elliptic curves of analytic rank $0$ or $1$ over $\mathbf{Q}$ in this setting. In particular, non-trivial $p$-torsion is allowed in the Mordell--Weil group. Using Hida families, we also prove an Iwasawa Main Conjecture for newforms of weight $2$ of multiplicative reduction at Eisenstein primes. In the above situations, we also get $p$-converse to the theorems of Gross--Zagier--Kolyvagin. The $p$-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a $3$-isogeny.

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arXiv2024

$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\mathbf{Q}$. As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$.

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