REVIEW
On the Iwasawa invariants of admissible quadratic twists of elliptic curves at ordinary Eisenstein primes
Number Theory · Submitted 30 Oct 2026 · 66 references
Pith's verdict: Real extension of Greenberg-Vatsal to ramified admissible twists, but Theorem 4.1.1 has a p=3 gap and the mu=0 proof depends on a companion preprint.
T0 · AI review
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Referee report
The paper
The paper claims
Admissible quadratic twists prove the cyclotomic Iwasawa Main Conjecture for elliptic curves whose mod-$p$ representation is reducible (an Eisenstein prime, $p\geq 3$), provided the twisted curve has no global $p$-torsion in its isogeny…
The reading
What carries the argument
The load-bearing object is the filtration $0\to\mathbb{F}_p(\varphi)\to E[p]\to\mathbb{F}_p(\psi)\to 0$ with $\varphi\psi=\omega$, the mod-$p$ cyclotomic character, together with the admissibility condition that the quadratic character $\chi$ have parity opposite to the unramified character among $\varphi,\psi$. Admissibility makes two choices agree: the Selmer group dictated by ramification (relaxed for the ramified character, strict for the unramified one) equals the Selmer group dictated by parity to which Mazur–Wiles's theorem applies. For the non-admissible $\mu=0$ case the key is Ribet's lemma, which lets one move to an isogenous curve in which the odd ramified character is a quotient rather than a submodule of $E[p]\otimes\chi$, converting a potentially positive $\mu$ in a quotient into vanishing for the whole Selmer group. The control theorem of Section 5 computes the constant term of the generator of the Selmer dual, and comparison with Kato's branched $p$-adic $L$-function at the $\omega^{(p-1)/2}$ branch gives the BSD formula.
Extended reading notes
Core claim
The central claim is Theorem 4.1.1: if $E$ has ordinary reduction at $p\geq 3$ with $E[p]$ reducible over $G_{\mathbb{Q}}$, and $\chi$ is a ramified quadratic admissible twist with $H^0(\mathbb{Q}, E'[p]\otimes\chi)=0$ for all $E'$ isogenous to $E$, then the cyclotomic Iwasawa Main Conjecture holds for the twist, i.e. $\mathrm{Char}_{\Lambda}(\mathrm{Sel}(E^{K_\chi}/\mathbb{Q}_{\mathrm{cyc}})^\vee)=(L_p(E,\chi))$ in $\Lambda=\mathbb{Z}_p\llbracket T\rrbracket$. The proof splits according to the sign of the admissible twist: in both the $+$ and $-$ admissible cases the residual Selmer group of $E[p]\otimes\chi$ sits in a short exact sequence between the relaxed Selmer group of an even character and the strict Selmer group of an odd character, so Mazur–Wiles vanishing of $\mu$ and equality of $\lambda$'s for characters transfer to the curve, and Kato's one-sided divisibility is upgraded to the equality. The $\mu=0$ statement for non-admissible twists is obtained separately, without comparing $\lambda$-invariants.
Load-bearing premise
The proof of Greenberg's $\mu=0$ conjecture for non-admissible twists rests on the claim that two specific kernels (in steps (i) and (ii) of Section 4.2) are finite, and this is delegated to a companion preprint [LY26] rather than shown here; if that finiteness argument does not carry over to the field $L=\mathbb{Q}(E[p])K_\chi$, Theorem A and the cotorsion corollary collapse.
Editorial extensions
If this is right
- For any ordinary elliptic curve with reducible $E[p]$, every admissible ramified quadratic twist satisfies the cyclotomic Iwasawa Main Conjecture, turning Kato's divisibility into an equality under the stated torsion hypothesis.
- Greenberg's $\mu=0$ conjecture holds for all quadratic twists of ordinary curves at Eisenstein primes, covering every potentially multiplicative curve and many potentially good ordinary curves, with an Euler-system-free proof.
- Mazur's cotorsion conjecture follows over any abelian number field for these twists, as a direct corollary of the $\mu=0$ result.
- The rank-$0$ $p$-part BSD formula holds for the twisted curves, including all potentially multiplicative curves at $p=3$ whose isogeny class has no global $3$-torsion.
- The $\lambda$-invariant of the twisted Selmer group is the sum of the $\lambda$-invariants of the two characters $\varphi\chi,\psi\chi$, giving an explicit additive formula in the admissible cases.
Reading between the lines
- The $\mu=0$ mechanism for non-admissible twists is more general than the statement here suggests: because its only obstruction is the relaxed Selmer group of an odd ramified character, the same Ribet-lemma quotient move should handle non-quadratic and higher-conductor twists, potentially covering all potentially ordinary Eisenstein primes.
- The hard boundary in this family is admissibility, not ordinariness: where admissibility fails, both the algebraic and analytic comparisons genuinely break (positive $\mu$, non-additive $\lambda$, failed congruences), so a full Main Conjecture for non-admissible twists likely needs a genuinely new input such as a twisted Beilinson–Flach method, which the paper states is unclear.
- The hypothesis $H^0(\mathbb{Q},E'[p]\otimes\chi)=0$ is the only point where the classical theory of the trivial character $\mathbb{F}_p(1)$ enters; removing it would extend the Main Conjecture and hence the BSD formula to a broader set of isogeny classes.
- The constant-term computation in Theorem 5.0.2, being valid with torsion and for arbitrary ramified quadratic twists, is a reusable tool: any future proof of the Main Conjecture for a wider class of twists would automatically upgrade to a rank-$0$ BSD formula for those twists.
Original abstract
We study the Iwasawa $\mu$- and $\lambda$-invariants of quadratic twists of rational elliptic curves at Eisenstein primes $p\geq 3$. We verify Greenberg's $\mu=0$ conjecture for all such twists, which gives an Euler system free proof of Mazur's cotorsion conjecture, and study the $\lambda$-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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study the Iwasawa $\mu$- and $\lambda$-invariants of quadratic twists of rational elliptic curves at Eisenstein primes $p\geq 3$. We verify Greenberg's $\mu=0$ conjecture for all such twists, which gives an Euler system free proof of Mazur's cotorsion conjecture, and study the $\lambda$-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the Iwasawa $\mu$- and $\lambda$-invariants of quadratic twists of rational elliptic curves at Eisenstein primes $p\geq 3$. | contribution 0.97 | 0.90 | 0.05 | asserted 0.83 |
| s1 | We verify Greenberg's $\mu=0$ conjecture for all such twists, which gives an Euler system free proof of Mazur's cotorsion conjecture, and study the $\lambda$-invariants when the twist is admissible in the sense of [GV00]. | result 0.61 | 0.93 | 0.45 | asserted 1.00 |
| s2 | 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. | result 0.92 | 0.95 | 0.59 | asserted 1.00 |
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References
Discussion
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