Specialisations of the Burungale-Castella-Skinner main conjecture to $\mathbb Z_p$-lines

Fabien Trihan; Ki-Seng Tan; Kwok-Wing Tsoi

REVIEW

Specialisations of the Burungale-Castella-Skinner main conjecture to $\mathbb Z_p$-lines

Ki-Seng Tan, Fabien Trihan, Kwok-Wing Tsoi

Number Theory · Submitted 27 Aug 2026 · 14 references

Pith's verdict: A genuinely new, clean structural result in Iwasawa theory: the BCS two-variable main conjecture specialises to every Z_p-line with an explicit local factor, and the exceptional-line analysis holds up under the stated hypotheses.

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arXiv ID2608.26791 (version 1)

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CitationFabien Trihan, Ki-Seng Tan, Kwok-Wing Tsoi. "Specialisations of the Burungale-Castella-Skinner main conjecture to $\mathbb Z_p$-lines." arXiv:2608.26791 (2026). https://pith.science/paper/ANAJET3N

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Editor's note

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This note does what it says and the new content is real. The BCS two-variable theorem and Tan's specialisation formula are imported, but the exceptional-line notion, its finiteness, the cyclotomic/anticyclotomic dichotomy, and the rank bound are proved here. The core idea that the global factor is trivial because E[p](L')=0 is short and convincing, and the proof chain from Theorem 2.3 and Theorem 3.3 to the linewise identity is explicit. I read the full text and did not find a hidden circular step or any fitted parameter. The self-citation to [Tan14] is to a published Math. Ann. theorem with independent standing, so the citation pattern is not a concern to me.

The main soft spot is the one the reader flagged. The central equality inherits the status of the very recent BCS theorem, and Proposition 4.2 applies BCS Thm 1.1.2(a) to the quadratic twist E_K. That application is plausible — p splits in K, so E and E_K are isomorphic over Q_p, and residual irreducibility is preserved under twist — but it is the one place where I would ask the authors to spell out the hypotheses in a sentence or two. It is a minor citation-context issue, not a mathematical gap. The other caveat is that Theorem 4.7's rank bound depends on finiteness of Sha and non-degeneracy of Schneider's cyclotomic height pairing; that is stated honestly and is consistent with the literature.

The paper is short on purpose. For a specialist in Iwasawa theory, it gives a clean structural statement: one two-variable main conjecture becomes a compatible family of one-variable main conjectures, and the exceptional lines are exactly the Selmer-non-torsion lines. It does not resolve a long-open problem by itself, but it supplies a useful new dimension-reduction tool. I would bring it to a reading group and I would cite it when discussing specialisations of higher-rank Iwasawa main conjectures.

Recommendation: this deserves a serious referee, not a desk reject. If the referee confirms the twist application in Proposition 4.2 and the BCS hypotheses are exactly as stated, the paper is acceptable in essentially this form.

Referee report

Major objections (2)

[§3, Theorem 3.3] The claim that T_p(E[p^∞](L')) = 0 follows from E[p](L') = 0. The authors argue that E[p](L') = 0 by residual surjectivity: if nonzero, G_K would act through the abelian group Gal(L'/K). This is correct, but the step that the Tate module vanishes because the p-primary torsion is trivial could be stated more explicitly (it is not automatic from the vanishing of the p-torsion alone, but a simple induction shows no element of p-power order exists). Including this detail would strengthen the argument.
[§4.3, Proposition 4.5] The deduction that F ∈ a^{e} because each ψ_i − 1 divides F and each ψ_i − 1 ∈ a uses that the primes (ψ_i − 1) are distinct height-one primes in the UFD Λ, so their product divides F. The conclusion that pr_+(F) = L_{E/L^+} belongs to (J_+)^e then follows. The reasoning is sound, but the coprimality of the distinct primes (ψ_i − 1) could be noted explicitly.

Minor objections (4)

[§3, Definition 3.1(ii)] The notation [w]_{L'} denotes the Frobenius element in Gal(L'/K). Since L'/K is a ℤ_p-extension, the Frobenius is well defined when the local extension is unramified. This is fine, but a brief remark that the definition depends on the choice of place above w could be added for clarity.
[§3.5, Theorem 3.5(i)] The argument that L_{E/L'} ∈ Λ' because (L_{E/L'}) = char_{Λ'}(X_{L'}) and the latter is an ideal of Λ' is valid but terse. It relies on the fact that the fractional ideal generated by L_{E/L'} equals an actual ideal, forcing the element to be integral. A short justification (e.g., 'since char_{Λ'}(X_{L'}) is contained in Λ', the generator of this principal fractional ideal must lie in Λ'') would help readers.
[§4.1, Proof of Proposition 4.2] The nonvanishing of L^{MSD}_p(E/Q) and L^{MSD}_p(E^K/Q) is attributed to BCS Thm. 1.1.2(a). Since that theorem itself depends on the same Assumption 2.1, it would be helpful to explicitly remark that the hypotheses for E^K also hold (especially the surjectivity condition for the twist). The brief comment that E^K[p] is irreducible because it is a quadratic twist of E[p] suffices, but the reader would benefit from a short verification.
[§4.3, Theorem 4.7] The semisimplicity statement 'J_+ X_{L^+} / J_+ = 0' is used to conclude that each factor module is of length one. The proof that length_R(X_+) = r and dim_{Q_p}(X_+/J_+X_+) = r indeed forces all a_i = 1. This is correct, but the phrase 'X_{L^+} is semisimple at J_+' might be replaced by a more descriptive statement to avoid potential confusion.

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The paper

The paper claims

Specialising the BCS two-variable main conjecture to every $ mathbb Z_p$-line yields a one-variable main conjecture with an explicit local-factor discrepancy.

The reading

This paper establishes that the two-variable Iwasawa main conjecture proved by Burungale, Castella, and Skinner specialises correctly to every one-dimensional $\mathbb Z_p$-line in the unique $\mathbb Z_p^2$-extension of an imaginary quadratic field. For each such line, the characteristic ideal of the dual Selmer group is the ideal generated by a linewise element obtained by projecting the two-variable Perrin–Riou $L$-function and dividing by an explicit local factor. Lines on which the projection vanishes are called exceptional, and the paper proves they are finite, that the cyclotomic line is non-exceptional, and that the anticyclotomic line is exceptional. Under finiteness of the $p$-primary Tate–Shafarevich group and non-degeneracy of the cyclotomic $p$-adic height pairing, the number of exceptional lines is at most the Mordell–Weil rank, with uniqueness in rank one. The result matters because it exhibits a precise sense in which a higher-dimensional main conjecture encodes a compatible family of one-variable main conjectures.

What carries the argument

The engine is the specialisation formula for characteristic ideals of dual Selmer groups over $\mathbb Z_p^d$-extensions, used in the form $\operatorname{pr}_{L'}(\operatorname{char}_{\Lambda}(X_L)) = \vartheta_{L/L'}\,\operatorname{char}_{\Lambda'}(X_{L'})$. The factor $\vartheta_{L/L'} = (\theta_{L/L'})$ is an explicit product of local factors: the $p$-parts of Tamagawa numbers at finite places where the decomposition subgroup meets $\Psi = \operatorname{Gal}(L/L')$, and Euler factors $(1-\alpha_w^{-1}[w])(1-\alpha_w^{-1}[w]^{-1})$ at primes above $p$ when the local extension is unramified. The global factor is trivial because residual irreducibility forces $E[p](L') = 0$. Feeding the two-variable equality $(L^{\mathrm{PR}}_{E/L}) = \operatorname{char}_{\Lambda}(X_L)$ through this formula and dividing by $\theta_{L/L'}$ produces the linewise element $L_{E/L'}$, whose integrality is part of the conclusion; the non-vanishing of $\theta_{L/L'}$ is what makes the division legitimate.

Extended reading notes

Core claim

On its own terms, the paper proves the following. Under Assumption 2.1, for every $\mathbb Z_p$-line $L'/K \subset L/K$, the element $L_{E/L'} = \theta_{L/L'}^{-1}\,\operatorname{pr}_{L'}(L^{\mathrm{PR}}_{E/L})$ lies in $\Lambda'$ and satisfies $$(L_{E/L'}) = \operatorname{char}_{\Lambda'}(X_{L'}),$$ where the characteristic ideal of a non-torsion module is declared to be the zero ideal. The global factor appearing in the general specialisation formula is trivial, so the entire discrepancy between the projected two-variable characteristic ideal and the one-variable characteristic ideal is the explicit principal ideal $\vartheta_{L/L'} = (\theta_{L/L'})$. Exceptional lines—those on which the projected Perrin–Riou element is zero—are exactly the lines on which $X_{L'}$ fails to be torsion. The cyclotomic line has $\theta_{L/L^+} = 1$ and is non-exceptional, recovering the product of the Mazur–Swinnerton-Dyer $p$-adic $L$-functions of $E$ and its quadratic twist, while the anticyclotomic line is exceptional because $X_{L^-}$ has rank one. The final results bound the number of exceptional lines by the cyclotomic augmentation order, show this order equals the Mordell–Weil rank under the stated finiteness and non-degeneracy assumptions, and imply uniqueness of the anticyclotomic exceptional line in rank one.

Load-bearing premise

The load-bearing premise is Assumption 2.1—residual surjectivity of $\bar\rho_{E,p}$, splitting of $p$ in $K$, the absence of ‘vexing primes’, and the remaining BCS hypotheses—because these conditions are needed both to import the integral two-variable main conjecture and to force $E[p](L') = 0$, making the global specialisation factor trivial.

Editorial extensions

If this is right

Over the cyclotomic line $L^+$, the local factor is $1$, so the characteristic ideal of $X_{L^+}$ equals the product of the Mazur–Swinnerton-Dyer $p$-adic $L$-functions of $E$ and its quadratic twist by $K$.
Over the anticyclotomic line $L^-$, the projected Perrin–Riou element is $0$ and $X_{L^-}$ is not torsion, so this line is always exceptional.
There are finitely many exceptional lines, each corresponding to a height-one prime ideal of the two-variable Iwasawa algebra containing the Perrin–Riou element.
If the $p$-primary Tate–Shafarevich group over $K$ is finite and Schneider's cyclotomic height pairing is non-degenerate, the number of exceptional lines is at most $\operatorname{rank}_{\mathbb Z} E(K)$, and in rank one the anticyclotomic line is the unique exceptional line.
For finite-order characters in the CGS-admissible range, the linewise element satisfies an explicit interpolation formula in terms of central values of complex Rankin $L$-functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The mechanism suggests a general principle: a two-variable main conjecture whose left side is a characteristic ideal and whose right side is a single $p$-adic $L$-element should specialise to one-variable main conjectures whenever the global factor is trivial, with non-torsion Selmer modules marking the exceptional one-dimensional slices.
If the identity is correct, exceptionality is a purely Selmer-theoretic phenomenon, so one could attempt to detect exceptional lines computationally by testing torsion of $X_{L'}$ through control theorems rather than by numerically evaluating the two-variable $p$-adic $L$-function.
A natural extension would be to remove the CGS-admissibility restriction on the interpolation formula: the present interpolation applies only to characters whose conductors are supported at the two primes above $p$, so a full interpolation over $\mathbb Z_p^2$ would require treating everywhere-unramified finite-order characters.

Original abstract

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,877 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.43 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension.	background 1.00	0.10	0.02	asserted 0.83
s1	In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor.	contribution 0.72	0.95	0.79	asserted 1.00
s2	With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero.	result 0.60	0.62	0.11	asserted 1.00
s3	We call such lines exceptional and prove that only finitely many occur.	result 0.94	0.95	0.12	asserted 1.00
s4	We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional.	result 0.99	0.95	0.15	asserted 1.00
s5	Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$.	result 0.96	0.95	0.24	asserted 1.00
s6	In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.	result 0.62	0.89	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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full text
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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (14)

[1]

D. Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, J. Reine Angew. Math. 698 (2015), 105--159

View on Pith 2015
[2]

A. Burungale, F. Castella, and C. Skinner, Base change and Iwasawa main conjectures for GL_2 , Int. Math. Res. Not. IMRN 2025, no. 8, Paper No. rnaf082, 15 pp

View on Pith 2025
[3]

F. Castella, G. Grossi, and C. Skinner, Mazur's main conjecture at Eisenstein primes, Math. Ann. 393 (2025), no. 2, 2451--2506

View on Pith 2025
[4]

B. Gross and D. Zagier, Heegner points and derivatives of L -series, Invent. Math. 84 (1986), no. 2, 225--320

View on Pith 1986
[5]

B. H. Gross, Kolyvagin's work on modular elliptic curves, in L -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser. 153, Cambridge Univ. Press, Cambridge, 1991, 235--256

View on Pith 1989
[6]

B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183--266

View on Pith 1972
[7]

T. Ochiai, Control theorem for Greenberg's Selmer groups of Galois deformations, J. Number Theory 88 (2001), no. 1, 59--85

View on Pith 2001
[8]

B. Perrin-Riou, Fonctions L p -adiques associ\'ees \`a une forme modulaire et \`a un corps quadratique imaginaire, J. London Math. Soc. (2) 38 (1988), no. 1, 1--32

View on Pith 1988
[9]

A. Ray, Arithmetic statistics and Diophantine stability for elliptic curves, Ramanujan J. 62 (2023), no. 1, 215--239

View on Pith 2023
[10]

P. Schneider, $p$-adic height pairings. I, Invent. Math. 69 (1982), no. 3, 401--409

View on Pith 1982
[11]

P. Schneider, p -adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329--374

View on Pith 1985
[12]

K.-S. Tan, Selmer groups over _p^d -extensions, Math. Ann. 359 (2014), no. 3--4, 1025--1075

View on Pith 2014
[13]

K.-S. Tan, p -adic L -functions for elliptic curves over global function fields, preprint, arXiv:2603.10576 (2026)

View on Pith Pith review arXiv 2026
[14]

K.-S. Tan, F. Trihan, and K.-W. Tsoi, Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields, preprint, arXiv:2603.11615 (2026)

View on Pith arXiv 2026

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, J. Reine Angew. Math. 698 (2015), 105--159

View on Pith 2015

[2] [2]

A. Burungale, F. Castella, and C. Skinner, Base change and Iwasawa main conjectures for GL_2 , Int. Math. Res. Not. IMRN 2025, no. 8, Paper No. rnaf082, 15 pp

View on Pith 2025

[3] [3]

F. Castella, G. Grossi, and C. Skinner, Mazur's main conjecture at Eisenstein primes, Math. Ann. 393 (2025), no. 2, 2451--2506

View on Pith 2025

[4] [4]

B. Gross and D. Zagier, Heegner points and derivatives of L -series, Invent. Math. 84 (1986), no. 2, 225--320

View on Pith 1986

[5] [5]

B. H. Gross, Kolyvagin's work on modular elliptic curves, in L -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser. 153, Cambridge Univ. Press, Cambridge, 1991, 235--256

View on Pith 1989

[6] [6]

B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183--266

View on Pith 1972

[7] [7]

T. Ochiai, Control theorem for Greenberg's Selmer groups of Galois deformations, J. Number Theory 88 (2001), no. 1, 59--85

View on Pith 2001

[8] [8]

B. Perrin-Riou, Fonctions L p -adiques associ\'ees \`a une forme modulaire et \`a un corps quadratique imaginaire, J. London Math. Soc. (2) 38 (1988), no. 1, 1--32

View on Pith 1988

[9] [9]

A. Ray, Arithmetic statistics and Diophantine stability for elliptic curves, Ramanujan J. 62 (2023), no. 1, 215--239

View on Pith 2023

[10] [10]

P. Schneider, \(p\)-adic height pairings. I, Invent. Math. 69 (1982), no. 3, 401--409

View on Pith 1982

[11] [11]

P. Schneider, p -adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329--374

View on Pith 1985

[12] [12]

K.-S. Tan, Selmer groups over _p^d -extensions, Math. Ann. 359 (2014), no. 3--4, 1025--1075

View on Pith 2014

[13] [13]

K.-S. Tan, p -adic L -functions for elliptic curves over global function fields, preprint, arXiv:2603.10576 (2026)

View on Pith Pith review arXiv 2026

[14] [14]

K.-S. Tan, F. Trihan, and K.-W. Tsoi, Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields, preprint, arXiv:2603.11615 (2026)

View on Pith arXiv 2026