The refined Tamagawa number conjectures for $\mathrm{GL}_2$

Chan-Ho Kim; Robert Pollack

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The refined Tamagawa number conjectures for $\mathrm{GL}_2$

Chan-Ho Kim, Robert Pollack

Number Theory · Submitted 14 May 2025 · 108 references

Pith's verdict: Original and technically rich, but the claimed equivalence between the localized IMC and non-vanishing of the Kolyvagin system / Kurihara numbers is false as stated, so the unconditional applications need repair.

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CitationChan-Ho Kim, Robert Pollack. "The refined Tamagawa number conjectures for $\mathrm{GL}_2$." arXiv:2505.09121 (2025). https://pith.science/paper/QLIAI5T2

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Editor's note

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Two things to know upfront. First, this is a genuinely original and technically rich paper with a plausible structure theorem. Second, the claimed bridge between the Iwasawa main conjecture and the non-vanishing of Kurihara numbers, Theorem 3.18(2)/(3), is false as stated, and that break takes the unconditional applications (Corollary 1.5 and the '(ord)' part of Theorem 1.2) with it.

The good parts are real. The 'relative' Kolyvagin system argument in Section 5 extracts not just coranks but the full O-module structure of Bloch–Kato Selmer groups from Kolyvagin derivatives of L-values. The connection between Kato's Euler system and Kurihara numbers for higher weight, via explicit reciprocity with torsion coefficients, is new and useful. The numerical appendix with Pollack gives concrete Selmer structures, several of which are out of reach of the BSD conjecture. The main theorem is cleanly stated and, conditional on its non-vanishing hypothesis, is independent of weight, local behavior, and analytic rank.

The soft spots, in order of severity. The equivalence in Theorem 3.18 is load-bearing. Since the Λ-adic Kolyvagin system module is free of rank one, write κ∞ = f·κ⁰. The specialization mod P vanishes exactly when f ∈ P. The localized IMC is ord_P(f) = ord_P(char Sel∨); this equality is perfectly compatible with ord_P(f) > 0. So the IMC can hold at P while the specialization is zero. The same logic invalidates the Λ-primitivity claim. The proof's note that '[MR04] claims one direction... the proof implies the equivalence' cannot work, because the missing converse would require ord_P(char Sel∨) = 0, which is not part of the IMC. Proposition 4.2 transfers this to the Kurihara numbers, so the claimed equivalence in Theorem 1.2 between e^{min,†} ≠ 0 and (IMC at 1) is not established. The conditional Theorem 1.1 may still be fine, but the unconditional ordinary case is currently over-sold.

Minor issues: the explicit reciprocity law is proved only up to a non-zero uniform constant; the paper acknowledges this and the constant cancels in the differences that matter, so that is acceptable. The proof also leans on the author's submitted paper [Kim] for integrality of Kato's zeta morphism, a dependency to watch.

Who is this for? Iwasawa theorists and anyone working on Selmer group structures, p-converses, or BSD-type formulas. It deserves serious referee time, but it needs a major revision: fix or replace Theorem 3.18 and then re-derive the applications. I'd send it to a specialist, not desk-reject.

Referee report

Significance

If the main results were correct, the paper would be a significant advance: it would give a structural BSD-type formula for Selmer groups that is insensitive to weight and local behavior at p, and it would upgrade known Iwasawa main conjecture results to exact structure theorems. The idea of using Kolyvagin derivatives of modular symbols rather than p-adic L-function derivatives is original and appealing, and the numerical appendix with Pollack is a useful, reproducible computational contribution. However, the central bridge between the Iwasawa main conjecture and non-vanishing of the Kurihara numbers, Theorem 3.18(2)–(3), is false as stated. This invalidates the unconditional applications in Theorem 1.2, Corollary 1.5, and Theorem 6.2, so the advertised significance is not achieved in the present version.

Major objections (3)

[§3.5.3, Theorem 3.18(2)–(3)] The asserted equivalences are false as stated. By Theorem 3.11, KS(Tf(k−r)⊗Λ) is free of rank one over Λ; write κ∞ = f·κ⁰. The reduction κ∞ mod P is zero exactly when f∈P. The localized IMC in Conjecture 3.17 is the equality ord_P(f) = ord_P(char Sel0(Q∞,Wf(r))∨). This equality is compatible with ord_P(f) > 0, in which case the specialization is zero while the IMC holds at P. Likewise, equality of full characteristic ideals (f) = (char Sel∨) can hold with f a non-unit, so κ∞ need not be Λ-primitive. The proof line in Theorem 3.18 that "the proof implies the equivalence" cannot supply the missing converse, because that converse would require ord_P(char Sel∨) = 0, which is not part of the IMC. Consequently, the implication '(IMC at 1) ⇒ e^{min,†} ≠ 0' in Theorem 1.2 is not established, and the unconditional consequences in Corollary 1.5 and in the proof of Theorem 6.2, which rely on that implication, do not follow. The converse direction (e^{min,†} ≠ 0 ⇒ IMC at 1) remains valid, since it follows from the one-sided divisibility in Theorem 3.18(1) once κ is known to be non-trivial.
[§3.1.1, Theorem 3.1] The integrality of Kato's zeta morphism over finite abelian extensions L is a load-bearing input: it is used to construct Kato's Kolyvagin system that underlies Theorems 1.1 and 1.2. The proof of Theorem 3.1 delegates a key step to the author's submitted paper [Kim]. Since [Kim] is not part of the published record, the referee cannot verify this step from the manuscript. Please either include a complete proof of the required integrality statement or state precisely which result of [Kim] is being used and its status. This dependency is especially concerning because the paper otherwise advertises independence from integral p-adic Hodge theory.
[§4, Proposition 4.2 and §5] The numerical invariants of the Kurihara numbers are used in §5 to compute ordπ(loc_s^p κKato) only up to a non-zero uniform constant C±_per and a fixed Euler factor. The manuscript asserts that this constant is cancelled in the differences that appear in Theorem 1.1. This cancellation is plausible for differences such as ordπ(e_n) − ordπ(e_{nℓ}), and the structure formula indeed uses only such differences. However, the paper should state this cancellation explicitly at the point where exact valuations of loc_s^p κKato are first identified with valuations of e^{min,†}, because Proposition 4.2 as written only gives non-vanishing, not equality of numerical invariants. Without an explicit statement, the transition from the Selmer-theoretic Kolyvagin system invariants to the modular-symbol invariants in §5.3 and §5.4 is incomplete.

Minor objections (4)

[§2.3.4, equation (2.3)] The functional equation for Kurihara numbers is stated without proof; a brief derivation or a precise reference to the modular-symbol functional equation would help the reader.
[§3.4.1, after Theorem 3.11] The notation KS and KS is introduced with a comment that they are not distinguished, but the two symbols are visually nearly identical; please use distinct typography or explicitly identify one with the other.
[§3.5.3, Theorem 3.19] The exception 'except πΛ' in the statement of Wan's theorem is potentially confusing: please clarify whether this exception is the prime ideal generated by a uniformizer of the coefficient ring, and confirm explicitly that the augmentation ideal is not excluded.
[References] The reference [Kur24] contains a typo: 'anayltic' should be 'analytic'.

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The paper

The paper claims

Kurihara numbers determine the exact Selmer group structure for modular forms with large image, yielding corank, elementary divisors, and the p-part of a BSD-type formula.

The reading

The paper claims that for a modular form whose attached Galois representation has large image, the full module structure of the central-critical Bloch–Kato Selmer group is determined not by the L-function and its analytic derivatives but by a discrete family of modular-symbol numbers, the Kurihara numbers. These numbers are built from special L-values at auxiliary primes, normalized by minimal integral periods, and the paper treats them as Kolyvagin derivatives of special L-values. If this family does not vanish, the paper proves an explicit isomorphism describing the Selmer group, giving its corank, its elementary divisors, and the p-part of a Birch–Swinnerton-Dyer type formula. The non-vanishing is shown to be equivalent to the Iwasawa main conjecture localized at the augmentation ideal, and for good ordinary primes with p-distinguished large-image representations the formula is unconditional. Consequences include a discrete analogue of the Beilinson–Bloch–Kato conjecture, the p-parity conjecture, and a higher-weight p-converse to the theorem of Gross–Zagier and Kolyvagin.

What carries the argument

The central objects are the Kurihara numbers $e^{\min,r} = \{ e^{\min,r}_n \in \mathcal{O}/I_n \mathcal{O} \}_{n \in \mathcal{N}_1}$, collections of modular-symbol values at $s = r$, normalized by minimal integral periods and reduced modulo the ideal $I_n$ generated by $\ell-1$ and $1 - a_\ell(f) + \psi^{-1}(\ell)\ell^{k-1}$ for primes $\ell$ dividing $n$. Their numerical invariants $\mathrm{ord}$ and $\partial^{(i)}$ mirror the order of vanishing and Taylor coefficients of $L(f,s)$ at $s = r$, which is why the paper calls them Kolyvagin derivatives of special L-values. The proof rests on three load-bearing mechanisms: an explicit reciprocity law with torsion coefficients that equates the image of Kato's Kolyvagin system with the Kurihara numbers up to a nonzero uniform constant; Mazur–Rubin's Kolyvagin system machinery, including the rigidity and rank-one property of Kato's system; and a relative Kolyvagin system argument that compares canonical and Bloch–Kato Selmer structures using global Poitou–Tate duality, Flach's generalized Cassels–Tate pairing, the self-duality of $T^\dagger_f$, and the functional equation of modular symbols, which supplies the parity constraint in the filtration.

Extended reading notes

Core claim

Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that $\rho_f$ has large image. Theorem 1.1 states: if the collection of Kurihara numbers $e^{\min,\dagger}$ for $f$ at the central critical point $s = k/2$ does not vanish, then Kato's Kolyvagin system $\kappa_{\mathrm{Kato},\dagger}$ for $T^\dagger_f$ is non-trivial, and there is an isomorphism of co-finitely generated $\mathcal{O}$-modules $\mathrm{Sel}(\mathbb{Q}, W^\dagger_f) \cong (F/\mathcal{O})^{\oplus \,\mathrm{ord}(e^{\min,\dagger})} \oplus \bigoplus_{i\geq 0} (\mathcal{O}/\pi^{e_i}\mathcal{O})^{\oplus 2}$, where $e_i = \tfrac{1}{2}\big(\partial^{(\mathrm{ord}(e^{\min,\dagger})+2i)}(e^{\min,\dagger}) - \partial^{(\mathrm{ord}(e^{\min,\dagger})+2i+2)}(e^{\min,\dagger})\big)$. Consequently the Selmer corank equals $\mathrm{ord}(e^{\min,\dagger})$ and the length of the quotient by the maximal divisible submodule equals $\partial^{(\mathrm{ord})}(e^{\min,\dagger}) - \partial^{(\infty)}(e^{\min,\dagger})$. Theorem 1.2 proves that the non-vanishing of $e^{\min,\dagger}$ is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, which makes the structure theorem unconditional in the good ordinary, p-distinguished, large-image case. The discovery is that discrete Kolyvagin derivatives of L-values, rather than complex analytic or p-adic derivatives, carry the exact bound of the Selmer group, so the result is insensitive to weight, to the local behaviour of $f$ at $p$, and to analytic rank.

Load-bearing premise

Everything hangs on the non-vanishing of the collection of Kurihara numbers at the central critical point; the paper proves this is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, so for forms outside the proven ordinary case the structure formula is conditional on that conjecture.

Editorial extensions

If this is right

If the Kurihara numbers do not vanish, Kato's Kolyvagin system is non-trivial; this is the cyclotomic analogue of Kolyvagin's conjecture and is proved unconditionally for good ordinary p-distinguished forms of large image at p >= 5.
The Selmer corank, all higher Fitting data, and the p-part of a Birch–Swinnerton-Dyer formula follow from the single family e^{min,†}, with no condition on analytic rank, weight, or the local automorphic representation at p.
For analytic rank zero, the length of the Selmer group is the normalized special L-value minus the analytic fudge factor ∂^{(∞)}(e^{min,†}), and this extends to all critical points r != k/2 without the Iwasawa main conjecture.
In the good ordinary, p-distinguished, large-image setting, the paper obtains the p-parity conjecture, the rank-one p-converse to Nekovář and Zhang, and computational upper bounds on Selmer ranks.
The appendix computes Kurihara numbers for elliptic curves of rank up to four and for higher weight modular forms, determining Selmer structures that go beyond what the Birch–Swinnerton-Dyer conjecture alone predicts.

Reading between the lines

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

The equivalence between e^{min,†} != 0 and the localized Iwasawa main conjecture suggests a computational route to numerical verification of that conjecture: a single non-vanishing Kurihara number at a suitable Kolyvagin prime would certify the localized main conjecture, and the appendix tables can be read as such numerical evidence.
Because the argument upgrades upper-bound-only Euler-system information to an exact structure formula, other Selmer-group results obtained from Euler systems likely admit the same discrete variation, turning one-sided divisibility into equalities where the corresponding localized Iwasawa main conjecture is known.
The analytic fudge factor ∂^{(∞)} is deliberately left uncontrolled, and the paper conjectures it is the product of local Tamagawa factors away from p; a natural testable extension is to compute both sides numerically for higher-weight modular forms with canonical periods, where the conjectural equality becomes a concrete, checkable congruence.
The relative Kolyvagin-system comparison between canonical and Bloch–Kato Selmer structures appears transplantable to other settings with an Euler system and a self-dual twist, such as anticyclotomic families or higher-rank groups, although such extensions would need new global duality input.

Original abstract

Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic rank. As consequences, we prove the non-vanishing of Kato's Kolyvagin system and complete a "discrete" analogue of the Beilinson-Bloch-Kato conjecture for modular forms at ordinary primes. We also obtain the higher weight analogue of the $p$-converse to the theorem of Gross-Zagier and Kolyvagin, the $p$-parity conjecture, and a new computational upper bound of Selmer ranks. We also discuss how to formulate the refined conjecture on the non-vanishing of Kato's Kolyvagin system for modular forms of general weight. In the appendix with Robert Pollack, we compute several numerical examples on the structure of Selmer groups of elliptic curves and modular forms of higher weight. Sometimes our computation provides a deeper understanding of Selmer groups than what is predicted by Birch and Swinnerton-Dyer conjecture.

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Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic rank. As consequences, we prove the non-vanishing of Kato's Kolyvagin system and complete a "discrete" analogue of the Beilinson-Bloch-Kato conjecture for modular forms at ordinary primes. We also obtain the higher weight analogue of the $p$-converse to the theorem of Gross-Zagier and Kolyvagin, the $p$-parity conjecture, and a new computational upper bound of Selmer ranks. We also discuss how to formulate the refined conjecture on the non-vanishing of Kato's Kolyvagin system for modular forms of general weight. In the appendix with Robert Pollack, we compute several numerical examples on the structure of Selmer groups of elliptic curves and modular forms of higher weight. Sometimes our computation provides a deeper understanding of Selmer groups than what is predicted by Birch and Swinnerton-Dyer conjecture.

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 $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image.	background 0.96	0.07	0.02	asserted 0.51
s1	We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal.	contribution 0.86	0.95	0.85	asserted 1.00
s2	Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic rank.	result 0.92	0.92	0.27	asserted 1.00
s3	As consequences, we prove the non-vanishing of Kato's Kolyvagin system and complete a "discrete" analogue of the Beilinson-Bloch-Kato conjecture for modular forms at ordinary primes.	result 0.94	0.95	0.23	asserted 1.00
s4	We also obtain the higher weight analogue of the $p$-converse to the theorem of Gross-Zagier and Kolyvagin, the $p$-parity conjecture, and a new computational upper bound of Selmer ranks.	result 0.94	0.95	0.11	asserted 1.00
s5	We also discuss how to formulate the refined conjecture on the non-vanishing of Kato's Kolyvagin system for modular forms of general weight.	contribution 0.51	0.82	0.04	hedged possibility 0.33
s6	In the appendix with Robert Pollack, we compute several numerical examples on the structure of Selmer groups of elliptic curves and modular forms of higher weight.	contribution 0.76	0.95	0.02	asserted 0.99
s7	Sometimes our computation provides a deeper understanding of Selmer groups than what is predicted by Birch and Swinnerton-Dyer conjecture.	interpretation 0.93	0.82	0.04	hedged possibility 0.89

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How the literature uses it

1 arXiv paper cite this in 1 sentence

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

What it cites · 17 sentences without a confident label

Cites as support 1 paper

For elliptic curves with good ordinary reduction, Conjecture [ref] is confirmed recently [ref] .

2312.09301 · The good ordinary or Fontaine–Laffaille case · 0.97

Cites as background 1 paper

We quickly review the notion of the canonical Kato's Euler system following [ref] .

Refined applications of Kato's Euler systems for modular forms · Canonical Kato's Euler systems · 0.98

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.

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