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REVIEW 3 major objections 5 minor 27 references

On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$

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Pith's one-line read For abelian varieties over characteristic-p function fields, the leading term of the Hasse–Weil–Artin L-function is computed by an explicit BSD-type formula, conditional on finiteness of the Tate–Shafarevich group and a local ramification…

desk verdict A serious conditional p-adic BSD formula for Hasse–Weil–Artin L-functions over function fields; the new equivariant Riemann–Roch and explicit local terms are real, but the p-part rests on an unpublished [BKK] input whose 'verbatim' extension needs scrutiny. read the letter →

arxiv 2411.12404 v1 pith:I2Z3MI65 submitted 2024-11-19 math.NT

Wansu Kim , Ki-Seng Tan , Fabien Trihan , Kwok-Wing Tsoi This is my paper · ORCID
classification math.NT MSC 11G4014G1011J9514C40
keywords BirchandSwinnerton-DyerconjectureHasse-Weil-ArtinL-functionsglobalfunctionfieldsequivariantRiemann-RochtheoremweaklyramifiedcoversTate-ShafarevichgroupNéronmodelscharacteristicp
topics Birch and Swinnerton-Dyer
open problems Birch and Swinnerton-Dyer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a Birch–Swinnerton-Dyer-type formula for the leading term at $s=1$ of the Hasse–Weil–Artin $L$-function attached to an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex Galois representation $\chi$. The formula expresses the $\lambda$-adic valuation of the normalized leading term as a sum of explicit arithmetic terms: a volume factor, a local ramification correction, the twisted regulator, and the length of a twisted Tate–Shafarevich module, with the $p$-part included when a local weak-ramification hypothesis holds. A sympathetic reader should care because this makes the equivariant refinement of BSD over function fields explicit enough to compute individual leading terms, not merely classes up to torsion, and it covers the troublesome $p$-part even for weakly wildly ramified characters. The main new engine is an equivariant Riemann–Roch theorem for weakly ramified covers of curves over finite fields.

What carries the argument

The central engine is an equivariant Riemann–Roch theorem for weakly ramified covers of curves over a perfect field of characteristic $p$ (Theorem 3.2). It computes the class $[\mathrm{R}\Gamma(C_F,E)] \in K_0(k[G])$ of the cohomology of a $G$-equivariant vector bundle $E$ on $C_F$, under the local condition (3.2a) that at wildly ramified points the completed stalk is a sum of fractional ideals $\mathfrak{p}_v^{-b_{v,i}}$ with $b_{v,i} \equiv -1 \bmod |P_v|$; the formula is written in terms of induced modules $\operatorname{Ind}_{G_v}^G(P_v(j))$ attached to the inertia action. The other load-bearing piece is Köck's local integral normal basis theorem (Theorem 2.11), which guarantees freeness of those fractional ideals under weak ramification. Applied to $E=\operatorname{Lie}(A_F)(-D_F)$, the theorem converts the incoherent-cohomology correction term of the equivariant BSD conjecture into an explicit local ramification sum, yielding the loc factor.

What would settle it

Compute both sides of Theorem 7.12 for a constant supersingular abelian variety over $\mathbb{F}_q(t)$ with an Artin–Schreier $p$-extension $F/K$: the left side is obtained from the Grothendieck–Lefschetz trace formula for the $L$-function, and the right side from the Néron-model fibre data and the regulator; any mismatch in the $p$-adic valuation at the place above $p$ would disprove the formula, and agreement for many such pairs would confirm the $p$-part computation.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 7.12: for an abelian variety $A$ over a global function field $K$ of characteristic $p>0$, a finite Galois extension $F/K$ with group $G$, and an irreducible complex character $\chi$ of $G$ defined over a number field $E$, if the Tate–Shafarevich group $\Sha(A/F)$ is finite and the local Assumption 7.2 holds, then for any place $\lambda$ of $E$ above a prime $\ell$ the normalized leading term $\mathcal{L}_S(A,\chi)$ satisfies the equality of fractional ideals $\mathcal{L}_S(A,\chi) O_{E,\lambda} = (\operatorname{vol}_{D_1}(A/K) \prod_{v \in D_2} |\operatorname{Lie}(A_F)(k_{\tilde v})^{G_{\tilde v}}|)^{\deg \chi} \cdot \operatorname{loc}_{D/F}(A,\chi) \cdot \frac{\operatorname{Reg}^\chi_\lambda}{|G|^{r_{\mathrm{alg}}(\chi)}} \cdot \operatorname{Char}_\lambda(X^\vee_{\chi,\lambda}(A/F))$. The paper's specific contribution is to make this explicit at $\ell=p$ by showing that the equivariant vector bundle $L=\operatorname{Lie}(A_F)(-D_F)$ satisfies the local freeness condition (3.2a) whenever $F/K$ is weakly ramified everywhere and tame at places of non-semistable reduction, so that the new equivariant Riemann–Roch theorem computes the coherent-cohomology factor directly.

Load-bearing premise

The load-bearing premise is that the Tate–Shafarevich group $\Sha(A/F)$ is finite and that, at the characteristic prime, the extension $F/K$ is weakly ramified everywhere, semistable at places where wild ramification occurs, and tame at places where the abelian variety has bad reduction; without these local conditions the explicit $p$-part formula is not proved.

Editorial extensions

If this is right

  • For every prime $\ell$ coprime to $|G|$, the $\lambda$-adic valuation of the normalized leading term is given by the explicit formula (7.13b), involving only volumes, the local correction, the twisted regulator, and characteristic ideals of the twisted Tate–Shafarevich group, with no torsion ambiguity.
  • When the character is trivial and $F=K$, the formula specializes to the $\ell$-part of the classical BSD formula over function fields, so the result is compatible with the known full BSD theorem.
  • The $p$-part of the leading term is computed for weakly wildly ramified characters, provided $A$ has semistable reduction everywhere in $F$, $F/K$ is weakly ramified, and $F/K$ is tame at the places where $A$ does not have semistable reduction.
  • The local ramification correction $\operatorname{loc}_{D/F}(A,\chi)$ is $1$ for $p$-extensions and unramified covers, and has a simple logarithmic expression in tame cyclic cases, giving a new explicit description even in the cyclic tame setting.
  • The theorem applies unconditionally (with $\Sha(A/F)$ known finite) to constant supersingular abelian varieties, to Ulmer elliptic curves with $p\equiv 2 \bmod 3$, and to Artin–Schreier extensions made from them; in these cases the $p$-part formula is fully explicit.

Reading between the lines

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

  • The paper's Assumption 7.2(3) is probably not optimal: if a wild place of $F/K$ meets a non-semistable fibre of $A$, the stable-subbundle construction mentioned in Remarks 6.7 and 7.17 suggests the correct $p$-part correction should be a local term attached to that subbundle; finding an explicit formula for it would extend the theorem verbatim.
  • Theorem 3.2 is a standalone equivariant Riemann–Roch theorem: any equivariant vector bundle satisfying the stalk condition (3.2a) has its Euler characteristic in $K_0(\mathbb{F}_p[G])$ computed by the displayed ramification formula, so the theorem should apply to other arithmetic bundles (e.g., sheaves of differentials or structure sheaves twisted by ramified divisors), not just Lie algebras of Né
  • Because Assumption 7.2 holds for all but finitely many primes, the theorem gives a complete proof of the leading-term BSD formula for every irreducible character away from a finite set; the only obstruction to a full theorem is the $p$-part, so the paper reduces the problem to understanding one local phenomenon.
  • The explicit $\operatorname{loc}$ term could be tested numerically: for cyclic tame extensions it is a sum over ramified places weighted by the character's inertia type, so one can compare it with the ratio between the $L$-value and the regulator in explicit Ulmer-curve examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proves an explicit BSD-type leading-term formula for Hasse-Weil-Artin L-functions attached to an abelian variety A over a global function field K of characteristic p>0 and an irreducible complex Galois representation chi, assuming finiteness of the relevant Tate-Shafarevich group and certain local hypotheses. The main result, Theorem 7.12 (with the simplified version Theorem 1.2), expresses the normalized leading term L_S(A,chi) at s=1 as a product of a volume term, a local ramification correction loc_{D/F}(A,chi), a chi-twisted regulator divided by |G|^{r_alg(chi)}, and a characteristic ideal of the chi-isotypic Tate-Shafarevich module. The proof combines an equivariant Riemann-Roch theorem for weakly ramified covers (Theorem 3.2), a detailed analysis of Lie algebras of Neron models under tame and weakly ramified base change (Section 5), and the equivariant BSD theorem of Burns-Kakde-Kim [BKK] quoted in Theorem 6.14. Section 8 gives examples, including Ulmer elliptic curves and an Artin-Schreier wild extension, where Assumption 7.2 holds for l=p.

Significance. If the result is correct, it is a substantial contribution: it provides the function-field analogue of Burns-Macias Castillo, extends the explicit leading-term formula to the p-primary part under natural hypotheses, and introduces a new local correction term loc_{D/F}(A,chi) that is explicit enough to compute in examples. The equivariant Riemann-Roch theorem of Section 3 generalizing Nakajima, Kock, and Fischbacher-Weitz-Kock is of independent interest. The paper is also honest about its hypotheses: it explicitly states that finiteness of X(A/F) remains open, that the explicit p-part relies on Assumption 7.2(3), and that without weak ramification only an inexplicit stable subbundle is available (Remarks 6.7 and 7.17). No parameters are fitted: the regulator, characteristic ideal, volume, and local terms are independently defined arithmetic objects. The principal weakness is the heavy reliance on the unpublished preprint [BKK], including an asserted 'verbatim' extension whose proof is not reproduced.

major comments (3)
  1. [Section 6, Theorem 6.14 and its proof] The proof of Theorem 6.14(2) states that a construction from the unpublished preprint [BKK], originally made for carefully chosen subgroups U_v inside A_F^0(p_v), extends 'verbatim' to the maximal choice U_v = A_F^0(p_v) and yields the coherent term chi_p^G(Lie(A_F)(-D_F)). This extension is load-bearing: it is exactly what converts the equivariant BSD theorem into Corollary 6.15, which is then used in Proposition 7.10 and Theorem 7.12. The manuscript does not reproduce the argument that the maximal choice is cohomologically trivial under Assumption 7.2(3), nor does it show that no hidden boundary or torsion term appears when comparing the Selmer complexes. Since [BKK] is an unpublished preprint, a reader cannot independently verify this step. If the 'verbatim' extension fails, every p-primary formula in Section 7 would need correction. I recommend that the authors either include a complete proof of this extension as a lemma or appendix, or explicitly reformulate the main theorem as conditional on a precisely stated theorem from [BKK] and make the relevant part of [BKK] available.
  2. [Section 7, Theorem 7.12 and Corollary 6.15] The p-primary part of Theorem 7.12 inherits the unresolved dependency described above, because Corollary 6.15 for l=p is stated under the hypotheses of Proposition 6.6(2), whose proof relies on the same 'verbatim' extension and on [BKK, Proposition 3.7(i)]. The paper does provide a detailed proof that the explicit line bundle Lie(A_F)(-D_F) satisfies condition (3.2a) under Assumption 7.2(3) via Corollary 5.8(3), but the Selmer-complex side is quoted rather than proved. This means the central p-primary claim is currently not checkable from the manuscript alone, even assuming the rest of the paper is correct.
  3. [Section 8, Examples] The examples are useful and do verify Assumption 7.2 in nontrivial settings, but they do not compute the new terms loc_{D/F}(A,chi) or the characteristic ideal in any concrete case, so no numerical or even explicit finite-field check of the main formula is presented. In particular, Example 8.6 contains a wild Artin-Schreier extension where the paper cannot verify finiteness of X(A/F); the authors state this, but as a consequence the example does not yield an unconditional application. A worked computation of loc in a tame cyclic or Artin-Schreier example would substantially increase confidence in the explicit local formula of Section 4.
minor comments (5)
  1. [Section 4, proof of Proposition 7.10] The text refers to a 'Hochschield-Serre-type spectral sequence'; the standard spelling is Hochschild-Serre.
  2. [Section 8, Example 8.6] There is a typo in the phrase 'a rational function field of characteristic of characterist ic p'; it should read 'of characteristic p'.
  3. [Section 7, Theorem 7.12] In equation (7.12a), the expression |Lie(A_F)(k_{\tilde v})^{G_{\tilde v}}| denotes the cardinality of a finite set; using absolute-value notation for cardinality is potentially confusing and should be explicitly defined.
  4. [Section 5, Proposition 5.5 and Corollary 5.8] The notation D_1, D_2, D and D_F is introduced only in the proof of Theorem 7.12; defining this decomposition earlier, near Proposition 5.5, would improve readability.
  5. [Throughout] The paper relies on the unpublished preprint [BKK] for several central inputs, including Theorem 6.14 and Proposition 6.6(2). It would be helpful to include the precise statement number of the relevant results from [BKK] in each citation rather than only 'cf. [BKK, Theorem 4.9]' and 'cf. [BKK, Proposition 3.7(i)]', so that a reader can locate the exact assertions being used.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the explicit formula's constants are independently defined, but the p-part proof leans on the authors' unpublished [BKK].

full rationale

The derivation chain is not circular in the operative sense. Theorem 7.12 is obtained by combining Corollary 6.15 (which is the equivariant-BSD-modulo-torsion input from Theorem 6.14) with the explicit equivariant Riemann-Roch computation of Corollary 4.15 and the Néron-model comparison of Corollary 5.8. The regulator, characteristic ideal, volume term, and loc term are all independently defined arithmetic objects; no parameter is fitted and no L-value is used to define the constants appearing on the right-hand side of (7.12d). The one notable support gap is that Theorem 6.14 is quoted from the unpublished preprint [BKK] by Burns, Kakde, and the first-named author, and in the proof the authors assert that the BKK construction 'can be extended verbatim' to the maximal choice U_v = A_F^0(p_v) and L_F = Lie(A_F)(-D_F). This makes the p-part formula depend on an unverified same-author result, and if that extension failed the p-primary identity would need correction. But this is an external-support and reproducibility risk, not a circularity: the final formula does not reduce by construction to its inputs, and the local computations in Sections 3–5 are carried out independently. Hence the score is 2 rather than higher.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theorem is conditional on an unproved finiteness statement and on local assumptions; the proof imports a theorem from the authors' unpublished preprint [BKK] and Köck's local integral normal basis theorem. No free parameters are fitted and no new entities are postulated.

assumptions (4)
  • domain assumption X(A/F) is finite, equivalently some l0-primary part is finite (Theorem 6.13).
    Used throughout: Proposition 6.3 for cohomology of the Selmer complex, Proposition 7.10 for the regulator term, and Theorem 7.12. Stated in Theorem 1.2; remains open in general.
  • domain assumption Assumption 7.2 holds for the chosen prime l.
    Requires no l-torsion in A(F), A^t(F) or local Néron fibres, and for l=p requires weak ramification everywhere, semistable reduction at all places of F, and tame ramification over the places where A does not have semistable reduction.
  • domain assumption BKK equivariant BSD theorem modulo torsion ([BKK, Theorem 4.10] and Proposition 5.6).
    Bridges the Selmer complex and coherent cohomology Euler characteristics to leading terms; cited as an unpublished preprint by Burns, Kakde and Kim, with the first-named author of the present paper also an author.
  • standard math Köck's local integral normal basis theorem ([Köc04, Theorem 1.1]).
    Used in Theorem 3.2 and Proposition 6.6 to obtain cohomological triviality and freeness of completed stalks at weakly wildly ramified places.

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Pith. "Pith review of On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$." pith.science (2026). https://pith.science/paper/I2Z3MI65

@misc{pith2026241112404,
  author       = {Pith},
  title        = {Pith review of: On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2Z3MI65}},
  note         = {Machine review of arXiv:2411.12404}
}
abstract

Given an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex continuous representation $\psi$ of the absolute Galois group of $K$, we obtain a BSD-type formula for the leading term of Hasse--Weil--Artin $L$-function for $(A,\psi)$ at $s=1$ under certain technical hypotheses. The formula we obtain can be applied quite generally; for example, it can be applied to the $p$-part of the leading term even when $\psi$ is weakly wildly ramified at some place under additional hypotheses. Our result is the function field analogue of the work of D. Burns and D. Macias Castillo, built upon the work on the equivariant refinement of the BSD conjecture by D. Burns, M. Kakde and the first-named author. To handle the $p$-part of the leading term, we need the Riemann--Roch theorem for equivariant vector bundles on a curve over a finite field generalising the work of S. Nakajima, B. K\"ock, and H. Fischbacher-Weitz and B. K\"ock, which is of independent interest.

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Reference graph

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