Pith. sign in

REVIEW 1 cited by

$p$-adic Asai and twisted triple product $L$-functions for finite slope families

T0 review · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper constructs two-variable p-adic Asai and twisted triple product L-functions for finite-slope families of Hilbert modular forms over real quadratic fields.

arxiv 2504.12066 v2 pith:KDN44DPG submitted 2025-04-16 math.NT

Ananyo Kazi , David Loeffler This is my paper · ORCID
classification math.NT
keywords modularproductconstructionfamilyfunctionhilberttripletwisted
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

Number theory attaches to modular forms functions called L-functions, whose values at special points are conjecturally tied to arithmetic objects like ranks of elliptic curves. p-adic L-functions are p-adic analytic objects that package infinitely many of these special values into a single function. This paper works with Hilbert modular forms, which are modular forms over a real quadratic field, and considers one-parameter families of such forms, varying the weight and a cyclotomic twist. The authors define two p-adic L-functions: one for the Asai, or twisted tensor, L-function, and one for a twisted triple product. The Asai construction generalizes an earlier construction of Grossi, Loeffler and Zerbes from ordinary (slope zero) families to finite-slope families. The triple product construction is a counterpart to one by the first author, but in a different range of weights in which the Hilbert modular form has the dominant weight. The technical engine is a new nearly-overconvergent version of higher Coleman theory for Hilbert modular surfaces. This allows the authors to push forward families of Eisenstein series into cohomology and pair them with cuspidal families. The paper states an interpolation theorem for the Asai function, reducing the proof to earlier work, and gives a more conditional description for the triple product, using Ichino's formula. A remark in the triple product section concedes that the interpolation may require choosing a different translate of the test vector, so the construction as written is not fully complete in that part. The Asai part is the more developed contribution.
Extended reading notes

Core claim

The paper defines a two-variable p-adic Asai L-function L^imp_{p,As}(Π)(κ_U,σ) on U×W, meromorphic in κ_U and holomorphic in σ, whose value at (a,s) with classical Π[a] and s in the critical interval equals E_p(As(Π[a]),s) times explicit gamma and period factors times the imprimitive Asai L-value (Theorem 7.6). It also claims a rigid-analytic function on U×U' whose values at (a,b) in the range k1+2a ≥ k2+ℓ+2b interpolate the square root of L(Π[a]×Σ[b],1/2).

Load-bearing premise

The existence of the nearly-overconvergent Eisenstein family on the chosen affinoids U×V (Theorem 6.3(2), based on [AI21, Theorem 4.3]) is load-bearing for the Asai L-function. If the analyticity conditions flagged in Remark 6.4 cannot be met without shrinking U and V, or if the bigger-sheaf construction is not carried out, the pairing that defines L^imp_p,As is not defined. Remark 6.4 leaves this to 'the interested reader'.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data fitting; the universal weight κ and cyclotomic variable σ are function variables, not free parameters. The slope bound h and level structures are hypotheses on the families, not fitted constants. No new physical entities are posited; the new nearly-overconvergent sheaves are explicitly constructed from integral de Rham structures, so no entity is pulled from a hat.

assumptions (6)
  • standard math Existence and basic properties of partial canonical subgroups on Hilbert modular surfaces (Theorem 3.3, [AIP18, Appendice A]).
    Used to define the p1-Hodge height strata X_v and the partial canonical subgroup H_can^n, foundational for the integral structures in Section 3.4.
  • standard math Integral structures omega^sharp_{A,1} and H^sharp_{A,1} exist with the stated filtration (Theorem 3.5 and Corollary 3.7, following [BP22, §5.5.1] and [Kaz24]).
    These give the vector bundles with marked sections from which the nearly-overconvergent sheaves are built.
  • domain assumption A nearly-overconvergent lift of the overconvergent Eisenstein family exists (Theorem 6.3(2), main result of [AI21, Theorem 4.3]).
    The pairing defining the Asai L-function uses this lift; Remark 6.4 notes it only holds under strong analyticity conditions, with details deferred.
  • standard math The control theorem for slope-finite parts of coherent cohomology (Corollary 4.16(2), after [BP22, Lemma 5.11]) ensures large-weight specialisations of the family are classical.
    Needed so that the p-adic L-function can interpolate classical automorphic L-values.
  • standard math Ichino's formula (Ich08; [BF20, Theorem 3.2]) relates the period integral to the central value of the degree-eight L-function.
    Used in Section 8.2; the interpolation proof for the triple product stops at this formula and does not resolve local test-vector issues.
  • domain assumption Setup hypotheses: p split in F, k1>k2>=1, k1=k2 mod 2, slope <= h < k1-1, and the central character and local root number conditions for the triple product.
    These restrict the families to which the construction applies; if violated, the classicality or self-duality claims fail.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $p$-adic Asai and twisted triple product $L$-functions for finite slope families." pith.science (2026). https://pith.science/paper/KDN44DPG

@misc{pith2026250412066,
  author       = {Pith},
  title        = {Pith review of: $p$-adic Asai and twisted triple product $L$-functions for finite slope families},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDN44DPG}},
  note         = {Machine review of arXiv:2504.12066}
}
abstract

We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' $L$-function, interpolating the central $L$-value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product $L$-function of arXiv:2401.13230, but differs in that it interpolates classical $L$-values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New perspectives on $p$-adic regulator formulae

    math.NT 2026-01 conditional novelty 6.0 of 10

    A finite-slope p-adic regulator formula for Asai–Flach classes is proved without finite-polynomial cohomology, and diagonal classes are reconstructed as pullback extension classes.

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages · cited by 1 Pith paper

  1. [52]

    Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

    url: https://doi.org/10.1215/00127094-2008-052 (cit. on p. 21). [Kat73] Nicholas M. Katz. “ p-adic properties of modular schemes and modular forms”. In: Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Math., Vol. 350. Springer, Berlin, 1973, pp. 69–190 (cit. on p. 4). [Kat78] Nicholas ...

    work page Pith review arXiv 1978

  2. [2023]

    Trilinear forms and the central values of triple product L-functions

    arXiv: 2311.14438 [math.NT] (cit. on pp. 6, 17). [GR24] Andrew Graham and Rob Rockwood. Nearly higher Coleman theory and p-adic L-functions for GSp(4)× GL(2) and GSp(4)× GL(2)× GL(2). 2024. arXiv: 2411.04559 [math.NT] (cit. on p. 2). [Ich08] Atsushi Ichino. “Trilinear forms and the central values of triple product L-functions”. In: Duke Math. J. 145.2 (20...

    work page arXiv 2008

  3. [2083]

    Le halo spectral

    doi: 10.1215/00127094-2020-0076 (cit. on pp. 2, 5, 6, 8, 11, 17). [AIP18] Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni. “Le halo spectral”. In: Ann. Sci. ´Ec. Norm. Sup´ er. (4)51.3 (2018), pp. 603–655. doi: 10.24033/asens.2362 (cit. on p. 4). [BF20] Iv´ an Blanco-Chac´ on and Michele Fornea. “Twisted triple product p-adic L-functions and Hirzeb...

    arXiv 2018

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.