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arxiv: 2604.05618 · v1 · submitted 2026-04-07 · 🧮 math.NT

On the computation of base-change lifts and lifts of Hida families

Iv\'an Blanco-Chac\'on , Luis Dieulefait , Antti Haavikko This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification 🧮 math.NT
keywords base-changeHida familiesHilbert modular formsHecke eigenvaluesL-functionsGalois extensionsnewformstotally real fields
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The pith

Explicit formula for Hecke eigenvalues of base-change lifts enables lifts of Hida families

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

The authors derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form that is the base-change lift of a classical newform to a totally real Galois number field. They show that for a totally real abelian number field the L-function of such a lifted form factors as a product of twisted L-functions corresponding to the characters of the field. They then use this formula to prove that Hida families of classical Hecke eigenforms admit base-change lifts to formal power series whose specializations are the individual base-change lifts. A reader would care because this turns the lifting process into an explicit computation rather than an existence statement alone.

Core claim

We derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form which is a base-change lift of a classical newform to a totally real Galois number field. We show that for a totally real abelian number field F the L-function of a base-change lifted form can be factorized as a product of twisted L-functions over the characters of F. Moreover, we use the formula for the Hecke eigenvalues of a base-change lift to prove the existence of a base-change lift of a Hida family. In particular, we show that a Hida family of classical Hecke eigenforms can be lifted to a formal power series that specializes to the base-change lifts of the Hida family of classical cusp forms.

What carries the argument

The explicit formula for the Hecke eigenvalues of base-change lifts of classical newforms.

If this is right

  • Direct computation of Hecke eigenvalues becomes possible for the lifted Hilbert modular forms.
  • Factorization of L-functions into twists holds for abelian base fields.
  • Base-change lifts exist for all members of a Hida family simultaneously via a single formal power series.
  • The lifted family preserves the specialization properties at each arithmetic point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This explicit control over the lifted eigenvalues may allow direct calculation of other quantities like Fourier coefficients in the family.
  • Similar techniques might be applied to study congruences between base-changed forms.
  • Computational checks for specific examples could confirm the formula in low-degree cases.

Load-bearing premise

The existence of the base-change lift as a Hilbert modular form with the expected compatibility under the Galois action and Hecke operators is taken as given from prior work.

What would settle it

If the Hecke eigenvalues computed from the formula for a known base-change lift do not agree with the eigenvalues obtained by other means, such as direct construction of the Hilbert modular form for small fields and weights, the formula would be falsified.

read the original abstract

We derive an explicit formula for the Hecke eigenvalues of a Hilbert modular form which is a base-change lift of a classical newform to a totally real Galois number field. We show that for a totally real abelian number field $F$ the $L$-function of a base-change lifted form can be factorized as a product of twisted $L$-functions over the characters of $F$. Moreover, we use the formula for the Hecke eigenvalues of a base-change lift to prove the existence of a base-change lift of a Hida family. In particular, we show that a Hida family of classical Hecke eigenforms can be lifted to a formal power series that specializes to the base-change lifts of the Hida family of classical cusp forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper derives an explicit formula for the Hecke eigenvalues of Hilbert modular forms arising as base-change lifts of classical newforms to totally real Galois number fields. For totally real abelian fields F it establishes a factorization of the associated L-function as a product of twisted L-functions over the characters of Gal(F/Q). It then applies the eigenvalue formula to construct a base-change lift of a Hida family, realized as a formal power series whose specializations recover the base-change lifts of the classical eigenforms in the family.

Significance. If the explicit eigenvalue formula and the subsequent Hida-family construction hold, the work supplies concrete computational tools for base change that are currently scarce in the literature. The L-function factorization for abelian extensions is a direct and verifiable consequence that may simplify analytic continuation arguments, while the Hida-family lift extends ordinary families to higher-dimensional settings and could support explicit Iwasawa-theoretic calculations.

minor comments (3)
  1. The introduction would benefit from a short paragraph recalling the standard construction of base-change lifts (via Langlands or via trace formulas) before stating the new explicit formula, to improve accessibility for readers outside the immediate subfield.
  2. Notation for the Hecke eigenvalues (e.g., the symbols a_p and the normalized eigenvalues) is introduced in §2 but used with slight variations in §4; a single consistent definition would prevent minor confusion.
  3. The factorization identity in the abelian case is stated without an equation number; numbering it would facilitate cross-references in the Hida-family argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. As the major comments section of the report is empty and no specific issues were raised, we have no points requiring point-by-point response.

Circularity Check

0 steps flagged

No circularity: explicit formulas and constructions build on external base-change theory

full rationale

The paper derives an explicit Hecke-eigenvalue formula for base-change lifts of classical newforms, proves an L-function factorization over Galois characters when the base field is totally real abelian, and applies the formula to construct a lift of a Hida family to a formal power series. These steps are presented as direct consequences of the eigenvalue formula together with standard prior results on base change and Hida theory; the existence of the classical base-change lift is assumed from the literature rather than derived internally. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear in the claimed derivation chain. The central results therefore remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on established domain assumptions from the theory of automorphic forms and Hida families without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Existence and basic properties of base-change lifts for classical newforms to Hilbert modular forms over totally real Galois fields
    Invoked to state the eigenvalue formula and L-function factorization.
  • domain assumption Existence and p-adic properties of Hida families of classical Hecke eigenforms
    Used to prove the lift to formal power series.

pith-pipeline@v0.9.0 · 5432 in / 1398 out tokens · 62596 ms · 2026-05-10T18:37:35.116124+00:00 · methodology

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Reference graph

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