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

This paper derives an explicit formula for the Hecke eigenvalues of Hilbert modular forms arising as base-change lifts of classical newforms, demonstrates the factorization of their $L$ -functions over abelian fields, and utilizes these results to prove the existence of base-change lifts for Hida families.

Original authors: Iván Blanco-Chacón, Luis Dieulefait, Antti Haavikko

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Iván Blanco-Chacón, Luis Dieulefait, Antti Haavikko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef who has perfected a specific, delicious recipe for a soup (let's call it Recipe A). This soup is famous in your home kitchen (the rational numbers, $\mathbb{Q}$ ). Now, imagine you want to take this recipe and cook it in a much larger, more complex kitchen with different ingredients and tools (a totally real number field, $F$ ).

The big question is: How do you translate the recipe? If you just guess, you might ruin the flavor. You need a precise translation guide that tells you exactly how the taste of the new soup (the Base-Change Lift) relates to the original soup.

This paper by Blanco-Chacón, Dieulefait, and Haavikko is essentially a translation manual and a construction kit for mathematicians. Here is the breakdown of what they did, using simple analogies:

1. The Problem: Translating the Recipe

In the world of numbers, "recipes" are called Modular Forms. They are complex mathematical objects that encode deep patterns in prime numbers.

Classical Modular Forms: The original soup cooked in the simple kitchen ( $\mathbb{Q}$ ).
Hilbert Modular Forms: The translated soup cooked in the complex kitchen ( $F$ ).

For a long time, mathematicians knew that if you had a recipe in the simple kitchen, a matching recipe must exist in the complex kitchen (thanks to previous work by Langlands, Arthur, Clozel, and Dieulefait). But they didn't know how to write down the new recipe. They knew the dish existed, but they couldn't tell you the exact ingredients (the Hecke Eigenvalues) needed to make it.

2. The Breakthrough: The "Magic Formula"

The authors' first major achievement is deriving an explicit formula.

The Analogy: Imagine you have a list of numbers that describe the flavor of the original soup at different temperatures. The authors found a mathematical "machine" that takes those numbers and instantly spits out the exact flavor numbers for the new soup.
How it works: If a prime number $p$ splits into $r$ pieces in the new kitchen, the flavor of the new soup is a specific combination of the original flavors at powers of $p$ . It's like saying, "To get the taste of the new dish at this specific point, take the original taste at $p^r$ , subtract a little bit of the original taste at $p^{r-2}$ , and you're done."

This allows mathematicians to compute the new recipe without having to guess or search blindly.

3. The "Family Portrait" (Hida Families)

The paper goes a step further. Usually, recipes come in families. You might have a soup that gets spicier as you add more pepper (changing the "weight" of the form).

Hida Families: Think of this as a continuous line of soup recipes where you can smoothly adjust the "spiciness" (weight) from one level to another.
The Challenge: Can you translate the entire family of recipes at once, rather than translating each one individually?
The Solution: The authors proved that yes, you can. They showed that the "translation machine" (the formula from step 2) works not just for one soup, but for the whole infinite family. They constructed a "Super-Recipe" (a formal power series) that contains all the translated soups inside it.

4. Why Does This Matter? (The "Why" of the Paper)

Why do we care about translating these recipes?

Connecting Worlds: It helps mathematicians connect the world of simple numbers to the world of complex number fields.
Solving Hard Problems: The paper uses this translation to solve a specific puzzle about "potentially diagonalizable" lifts.
- The Metaphor: Imagine trying to prove that a complex, tangled knot can be untangled into a simple straight line under certain conditions. The authors used their "translation machine" to show that if you have a knot in the simple kitchen, you can find a version of it in the complex kitchen that is much easier to untangle (diagonalize).
Removing Restrictions: Previous methods required very strict conditions (like the kitchen being very small or the soup being very mild). This new method removes those restrictions, allowing the translation to work in much broader, more difficult scenarios.

5. The Computer Code

Finally, the authors didn't just write theory; they wrote computer code (in a program called Magma).

The Analogy: They didn't just describe the translation machine; they built it and gave you the remote control.
The Result: If you give the computer a specific recipe (a classical modular form) and a target kitchen (a number field), the code will instantly calculate the exact ingredients for the new recipe and verify that it matches a known Hilbert modular form.

Summary

In short, this paper is a bridge.

It gives a formula to translate a mathematical object from a simple world to a complex one.
It proves this translation works for entire families of objects, not just single ones.
It uses this to solve hard puzzles about the structure of numbers.
It provides software so anyone can do the translation themselves.

It turns a mysterious, non-constructive existence proof ("It exists!") into a practical, computational tool ("Here is exactly how to build it!").

1. Problem Statement

The paper addresses two primary challenges within the Langlands program and the theory of automorphic forms:

Explicit Computation of Base-Change Lifts: While the existence of base-change lifts (lifting automorphic representations from a number field $F$ to an extension $L$ ) was established non-constructively by Langlands, Arthur, Clozel, and Dieulefait, there was a lack of explicit formulas to compute the Hecke eigenvalues of the lifted form, particularly for totally real Galois extensions. Previous explicit results by Doi, Naganuma, and Saito were limited to quadratic fields or cyclic extensions of prime order with specific class number conditions.
Base-Change for Hida Families: There was no established framework for lifting entire families of modular forms (Hida families) to Hilbert modular forms over totally real fields. This is crucial for studying deformations of Galois representations and proving automorphy of symmetric powers and tensor products over general totally real fields.

2. Methodology

The authors employ a combination of Galois representation theory, explicit trace formulas, and deformation theory:

Galois Representation Analysis: The authors utilize the correspondence between classical newforms and 2-dimensional Galois representations ( $\rho_{f,q}$ ). They analyze the behavior of Frobenius elements under restriction to a totally real Galois extension $F/\mathbb{Q}$ .
Recursive Trace Formulas: By studying the characteristic polynomial of Frobenius matrices and utilizing the recursion relations of Fourier coefficients (Newton sums), they derive a closed-form expression for the trace of the $r$ -th power of a Frobenius element, where $r$ is the residual degree of a prime in the extension.
L-function Factorization: They prove the factorization of the $L$ -function of the lifted form into a product of twisted $L$ -functions of the original form, utilizing properties of abelian Galois groups and Dirichlet characters.
Hida Theory Construction: The authors construct the base-change lift of a Hida family by applying the derived eigenvalue formulas to the Iwasawa coefficients (the formal power series coefficients) of the family. They demonstrate that these specialized series correspond to the base-change lifts of the classical specializations.
Algorithmic Implementation: The theoretical formulas are implemented in Magma to computationally verify the lifts and prune the search space for Hilbert modular forms.

3. Key Contributions and Results

A. Explicit Formula for Hecke Eigenvalues (Theorem 3.1)

The paper derives a general formula for the Hecke eigenvalue $C(\mathfrak{p})$ of a base-change lift $h$ of a classical newform $f$ at a prime $\mathfrak{p} \subset \mathcal{O}_F$ lying above a rational prime $p$ .
Let $r = f(\mathfrak{p}|p)$ be the residual degree. The eigenvalue is given by:
$C(\mathfrak{p}) = \begin{cases} a(p) & \text{if } r = 1 \\ a(p^r) - \chi(p)p^{k-1}a(p^{r-2}) & \text{if } r > 1 \end{cases}$
where $a(n)$ are the Fourier coefficients of $f$ , $\chi$ is the nebentypus, and $k$ is the weight. This generalizes the Doi-Naganuma formula to any totally real Galois number field.

B. Factorization of L-functions (Theorem 4.1)

For a totally real abelian number field $F$ of degree $n$ , the $L$ -function of the base-change lift $h$ factorizes as:
$L(h, s) = \prod_{i=0}^{n-1} L(f, s, \chi_i)$
where $\chi_i$ are the Dirichlet characters associated with the abelian extension $F/\mathbb{Q}$ . The authors provide an elementary, constructive proof of this factorization, extending previous results by Saito and Arthur-Clozel.

C. Base-Change of Hida Families (Theorem 5.3)

The authors define and prove the existence of a base-change lift for a Hida family of ordinary classical cusp forms to a Hida family of ordinary Hilbert modular forms.

Construction: They define the lift by applying the eigenvalue formula from Theorem 3.1 to the coefficients of the formal power series defining the Hida family.
Result: The lifted formal power series specializes to the base-change lifts of the classical specializations of the original family.

D. Applications to Automorphic Lifts (Theorems 1.1, 1.2, 5.5, 5.6)

Using the machinery of Hida families, the authors generalize results regarding potentially diagonalizable automorphic lifts.

Ordinary Case (Theorem 5.5): For a $p$ $p$ -ordinary cuspidal Hecke eigenform $f$ $f$ and a totally real Galois field $F$ $F$ , there exists a sequence of weights $\{k_r\}$ ${k_{r}}$ such that the restriction of the Galois representation $\rho_{f,p}|_{G_F}$ $ρ_{f, p} ∣_{G_{F}}$ admits a potentially diagonalizable automorphic lift.
- Significance: This removes the restrictive condition $p > \max\{k, 6\}$ found in previous literature (e.g., [4]), as the Hida family approach allows for weight variation that bypasses this bound.
Non-Ordinary Case (Theorem 5.6): A similar result is proven for non-ordinary forms, provided the residual representation has a large image and $p > \max\{k, 6\}$ .

4. Significance

Computational Accessibility: The paper transforms the base-change problem from a purely existential theorem into a computational tool. The provided Magma algorithms allow researchers to explicitly construct lifts and verify them against known Hilbert modular forms.
Removal of Restrictions: By utilizing Hida families, the authors remove the "large prime" condition ( $p > \max\{k, 6\}$ ) for ordinary forms, which was a significant bottleneck in previous automorphy lifting theorems.
Future Directions: The work lays the groundwork for proving the automorphy of symmetric powers and tensor products of modular forms over totally real fields (GL2/F). This is a critical step toward the full Langlands functoriality conjecture for these groups, as it allows the transfer of automorphic properties from $\mathbb{Q}$ to $F$ .
Unification: It unifies the classical modular form theory with Hilbert modular form theory via explicit eigenvalue formulas, bridging the gap between the work of Doi-Naganuma (quadratic fields) and the general Langlands functoriality.

5. Conclusion

This paper provides a comprehensive solution to the explicit computation of base-change lifts for totally real Galois fields and extends this theory to Hida families. By deriving explicit Hecke eigenvalue formulas and proving the existence of lifted families, the authors enable new applications in the study of Galois representations, specifically regarding potentially diagonalizable lifts and the automorphy of higher-dimensional representations over totally real fields. The inclusion of Magma implementations ensures these theoretical results are immediately applicable for computational number theory research.