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Abelian varieties genuinely of $\mathrm{GL}_n$ -type

This paper extends the theory of abelian varieties of $\mathrm{GL}_n$ -type by generalizing Ribet's classical notion to a "genuine" setting, developing associated tools like building blocks and inner twists, broadening existing results on Galois representations under weaker assumptions, and providing an explicit construction of a family of genuinely $\mathrm{GL}_4$ -type abelian fourfolds.

Original authors: Francesc Fité, Enric Florit, Xavier Guitart

Published 2026-07-29

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CC BY 4.0

Original authors: Francesc Fité, Enric Florit, Xavier Guitart

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 detective trying to solve a mystery about the hidden shapes of the universe. In the world of mathematics, specifically a branch called number theory, there are objects called "abelian varieties." Think of these not as physical shapes you can hold, but as complex, multi-dimensional puzzles that exist in the realm of numbers. They are like intricate musical instruments; just as a violin has strings that vibrate to create sound, these mathematical objects have hidden "strings" (called endomorphisms) that vibrate in specific ways when you pluck them with different numbers.

For a long time, mathematicians have been trying to understand how these instruments are built. They discovered that some of these puzzles are made of smaller, simpler pieces, while others are "simple" in the sense that they cannot be broken down into smaller, independent puzzles. A key question has been: when we look at these simple puzzles, what kind of "music" (or symmetry) do they play? Some play a specific type of tune called "symplectic," which is like a dance where partners always move in perfect, mirrored opposition. Others play an "orthogonal" tune, where movements are more like a rigid, straight-line march. For a long time, we only knew how to identify these tunes for a specific, smaller class of puzzles. The big mystery was whether this rule held true for a much larger, more complex family of puzzles, even when the rules of the game seemed a bit looser.

This paper, written by Francesc Fit´e, Enric Florit, and Xavier Guitart, dives deep into this mystery. They focus on a special group of these mathematical puzzles called "abelian varieties of GLn-type." You can think of "GLn-type" as a label that tells us how many "strings" the instrument has and how they are tuned. The authors are particularly interested in a rare breed of these puzzles that are "genuinely" of this type—meaning they are so complex and unique that they cannot be broken down into any smaller, simpler puzzles of a different type.

The authors prove a significant result: even when the rules are relaxed (specifically, when the central "core" of the puzzle isn't as strictly defined as previously thought), these complex puzzles still play the same two types of tunes: the symplectic dance or the orthogonal march. They didn't just guess this; they built a rigorous mathematical theory to prove it. They introduced new tools, like "building blocks" (the fundamental pieces these puzzles are made of) and "inner twists" (a way of seeing how the puzzle changes when you look at it from a different angle), to show that the underlying music remains consistent.

To make sure their theory wasn't just a beautiful idea with no real-world (or number-world) examples, they didn't stop at theory. They constructed a specific, explicit family of these puzzles. They started with a curve (a shape defined by an equation) that looks like a twisted loop, specifically a "genus 2 curve" defined over a quadratic field (a number system involving the square root of 2). They showed that the "Jacobian" of this curve (a specific mathematical object associated with the curve that acts like its fingerprint) is a perfect example of one of these "genuinely" complex puzzles. In fact, they provided a concrete equation for such a curve and proved that its associated puzzle is indeed a four-dimensional object (a "fourfold") that plays the symplectic or orthogonal tune, confirming their theory with a tangible example.

In short, the paper confirms that a beautiful pattern in the mathematical universe holds true even in more complex, less restricted situations. They took a known rule for simple cases, proved it works for a much wider and trickier set of cases, and then built a specific, working model to prove it really works in practice. They didn't just suggest it might be true; they proved it with the full weight of mathematical logic and backed it up with a concrete example they could point to.

Technical Summary: Abelian Varieties Genuinely of GLn-Type

Problem Statement
The paper addresses the classification and arithmetic properties of simple abelian varieties $A$ defined over a number field $k$ that are "of $GL_n$ -type." An abelian variety is of $GL_n$ -type if there exists a number field $E$ of degree $2\dim(A)/n$ embedded as a subalgebra of its endomorphism algebra $\text{End}_0(A)$ . The authors focus specifically on varieties that are genuinely of $GL_n$ -type, meaning $A$ is simple, of $GL_n$ -type, and its base change $A_{\bar{k}}$ contains no isogeny factor of $GL_m$ -type for $m < n$ .

The central problem is to generalize the theory of abelian varieties of $GL_2$ -type (introduced by Ribet) to higher $n$ . Specifically, the authors aim to extend results by Chi, Banaszak, Gajda, and Krasoń regarding the image of $\ell$ -adic Galois representations attached to these varieties. Previous results established that if the center $H$ of $\text{End}_0(A)$ is totally real, the associated Galois representations are symplectic or orthogonal. The paper seeks to weaken this hypothesis: can the symplectic or orthogonal nature of the representations be recovered if only the center $F$ of $\text{End}_0(A_{\bar{k}})$ (the endomorphism algebra of the geometric base change) is totally real, rather than $H$ ?

Methodology
The authors develop a structural theory involving "building blocks," "inner twists," and "nebentypes" to analyze these varieties.

Building Blocks: The authors establish that if $A$ is genuinely of $GL_n$ -type, then $A_{\bar{k}}$ is isogenous to a power $B^r$ of a simple abelian variety $B$ defined over $\bar{k}$ . This $B$ is termed the building block associated to $A$ . They prove that $B$ is a $GL_n$ -type building block, meaning $\text{End}_0(B)$ is a central division algebra over a field $F$ with Schur index $t_B$ satisfying $n t_B [F:\mathbb{Q}] = 2\dim(B)$ .
Brauer Group Relations: A key algebraic step involves relating the Brauer classes of $\text{End}_0(A)$ and $\text{End}_0(B)$ . The authors show that $[\text{End}_0(A)] = [\text{End}_0(B) \otimes_F H]$ in the Brauer group of $H$ . This allows them to relate the Albert types (classification of division algebras with positive involution) of $A$ and $B$ .
Inner Twists and Nebentypes: The authors adapt the theory of inner twists (originally for $GL_2$ ) to the $GL_n$ setting. They construct a character $\varepsilon$ (the nebentype) taking values in $H^\times$ such that the Galois representation $\varrho_\lambda$ is isomorphic to $\varepsilon \otimes \varrho_\lambda$ (twisted by complex conjugation). This construction relies on the fact that if $A$ is geometrically of the first kind (i.e., $F$ is totally real), the extension $H/F$ is abelian.
Descent of Representations: Using cohomological methods (specifically analyzing 2-cocycles in $H^2(\text{Gal}(E/H), E^\times)$ ), the authors prove that the $\ell$ -adic representations descend to the field $H_\lambda$ under the condition that the prime $\lambda$ splits the endomorphism algebra.
Pairing Construction: By exploiting the absolute irreducibility of the representations and Schur's lemma, they construct a non-degenerate, $G_k$ -equivariant pairing $\psi_\lambda$ on the representation space. The symmetry (alternating or symmetric) of this pairing is determined by the Albert type of the building block $B$ .

Key Contributions and Results

Generalization of Chi et al. Results: The Main Theorem (Theorem 7.5) states that if $A$ is genuinely of $GL_n$ -type and geometrically of the first kind (meaning the center $F$ of $\text{End}_0(A_{\bar{k}})$ is totally real), then for primes $\lambda$ where $\text{End}_0(A)$ splits, there exists an absolutely irreducible representation $\varrho_\lambda: G_k \to \text{Aut}(W_\lambda(A))$ of degree $n$ . Furthermore, there exists a $G_k$ -equivariant pairing $\psi_\lambda$ with values in $H_\lambda(\varepsilon \chi_\ell)$ , where $\varepsilon$ is the nebentype character and $\chi_\ell$ is the cyclotomic character.
- If the building block $B$ has Albert type I or II, $\psi_\lambda$ is alternating (symplectic).
- If $B$ has Albert type III, $\psi_\lambda$ is symmetric (orthogonal).
- This result holds even if the center $H$ of $\text{End}_0(A)$ is not totally real, provided $F$ is.
Characterization of Genuine $GL_n$ -type: The paper provides a criterion (Proposition 2.9) characterizing genuine $GL_n$ -type varieties: a simple abelian variety $A$ is genuinely of $GL_n$ -type if and only if $\text{End}_0(A_{\bar{k}})$ is simple and there exists a maximal subfield of $\text{End}_0(A)$ that is also a maximal subfield of $\text{End}_0(A_{\bar{k}})$ .
Explicit Construction of Examples: In Section 8, the authors construct an explicit family of abelian fourfolds genuinely of $GL_4$ -type. These are obtained as restrictions of scalars of Jacobians of genus 2 curves defined over a quadratic field $k$ . The curves are constructed to be isogenous to their Galois conjugates via a Richelot isogeny.
- Example 8.6: They provide a specific curve $C$ over $k = \mathbb{Q}(\sqrt{2})$ such that its Jacobian $J$ has trivial endomorphism ring over $\mathbb{Q}$ (i.e., $\text{End}(J_{\mathbb{Q}}) = \mathbb{Z}$ ) but is isogenous to its Galois conjugate. The restriction of scalars $A = \text{Res}_{k/\mathbb{Q}}(J)$ is an abelian fourfold genuinely of $GL_4$ -type with $\text{End}_0(A) \cong \mathbb{Q}(\sqrt{-2})$ .
- This example is significant because the center $H = \mathbb{Q}(\sqrt{-2})$ is a CM field, not totally real. Thus, the Main Theorem applies, but the previous results by Chi, Banaszak, Gajda, and Krasoń (which required $H$ to be totally real) do not.
Criterion for Trivial Endomorphisms: Section 9 provides a computational criterion (Proposition 9.3) to verify that an abelian variety has a trivial endomorphism ring over $\mathbb{Q}$ . This involves checking the stable irreducibility of the characteristic polynomials of Frobenius at two distinct primes and ensuring their splitting fields are linearly disjoint.

Significance and Claims
The paper claims to extend the theory of abelian varieties of $GL_n$ -type by relaxing the condition on the center of the endomorphism algebra from being totally real to requiring only that the center of the geometric endomorphism algebra be totally real. This allows for the inclusion of varieties where the endomorphism algebra over the base field is a CM field, a case previously excluded from the symplectic/orthogonal classification in the context of $GL_n$ -type varieties.

The authors emphasize that their construction of the family of abelian fourfolds in Section 8 provides the first explicit examples of varieties genuinely of $GL_4$ -type that are "of the second kind" (in the sense of the base field's endomorphism algebra) but "geometrically of the first kind." This demonstrates the necessity and utility of their generalized framework. The work relies on and generalizes the foundational techniques of Ribet, Pyle, and Chi et al., adapting them to the higher-dimensional $GL_n$ context through the systematic use of building blocks and inner twists.

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