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Explicit Brauer-Manin obstructions on plane quartics

This paper introduces an improved method for detecting the absence of rational points and low-degree divisors on plane quartics and other smooth projective curves over number fields by utilizing explicit Brauer-Manin obstructions that avoid the computationally expensive calculation of full $S$ -unit groups, thereby enabling the determination of indices (such as 2 or 4) that exceed the maximum local index.

Original authors: Nils Bruin, Brendan Creutz

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nils Bruin, Brendan Creutz

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: Does a specific geometric shape (a "plane quartic" curve) have any hidden "rational points" (solutions made of simple fractions)?

Sometimes, this shape looks like it has solutions everywhere you look locally (in every neighborhood of the number line), but when you zoom out to the whole picture, there are actually no solutions at all. This is a famous puzzle in mathematics called the "Hasse Principle," and this paper provides a new, sharper magnifying glass to catch these "ghost" shapes that trick us.

Here is how the authors, Nils Bruin and Brendan Creutz, solve the mystery, explained through everyday analogies:

1. The Setup: The Shape and the Clues

Think of the curve as a complex, twisted wire sculpture floating in space.

The Goal: Find a point on this wire that has "rational" coordinates (like 1/2 or 3/4).
The Trap: The wire might pass through every local neighborhood (you can find points with decimal approximations everywhere), but it might never actually touch a "rational" spot.
The Old Method: Previous detectives used a technique called "2-cover descent." Imagine trying to find the wire by checking every single possible key in a giant keyring (the "S-unit group"). This was like trying to open a safe by trying every combination in the universe. It was slow, expensive, and often required knowing the "class group" (a complex map of the number system) perfectly, which is sometimes impossible to calculate.

2. The New Method: The "Pairing" Test

The authors introduce a smarter way to check for solutions without needing the entire keyring.

The Analogy: The Secret Handshake
Imagine the curve has 28 special "bitangents" (lines that just barely touch the curve at two points). These lines act like 28 unique clues or keys.

The authors create a mathematical "handshake" (called a pairing) between these clues and the potential solutions.
Instead of checking every possible key, they only need to find a small, specific subset of keys that are "square-normed" (a fancy way of saying they fit a specific mathematical pattern).
They use these keys to test the local clues (the points found in neighborhoods). If the local clues fail to "shake hands" correctly with the specific subset of keys, the authors know for a fact that no global solution exists.

The Big Win:
The old method required knowing the entire list of keys to be sure. The new method says, "We don't need the whole list. If we just have a few specific keys that create a contradiction, we can prove the solution is impossible." This saves a massive amount of computing power.

3. The "Index" Mystery: How Many Points Do We Need?

Sometimes, the curve doesn't have a single rational point, but it might have a "cluster" of points that act like one. Mathematicians measure this with an Index.

Index 1: There is at least one rational point. (The mystery is solved; the wire is real).
Index 2: There are no single points, but there is a pair of points that work together.
Index 4: You need a group of four points to make sense of the shape.

The authors show how their method can prove that a curve has an Index of 2 or 4, even when local checks suggest it should have an Index of 1.

Analogy: Imagine you are looking for a specific type of fruit. You check every local market and find the fruit everywhere (local index 1). But your new "pairing test" reveals that the fruit is actually a hybrid that only exists in pairs or groups of four. The curve is "locally full" but "globally empty" of single points.

4. The "Brauer-Manin" Connection: The Invisible Wall

The paper explains that their method is mathematically equivalent to hitting an invisible wall known as the Brauer-Manin obstruction.

Think of the rational points as trying to walk through a maze.
Locally, the path looks clear.
But there is an invisible force field (the Brauer group) blocking the path.
The authors' "pairing test" is essentially a detector that tells you, "You can't pass here because the invisible wall is in the way." They prove that their specific algebraic test is just a different way of measuring this same invisible wall.

5. Real-World Proof: The Database

To prove their method works, they tested it on a database of over 80,000 curves.

They found 135 curves that looked like they had solutions everywhere but were actually empty.
They proved these curves had Index 2 or 4, meaning they are "almost" solvable but not quite.
Crucially, they did this unconditionally. They didn't have to make any "guesses" (like assuming a famous unproven hypothesis called GRH) to get the answer. They just needed a few specific keys, not the whole keyring.

Summary

In short, this paper gives mathematicians a lighter, faster, and more reliable tool to prove that certain geometric shapes have no rational solutions. Instead of trying to map the entire universe of numbers (which is hard), they use a clever "spot-check" system that detects invisible barriers, proving that some shapes are mathematically impossible to solve, even though they look possible from every angle.

Technical Summary: Explicit Brauer-Manin Obstructions on Plane Quartics

Problem Statement
The paper addresses the problem of determining whether a smooth plane quartic curve $X$ over a number field $k$ possesses rational points ( $X(k) \neq \emptyset$ ) or divisors of specific degrees (specifically degree 1 or 2), particularly in cases where the curve is locally soluble everywhere (i.e., $X(k_v) \neq \emptyset$ for all completions $k_v$ ). While the Hasse principle often fails for such curves due to Brauer-Manin obstructions, previous computational methods for detecting these obstructions via 2-cover descent were limited by the requirement to compute the full $S$ -unit group of the relevant étale algebras. This computation is often infeasible for high-degree algebras, such as the degree 28 algebra associated with the bitangents of a plane quartic.

Methodology
The authors develop a practical algorithm to detect obstructions to rational points and divisors without requiring the full determination of $S$ -unit groups. The method relies on 2-cover descent and the theory of Brauer-Manin obstructions, generalized from previous work on hyperelliptic curves.

Étale Algebras and Pairings: The approach utilizes the 28 bitangents of the plane quartic $X$ , which define an étale $k$ -algebra $L$ of degree 28. The authors construct a bilinear pairing $\langle \cdot, \cdot \rangle_S$ between local images of the curve and a subgroup of $S$ -units in $L$ with square norm. This pairing is derived from Hilbert symbols and imposes constraints on the image of the localization map.
Partial $S$ -Unit Information: A key innovation is that the algorithm does not require the full $S$ -unit group $(L^\times/L^{\times 2})_S$ . Instead, it suffices to compute a subgroup generated by explicit $S$ -units (obtained via relation matrices from a subset of primes) that is sufficiently large to constrain the image of the localization map. The method leverages the exactness of a sequence involving the restriction map and the pairing to show that if the image of the localization map avoids the local images of the curve at primes in $S$ , then no global points exist.
Algorithmic Implementation: The algorithm (Algorithm 3.4) computes a set $V_{B,S}(Y)$ , which represents the intersection of the orthogonal complement of a subgroup $B \subset (L^\times/L^{\times 2})_{S, N=1}$ and the product of local images $\prod_{v \in S} C_f(Y(k_v))$ . If this set is empty, the authors conclude that $Y(k)$ is empty.
Generalization: The method is generalized to arbitrary smooth projective curves of genus $g \geq 2$ . For hyperelliptic curves, it aligns with existing descent methods; for non-hyperelliptic curves, it utilizes the scheme of odd theta characteristics (or bitangents for quartics) to define the descent setup.

Key Contributions

Unconditional Results without Full Unit Groups: The primary contribution is a method to obtain unconditional results regarding the non-existence of rational points or divisors without computing the full $S$ -unit group of the étale algebra. This significantly lowers the computational barrier for curves where the algebra degree is high (e.g., degree 28 for plane quartics).
Index Determination: The paper provides a framework to determine the index of a plane quartic (the greatest common divisor of the degrees of closed points). Specifically, it can prove that a curve has index 2 or 4 even when the maximum local index is strictly smaller (e.g., local index 1 or 2), by showing the non-existence of rational divisors of degree 1 or 2.
Simplified Proof of Correctness: The authors present a simplified proof of correctness for the descent algorithm compared to previous works, relying on Hilbert reciprocity and Poitou-Tate duality.
Optimization of Local Computations: The paper describes techniques to reduce the set of primes $S$ required for computation, particularly by identifying primes where the local image is unramified (e.g., primes with odd Tamagawa numbers), thereby avoiding expensive local image computations.

Results and Examples
The authors implemented the algorithm in Magma and applied it to several datasets:

Revisiting a Conditional Example: They re-examined a curve from previous literature ( $x^4 + y^4 + x^2yz + 2xyz^2 - y^2z^2 + z^4 = 0$ ) which was previously known to have no rational points only conditionally on the Generalized Riemann Hypothesis (GRH). Using their method, they proved $X(\mathbb{Q}) = \emptyset$ unconditionally, avoiding the need to compute the class group of a degree 28 number field with a Minkowski bound exceeding $10^{22}$ .
Database Analysis: Applying the method to a database of 82,241 smooth plane quartics over $\mathbb{Q}$ with small discriminant, they identified 135 curves that are counterexamples to the Hasse principle (everywhere locally soluble but globally empty). For over 90% of the locally soluble curves in the database lacking obvious rational points, the method successfully found an obstruction.
Index 2 and 4 Examples: The authors provided explicit examples of curves with global index 2 or 4 where the maximum local index was 1 or 2, respectively. For instance, they proved a specific curve has index 4 despite having local index 2 everywhere, by showing the non-existence of rational divisors of degree 2.

Significance and Claims
The paper claims that its approach "significantly improves on the applicability over previous 2-cover descent methods" by removing the bottleneck of computing full $S$ -unit groups. The authors emphasize that the method is practical and has been successfully applied to curves where previous techniques were computationally intractable.

The paper explicitly states that the obstruction computed is equivalent to the finite abelian descent obstruction and the Brauer-Manin obstruction for suitably large sets $S$ (Corollary 6.7). However, the authors are modest regarding the extension of their computational advantages to bounding the rank of the Jacobian; they note that while their method computes a subset of the Selmer set, determining the size of the kernel (and thus the rank) still requires full information on the class and unit groups of $L$ , which their method does not provide.

The work demonstrates that explicit Brauer-Manin obstructions can be computed effectively for plane quartics, resolving cases of the Hasse principle and determining indices that were previously out of reach for unconditional proofs.