Counterexample to the Hasse Principle

• Counterexample to the Hasse Principle is an arithmetic phenomenon where varieties are locally solvable at every completion yet lack global rational points due to a nontrivial Brauer–Manin obstruction.
• Explicit families defined by equations such as (x² - a y²)(z² - b t²)(u² - ab w²) = c illustrate failures linked to the arithmetic of coflasque tori and precise local conditions.
• Quantitative density results reveal that these counterexamples, while infinite and explicitly constructed, are sparse among all varieties, highlighting the nuanced limits of local-to-global principles.

A counterexample to the Hasse principle is a variety, curve, or equation defined over a global field (typically ℚ) that possesses a solution over every local completion (i.e., it is everywhere locally solvable) but does not have a rational solution (a global point). Such failures directly contradict the naive expectation that local solvability at all completions should guarantee global solvability. Counterexamples have been constructed via diverse mechanisms, frequently revealing the subtle arithmetic of algebraic varieties, class groups, and higher cohomological obstructions.

1. Explicit Families of Counterexamples

Several infinite families of varieties demonstrate the failure of the Hasse principle in a controlled and explicit manner. One notable construction is a family of varieties that are principal homogeneous spaces under coflasque tori, each given by the equation

$$(x^2 - a y^2)(z^2 - b t^2)(u^2 - ab w^2) = c,\quad a, b, c \in \mathbb{Q}^*,$$

where $a, b, c$ must satisfy certain coprimality and square-freeness restrictions. For every place $v$ of $\mathbb{Q}$, these varieties are locally solvable:

$Y(\mathbb{Q}_v) \neq \emptyset.$

Nevertheless, for specific choices of parameters $a, b, c$, there are no rational points:

$Y(\mathbb{Q}) = \emptyset.$

These varieties provide explicit counterexamples to the Hasse principle, directly linking this phenomenon to precise arithmetic and structural features—specifically, to their interpretation as principal homogeneous spaces of certain coflasque tori.

2. The Brauer–Manin Obstruction as the Only Obstruction

The underlying mechanism responsible for these failures is the Brauer–Manin obstruction. For the varieties described, the only obstruction to the Hasse principle arises from the Brauer group of a smooth compactification $Y^c$:

$$\operatorname{Br}(Y^c)/\operatorname{Br}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}$$

(assuming $a, b, ab$ are all non-squares in $\mathbb{Q}^*$). An explicit generator of this group is the quaternion algebra

$$\mathcal{A} = (x^2 - a y^2, b) \in \operatorname{Br}(\mathbb{Q}(Y)).$$

The Brauer–Manin pairing between adelic points and this algebra evaluates local invariants $[c, b]_p$, with the necessary and sufficient obstruction condition

$$\sum_{p: a \notin \mathbb{Q}_p^{*2}} [c, b]_p \equiv 1 \pmod{2}.$$

If this congruence holds, then the global adelic sum is nontrivial, so

$Y(\mathbb{Q})^{\operatorname{Br}} = \emptyset,$

and the Hasse principle fails. In these cases, there is no obstruction at any single place, but the global sum of local invariants of $\mathcal{A}$ is nonzero, ruling out $\mathbb{Q}$-points.

3. Quantitative Density Results

The paper quantifies how frequently these counterexamples arise in their family. For parameters $a, b, c$ restricted to a subset $S \subset \mathbb{Z}^3$ with additional square-freeness and coprimality constraints, and the parameter box

$$S(P) = \{(a, b, c) \in S : \max\{|a|, |b|, |c|\} \leq P\},$$

one defines

• $N_{\text{fail}}(P)$: the count of parameter triples such that $Y(\mathbb{Q}) = \emptyset$
• $N_{\text{total}}(P)$: the total number of parameter triples in $S(P)$

The asymptotics are:

$$N_{\text{fail}}(P) = \frac{\tau_1 P^3}{\log P} - \frac{\tau_2 P^3}{(\log P)^{3/2}}(1 + o(1)),$$

$$N_{\text{total}}(P) = \frac{864}{\pi^6} P^3 + O(P^3 / \log P).$$

Thus, as $P \to \infty$, the proportion of counterexamples shrinks to zero, establishing that these counterexamples are "thin": almost all varieties in this family have a rational point, yet the counterexamples are explicit, infinite, and arithmetically meaningful.

4. Implications for the Hasse Principle

These results have far-reaching implications:

• They yield explicit families of varieties where the Hasse principle fails for arithmetic reasons traceable to a single nontrivial Brauer class.
• They strongly substantiate the philosophy that, for "sufficiently nice" varieties (e.g., principal homogeneous spaces of coflasque tori), the Brauer–Manin obstruction is the only obstruction—no further obstructions exist (e.g., from covering spaces).
• The investigation combines algebraic techniques (structure of the Brauer group computed via tori and their compactifications) with analytic number theory, via detailed estimates of the density and asymptotic counts of parameter sets leading to failures.

A crucial point is the naturality of the examples: the varieties are given by explicit equations, the Brauer group and its generator are computable, and the arithmetic function $h(a, b, c) = \prod_p (c, b)_p$, and corresponding local conditions, can be explicitly checked.

5. Summary of Key Formulas

Key formulas summarizing the arithmetic of these counterexamples include:

• The variety:

$(x^2 - a y^2)(z^2 - b t^2)(u^2 - ab w^2) = c$

• Generator of the Brauer group:

$$\mathcal{A} = (x^2 - a y^2, b) \in \operatorname{Br}(\mathbb{Q}(Y))$$

• Obstruction criterion:

$$\sum_{p: a \notin \mathbb{Q}_p^{*2}} [c, b]_p \equiv 1 \pmod{2} \implies Y(\mathbb{Q}) = \emptyset$$

• Density asymptotics:

$$N_{\text{fail}}(P) = \frac{\tau_1 P^3}{\log P} - \frac{\tau_2 P^3}{(\log P)^{3/2}}(1 + o(1))$$

6. Broader Context and Significance

This construction—and its quantitative analysis—refines our understanding of the Hasse principle's limitations. It exemplifies how arithmetic structures within the Brauer group, as detected by explicit quaternion algebras and Hilbert symbol calculations, can entirely account for the absence of rational points even amid universal local solubility. The precise upper and lower bounds for the density of these counterexamples reveal their arithmetic rarity but also their philosophical importance, illustrating both the reach and the boundaries of the local-to-global principle in arithmetic geometry.