# Hasse principle

The **Hasse principle**, also called the **local-global principle**, is a statement in number theory that an equation or variety has a solution over a global field, such as the rational numbers, whenever it has a solution over every local completion of that field, including the real numbers and the p-adic numbers (number systems extending the rationals, one per prime) for every prime p.<sup>[1](https://mathworld.wolfram.com/HassePrinciple.html)</sup><sup> • </sup><sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup> Solvability over all local fields is a necessary condition for a rational solution, since a rational solution yields one in every completion; the principle asks when this necessary condition is also sufficient.<sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup>

| Key fact | Detail |
|---|---|
| Statement | A collection of equations satisfies the Hasse principle if solvability in R and in all Q_p implies solvability in Q; for homogeneous equations the trivial all-zero solution is usually excluded.<sup>[1](https://mathworld.wolfram.com/HassePrinciple.html)</sup> |
| Origin | Originally discovered by Minkowski (1890) and Hasse (1923, 1924) in the classification of quadratic forms over algebraic number fields.<sup>[3](https://webhomes.maths.ed.ac.uk/~v1ranick/books/dublin.pdf)</sup> |
| Hasse-Minkowski theorem | Hasse proved it over Q in his 1921 Ph.D. thesis and extended it to quadratic forms over all number fields in 1924.<sup>[4](https://digitalcommons.lib.uconn.edu/srhonors_theses/17/)</sup> |
| Norm theorem | For a cyclic extension K/k of number fields, an element b of k that is a norm everywhere locally is a norm globally.<sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup> |
| First failures | Counterexamples due to Lind and, independently a bit later, Reichardt.<sup>[5](https://www.sciencedirect.com/science/article/abs/pii/S0021869323006294)</sup> |
| Famous counterexample | Selmer (1951): the plane cubic 3x^3 + 4y^3 + 5z^3 = 0 has a point over every Q_v but no rational point.<sup>[6](https://math.berkeley.edu/~fengt/4Bhargava_Rubinfest.pdf)</sup> |
| Interpretation | Failures for genus 1 curves are represented by nontrivial elements of the Tate-Shafarevich group of the Jacobian.<sup>[7](https://www.jmilne.org/math/Books/EC2.pdf)</sup> |

## The local-global framework

A global field is an algebraic number field, a finite extension of Q, or a function field over a finite field. Each place v of such a field gives a completion k_v, and the completions include the real or complex numbers at archimedean places and p-adic fields at the finite places.<sup>[8](https://www.math.uni-sb.de/ag/schulze/rsp_lecturenotes_quadraticforms.pdf)</sup> The p-adic numbers, introduced by [Kurt Hensel](https://www.edgepedia.org/kurt-hensel), extend the rationals in a different direction from the real numbers, one system for each prime; Hasse (1898-1979) built the framework of modern number theory on these extensions.<sup>[9](https://ocw.u-tokyo.ac.jp/lecture_files/gf_15/3/notes/en/03saito_eng.pdf)</sup>

When the existence of solutions everywhere locally implies the existence of a global solution, the equation is said to satisfy a Hasse principle.<sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup> Many later local-global questions follow the same pattern: given a global field k, does validity of a property for all, or for all but finitely many, local fields k_v ensure validity for k.<sup>[10](https://link.springer.com/article/10.1007/s40879-022-00567-z)</sup>

## The Hasse-Minkowski theorem

According to the Hasse-Minkowski theorem, a longstanding result, a quadratic form defined over a number field has a nontrivial zero whenever every completion of the field, taken at each of its places, contains a nontrivial zero of that form.<sup>[11](https://mathtube.org/lecture/notes/local-global-principles-quadratic-forms)</sup> In the form stated by Cassels: if F is a quadratic form over an algebraic number field k and F = 0 has a nontrivial solution everywhere locally, then it has a nontrivial solution in k.<sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup> Over the rationals, the theorem says a quadratic form with rational coefficients represents zero in Q if and only if it represents zero in R and in every Q_p.<sup>[12](https://www-users.cse.umn.edu/~garrett/students/reu/hasse_minkowski.pdf)</sup>

Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921 and extended it to quadratic forms over all number fields in 1924; historically it was the first notable application of p-adic fields that caught the attention of a wide mathematical audience.<sup>[4](https://digitalcommons.lib.uconn.edu/srhonors_theses/17/)</sup> The proof over Q divides by the number of variables n, treating n = 2, n = 3, n = 4, and n at least 5 separately after reducing to diagonal forms.<sup>[12](https://www-users.cse.umn.edu/~garrett/students/reu/hasse_minkowski.pdf)</sup> The same local-global idea gives Hasse's norm theorem: for a cyclic extension K/k, an element of k that is a norm everywhere locally is a norm globally.<sup>[2](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)</sup>

A second classical local-global theorem of the period is the Albert-Brauer-Hasse-Noether theorem of 1932 on central simple algebras: a central simple algebra over a number field splits over k if and only if it splits over every completion k_v.<sup>[10](https://link.springer.com/article/10.1007/s40879-022-00567-z)</sup>

## Counterexamples

The local condition is not sufficient in general. The first examples of varieties violating the local-global principle are due to Lind and, independently but a bit later, to Reichardt.<sup>[5](https://www.sciencedirect.com/science/article/abs/pii/S0021869323006294)</sup> Aitken and Lemmermeyer developed counterexamples of this type using only undergraduate number theory and algebra, extending the classical Lind-Reichardt counterexample.<sup>[13](https://arxiv.org/abs/1108.6310)</sup>

**Selmer's cubic.** In 1951 Selmer showed that the plane cubic 3x^3 + 4y^3 + 5z^3 = 0 has a Q_v-point for every place v but no Q-point; it represents a nontrivial element of the Tate-Shafarevich group of its Jacobian.<sup>[6](https://math.berkeley.edu/~fengt/4Bhargava_Rubinfest.pdf)</sup> For an elliptic curve E over Q, the Tate-Shafarevich group is the kernel of the map from H^1(Q, E) to the product of the local groups H^1(Q_p, E), so its nonzero elements are represented by curves with Jacobian E that have local points everywhere yet fail the Hasse principle.<sup>[7](https://www.jmilne.org/math/Books/EC2.pdf)</sup>

**Ubiquity of failures.** Failures are not isolated. In response to a question of Bjorn Poonen, it was shown that for each global field k there is an algebraic curve over k violating the Hasse principle, with examples found among Atkin-Lehner twists of elliptic modular curves and, in positive characteristic, Drinfeld modular curves.<sup>[14](https://arxiv.org/abs/0905.3459)</sup> For any number field K there is a genus one curve violating the Hasse principle that can be written as an intersection of two quadrics in P^3, proved via an elliptic curve over K whose Tate-Shafarevich group has a nontrivial 2-torsion subgroup.<sup>[5](https://www.sciencedirect.com/science/article/abs/pii/S0021869323006294)</sup> It is conjectured that infinitely many Fermat curves ax^p + by^p + cz^p = 0 of exponent p are counterexamples.<sup>[15](https://arxiv.org/abs/1601.07698)</sup>

## Obstructions and positive results

**Brauer-Manin obstruction.** Manin showed that in many cases the failure of the Hasse principle for rational points on a variety can be explained by a reciprocity law imposed by the [Brauer group](https://www.edgepedia.org/brauer-group) on the set of adelic points.<sup>[10](https://link.springer.com/article/10.1007/s40879-022-00567-z)</sup> The obstruction is not a complete explanation: working over any global field whose characteristic is not 2, Poonen built a variety X having no k-points, yet its lack of points is not accounted for by the Brauer-Manin obstruction, nor even when that obstruction is applied to finite étale covers.<sup>[16](https://annals.math.princeton.edu/2010/171-3/p17)</sup> For Enriques surfaces, an example was constructed with empty étale-Brauer set, hence no rational points, for which there is no algebraic Brauer-Manin obstruction to the Hasse principle.<sup>[17](https://www.sciencedirect.com/science/article/pii/S0001870810004536)</sup>

**Positive results in high dimension.** The principle holds for broad classes of varieties. Heath-Brown established the Hasse principle for smooth intersections of two quadrics in projective space of dimension 7, and Molyakov (2023) generalized this to non-conical, possibly singular intersections of two quadrics in P^7 over a number field.<sup>[18](https://arxiv.org/abs/2305.00313)</sup> For systems of two cubic forms defining a smooth projective variety in P^(n-1) over Q, the Hasse principle holds when n is at least 39.<sup>[19](https://durham-repository.worktribe.com/output/1746144/on-the-hasse-principle-for-complete-intersections)</sup> A single diagonal quartic equation obeys the Hasse principle when it has at least 12 variables, by work of Vaughan, while the case of 11 variables remains open.<sup>[20](https://academic.oup.com/qjmath/article/74/1/101/6633716)</sup>

## References

1. [Hasse Principle, Wolfram MathWorld](https://mathworld.wolfram.com/HassePrinciple.html)
2. [J. W. S. Cassels, Diophantine equations, Chapter 11 of Local Fields, Cambridge University Press](https://www.cambridge.org/core/books/abs/local-fields/diophantine-equations/9256587C881E8A33770F3A4F09258855)
3. [On the history of the algebraic theory of quadratic forms](https://webhomes.maths.ed.ac.uk/~v1ranick/books/dublin.pdf)
4. [Adam Gamzon, The Hasse-Minkowski Theorem, University of Connecticut Honors Scholar Theses (2006)](https://digitalcommons.lib.uconn.edu/srhonors_theses/17/)
5. [Han Wu, There are genus one curves violating Hasse principle over every number field, Journal of Algebra 643 (2024)](https://www.sciencedirect.com/science/article/abs/pii/S0021869323006294)
6. [Manjul Bhargava, A positive proportion of plane cubics fail/satisfy the Hasse principle, notes by Tony Feng](https://math.berkeley.edu/~fengt/4Bhargava_Rubinfest.pdf)
7. [James S. Milne, Elliptic Curves, The Arithmetic of Elliptic Curves](https://www.jmilne.org/math/Books/EC2.pdf)
8. [Quadratic Forms over Global Fields and their Integers, lecture notes, Universität des Saarlandes](https://www.math.uni-sb.de/ag/schulze/rsp_lecturenotes_quadraticforms.pdf)
9. [Local-global principle (Hasse principle 1921), University of Tokyo OCW lecture notes](https://ocw.u-tokyo.ac.jp/lecture_files/gf_15/3/notes/en/03saito_eng.pdf)
10. [Local-global questions for divisibility in commutative algebraic groups, European Journal of Mathematics (2022)](https://link.springer.com/article/10.1007/s40879-022-00567-z)
11. [Local-global principles for quadratic forms, mathtube.org](https://mathtube.org/lecture/notes/local-global-principles-quadratic-forms)
12. [The Hasse-Minkowski Theorem, University of Minnesota REU notes](https://www-users.cse.umn.edu/~garrett/students/reu/hasse_minkowski.pdf)
13. [Wayne Aitken and Franz Lemmermeyer, Counterexamples to the Hasse principle, arXiv:1108.6310](https://arxiv.org/abs/1108.6310)
14. [Curves over global fields violating the Hasse Principle, arXiv:0905.3459](https://arxiv.org/abs/0905.3459)
15. [Contre-exemples au principe de Hasse pour les courbes de Fermat, arXiv:1601.07698](https://arxiv.org/abs/1601.07698)
16. [Bjorn Poonen, Insufficiency of the Brauer-Manin obstruction applied to étale covers, Annals of Mathematics 171 (2010)](https://annals.math.princeton.edu/2010/171-3/p17)
17. [Failure of the Hasse principle for Enriques surfaces, Advances in Mathematics](https://www.sciencedirect.com/science/article/pii/S0001870810004536)
18. [Alexander Molyakov, Le principe de Hasse pour les intersections de deux quadriques dans P^7, arXiv:2305.00313](https://arxiv.org/abs/2305.00313)
19. [On the Hasse Principle For Complete Intersections, Compositio Mathematica 160 (2024)](https://durham-repository.worktribe.com/output/1746144/on-the-hasse-principle-for-complete-intersections)
20. [Jörg Brüdern and Trevor D. Wooley, Pairs of diagonal quartic forms: the non-singular Hasse principle, Quarterly Journal of Mathematics 74 (2023)](https://academic.oup.com/qjmath/article/74/1/101/6633716)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Arithmetic geometry › Rational and integral points*

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