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In mathematics, `!arithmetic geometry`! is roughly the application of techniques from `F33f`_`[algebraic geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_geometry]`_`f to problems in `F33f`_`[number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_theory]`_`f.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] Arithmetic geometry is centered around `F33f`_`[Diophantine geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diophantine_geometry]`_`f, the study of `F33f`_`[rational points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_point]`_`f of `F33f`_`[algebraic varieties`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_variety]`_`f.`:cite-ref-quanta-2-0[`F5bf`_`[2`#cite-note-quanta-2]`_`f]`:cite-ref-poonen-notes-3-0[`F5bf`_`[3`#cite-note-poonen-notes-3]`_`f]

In more abstract terms, arithmetic geometry can be defined as the study of `F33f`_`[schemes`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Scheme_(mathematics)]`_`f of `F33f`_`[finite type`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finite_morphism]`_`f over the `F33f`_`[spectrum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Spectrum_of_a_ring]`_`f of the `F33f`_`[ring of integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_of_integers]`_`f.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

>>Contents

• `F0af`_`[Overview`#overview]`_`f

• `F0af`_`[History`#history]`_`f

• `F0af`_`[19th century: early arithmetic geometry`#19th-century-early-arithmetic-geometry]`_`f

• `F0af`_`[Early-to-mid 20th century: algebraic developments and the Weil conjectures`#early-to-mid-20th-century-algebraic-developments-and-the-weil-conjectures]`_`f

• `F0af`_`[Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond`#mid-to-late-20th-century-developments-in-modularity-p-adic-methods-and-beyond]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Overview

The classical objects of interest in arithmetic geometry are rational points: `F33f`_`[sets of solutions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Solution_set]`_`f of a `F33f`_`[system of polynomial equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=System_of_polynomial_equations]`_`f over `F33f`_`[number fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_field]`_`f, `F33f`_`[finite fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finite_field]`_`f, `F33f`_`[p-adic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=P-adic_field]`_`f, or `F33f`_`[function fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_function_field]`_`f, i.e. `F33f`_`[fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Field_(mathematics)]`_`f that are not `F33f`_`[algebraically closed`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraically_closed]`_`f excluding the `F33f`_`[real numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f. Rational points can be directly characterized by `F33f`_`[height functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Height_function]`_`f which measure their arithmetic complexity.`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]

The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, `F33f`_`[étale cohomology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Étale_cohomology]`_`f provides `F33f`_`[topological invariants`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_property]`_`f associated to algebraic varieties.`:cite-ref-grothendieck-cohomology-6-0[`F5bf`_`[6`#cite-note-grothendieck-cohomology-6]`_`f] `F33f`_`[p-adic Hodge theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=P-adic_Hodge_theory]`_`f gives tools to examine when cohomological properties of varieties over the `F33f`_`[complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f extend to those over `F33f`_`[p-adic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=P-adic_field]`_`f.`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f]

>>History

>>>19th century: early arithmetic geometry

In the early 19th century, `F33f`_`[Carl Friedrich Gauss`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Carl_Friedrich_Gauss]`_`f observed that non-zero `F33f`_`[integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f solutions to `F33f`_`[homogeneous polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Homogeneous_polynomial]`_`f equations with `F33f`_`[rational`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_number]`_`f coefficients exist if non-zero rational solutions exist.`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f]

In the 1850s, `F33f`_`[Leopold Kronecker`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Leopold_Kronecker]`_`f formulated the `F33f`_`[Kronecker–Weber theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kronecker–Weber_theorem]`_`f, introduced the theory of `F33f`_`[divisors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divisor_(algebraic_geometry)]`_`f, and made numerous other connections between number theory and `F33f`_`[algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebra]`_`f. He then conjectured his "`F33f`_`[liebster Jugendtraum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kronecker's_Jugendtraum]`_`f" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his `F33f`_`[twelfth problem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hilbert's_problems]`_`f, which outlines a goal to have number theory operate only with rings that are quotients of `F33f`_`[polynomial rings`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial_ring]`_`f over the integers.`:cite-ref-princeton-9-0[`F5bf`_`[9`#cite-note-princeton-9]`_`f]

>>>Early-to-mid 20th century: algebraic developments and the Weil conjectures

In the late 1920s, `F33f`_`[André Weil`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=André_Weil]`_`f demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the `F33f`_`[Mordell–Weil theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mordell–Weil_theorem]`_`f which demonstrates that the set of rational points of an `F33f`_`[abelian variety`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abelian_variety]`_`f is a `F33f`_`[finitely generated abelian group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finitely_generated_abelian_group]`_`f.`:cite-ref-10[`F5bf`_`[10`#cite-note-10]`_`f]

Modern foundations of algebraic geometry were developed based on contemporary `F33f`_`[commutative algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Commutative_algebra]`_`f, including `F33f`_`[valuation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Valuation_theory]`_`f and the theory of `F33f`_`[ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ideal_(ring_theory)]`_`f by `F33f`_`[Oscar Zariski`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oscar_Zariski]`_`f and others in the 1930s and 1940s.`:cite-ref-11[`F5bf`_`[11`#cite-note-11]`_`f]

In 1949, `F33f`_`[André Weil`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=André_Weil]`_`f posed the landmark `F33f`_`[Weil conjectures`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weil_conjectures]`_`f about the `F33f`_`[local zeta-functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_zeta-function]`_`f of algebraic varieties over finite fields.`:cite-ref-12[`F5bf`_`[12`#cite-note-12]`_`f] These conjectures offered a framework between algebraic geometry and number theory that propelled `F33f`_`[Alexander Grothendieck`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Alexander_Grothendieck]`_`f to recast the foundations making use of `F33f`_`[sheaf theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sheaf_theory]`_`f (together with `F33f`_`[Jean-Pierre Serre`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jean-Pierre_Serre]`_`f), and later scheme theory, in the 1950s and 1960s.`:cite-ref-13[`F5bf`_`[13`#cite-note-13]`_`f] `F33f`_`[Bernard Dwork`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bernard_Dwork]`_`f proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.`:cite-ref-14[`F5bf`_`[14`#cite-note-14]`_`f] Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with `F33f`_`[Michael Artin`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Michael_Artin]`_`f and `F33f`_`[Jean-Louis Verdier`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jean-Louis_Verdier]`_`f) by 1965.`:cite-ref-grothendieck-cohomology-6-1[`F5bf`_`[6`#cite-note-grothendieck-cohomology-6]`_`f]`:cite-ref-15[`F5bf`_`[15`#cite-note-15]`_`f] The last of the Weil conjectures (an analogue of the `F33f`_`[Riemann hypothesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann_hypothesis]`_`f) would be finally proven in 1974 by `F33f`_`[Pierre Deligne`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre_Deligne]`_`f.`:cite-ref-16[`F5bf`_`[16`#cite-note-16]`_`f]

>>>Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond

Between 1956 and 1957, `F33f`_`[Yutaka Taniyama`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Yutaka_Taniyama]`_`f and `F33f`_`[Goro Shimura`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Goro_Shimura]`_`f posed the `F33f`_`[Taniyama–Shimura conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modularity_theorem]`_`f (now known as the modularity theorem) relating `F33f`_`[elliptic curves`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_curves]`_`f to `F33f`_`[modular forms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modular_forms]`_`f.`:cite-ref-17[`F5bf`_`[17`#cite-note-17]`_`f]`:cite-ref-18[`F5bf`_`[18`#cite-note-18]`_`f] This connection would ultimately lead to `F33f`_`[the first proof`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Wiles's_proof_of_Fermat's_Last_Theorem]`_`f of `F33f`_`[Fermat's Last Theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fermat's_Last_Theorem]`_`f in number theory through algebraic geometry techniques of `F33f`_`[modularity lifting`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lift_(mathematics)]`_`f developed by `F33f`_`[Andrew Wiles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Andrew_Wiles]`_`f in 1995.`:cite-ref-wiles1995-19-0[`F5bf`_`[19`#cite-note-wiles1995-19]`_`f]

In the 1960s, Goro Shimura introduced `F33f`_`[Shimura varieties`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Shimura_variety]`_`f as generalizations of `F33f`_`[modular curves`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modular_curve]`_`f.`:cite-ref-20[`F5bf`_`[20`#cite-note-20]`_`f] Since the 1979, Shimura varieties have played a crucial role in the `F33f`_`[Langlands program`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Langlands_program]`_`f as a natural realm of examples for testing conjectures.`:cite-ref-21[`F5bf`_`[21`#cite-note-21]`_`f]

In papers in 1977 and 1978, `F33f`_`[Barry Mazur`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Barry_Mazur]`_`f proved the `F33f`_`[torsion conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Torsion_conjecture]`_`f giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain `F33f`_`[modular curves`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modular_curve]`_`f.`:cite-ref-22[`F5bf`_`[22`#cite-note-22]`_`f]`:cite-ref-23[`F5bf`_`[23`#cite-note-23]`_`f] In 1996, the proof of the torsion conjecture was extended to all number fields by `F33f`_`[Loïc Merel`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Loïc_Merel]`_`f.`:cite-ref-24[`F5bf`_`[24`#cite-note-24]`_`f]

In 1983, `F33f`_`[Gerd Faltings`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gerd_Faltings]`_`f proved the `F33f`_`[Mordell conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Faltings's_theorem]`_`f, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates `F33f`_`[finite generation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finitely_generated_abelian_group]`_`f of the set of rational points as opposed to finiteness).`:cite-ref-25[`F5bf`_`[25`#cite-note-25]`_`f]`:cite-ref-26[`F5bf`_`[26`#cite-note-26]`_`f]

In 2001, the proof of the `F33f`_`[local Langlands conjectures for GLn`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_Langlands_conjectures]`_`f was based on the geometry of certain Shimura varieties.`:cite-ref-27[`F5bf`_`[27`#cite-note-27]`_`f]

In the 2010s, `F33f`_`[Peter Scholze`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Peter_Scholze]`_`f developed `F33f`_`[perfectoid spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Perfectoid_space]`_`f and new cohomology theories in arithmetic geometry over p-adic fields with application to `F33f`_`[Galois representations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Galois_representations]`_`f and certain cases of the `F33f`_`[weight-monodromy conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weight-monodromy_conjecture]`_`f.`:cite-ref-28[`F5bf`_`[28`#cite-note-28]`_`f]`:cite-ref-29[`F5bf`_`[29`#cite-note-29]`_`f]

>>See also

• `F33f`_`[Anabelian geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Anabelian_geometry]`_`f

• `F33f`_`[Frobenioid`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Frobenioid]`_`f

• `F33f`_`[Arithmetic dynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_dynamics]`_`f

• `F33f`_`[Arithmetic of abelian varieties`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_of_abelian_varieties]`_`f

• `F33f`_`[Birch and Swinnerton-Dyer conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Birch_and_Swinnerton-Dyer_conjecture]`_`f

• `F33f`_`[Moduli of algebraic curves`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Moduli_of_algebraic_curves]`_`f

• `F33f`_`[Siegel modular variety`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Siegel_modular_variety]`_`f

• `F33f`_`[Siegel's theorem on integral points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Siegel's_theorem_on_integral_points]`_`f

• `F33f`_`[Category theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Category_theory]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefsutherland2013`aSutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.

`:cite-note-quanta-2`!2.`! `F0af`_`[↑`#cite-ref-quanta-2-0]`_`f `:citerefklarreich2016`aKlarreich, Erica (June 28, 2016). "Peter Scholze and the Future of Arithmetic Geometry". Retrieved March 22, 2019.

`:cite-note-poonen-notes-3`!3.`! `F0af`_`[↑`#cite-ref-poonen-notes-3-0]`_`f `:citerefpoonen2009`a`F33f`_`[Poonen, Bjorn`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bjorn_Poonen]`_`f (2009). "Introduction to Arithmetic Geometry" (PDF). Retrieved March 22, 2019.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f Arithmetic geometry at the `F33f`_`[nLab`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=NLab]`_`f

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citereflang1997`a`F33f`_`[Lang, Serge`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Serge_Lang]`_`f (1997). `*Survey of Diophantine Geometry`*. `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f. pp. 43–67. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-61223-8. `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0869.11051.

`:cite-note-grothendieck-cohomology-6`!6.`! `F0af`_`[↑`#cite-ref-grothendieck-cohomology-6-0]`_`f `:citerefgrothendieck1960`a`F33f`_`[Grothendieck, Alexander`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Alexander_Grothendieck]`_`f (1960). "The cohomology theory of abstract algebraic varieties". `*Proc. Internat. Congress Math. (Edinburgh, 1958)`*. `F33f`_`[Cambridge University Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cambridge_University_Press]`_`f. pp. 103–118. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0130879.

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