Skip to main content

Computational Number Theory, Past, Present, and Future

  • Chapter
  • First Online:
Mathematics Going Forward

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2313))

  • 1413 Accesses

Abstract

A survey of computational number theory viewed through the prism of my own experience and the Pari/GP software.

For Catriona Byrne, with thanks

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Ann. Math.160, 781–793 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Akhilesh. Double tails of multiple zeta values. J. Number Theory170, 228–249 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  3. Algorithmic Number Theory Symposia I to IX. Lecture Notes in Computer Science 877 (1994), 1122 (1996), 1423 (1998), 1838 (2000), 2369 (2002), 3076 (2004), 4076 (2006), 5011 (2008), 6197 (2010), Springer-Verlag; X, XIII, and XIV, Open Book Series 1 (2012), 2 (2018), 4 (2020), XI and XII, J. of Computation and Math.17A (2014), 19A (2016), London Math. Soc.

    Google Scholar 

  4. B. Allombert. An efficient algorithm for the computation of Galois automorphisms. Math. Comp.73, 359–375 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Allombert. Explicit computation of isomorphisms between finite fields. Finite Fields Appl.8, 332–342 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Anderson et al., Improved bounds on number fields of small degree, arXiv:2204.01651 (2022).

    Google Scholar 

  7. J. Arias de Reyna. High precision computation of Riemann’s zeta function by the Riemann–Siegel formula, I. Math. Comp.80 (2011), 995–1009, and II, arXiv:2201.00342 (2022).

    Google Scholar 

  8. O. Atkin and J. Lehner. Hecke operators on Γ0(m). Math. Annalen185, 134–160 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  9. O. Atkin and F. Morain. Elliptic curves and primality proving. Math. Comp.61, 29–68 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  10. O. Atkin and P. Swinnerton-Dyer. Modular forms on congruence subgroups. Proc. Sympos. Pure Math. 19, American Math. Soc. 1–25 (1971).

    Google Scholar 

  11. A. Baily. On the density of discriminants of quartic fields. J. reine angew. Math.315, 190–210 (1980).

    MathSciNet  MATH  Google Scholar 

  12. R. Balasubramanian, J.-M. Deshouillers and F. Dress. Problème de Waring pour les bicarrés. I. Comptes Rendus Acad. Sci Paris303 85–88, and II 161–163 (1986).

    Google Scholar 

  13. A. Bartel, H. Johnston and H. W. Lenstra, Jr. Galois module structure of oriented Arakelov class groups. arXiv:2005.11533 (2020).

    Google Scholar 

  14. F. Beukers, H. Cohen and A. Mellit. Finite hypergeometric functions. Pure Appl. Math. Q.11, 559–589 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Belabas. A fast algorithm to compute cubic fields. Math. Comp.66, 1213–1237 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Belabas. A relative van Hoeij algorithm over number fields. J. Symbolic Computation37, 641–668 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Belabas and H. Cohen. Modular forms in Pari/GP. In: Research in the Math. Sciences5, Springer, 46–64 (2018).

    Google Scholar 

  18. K. Belabas and H. Cohen. Numerical Algorithms for Number Theory Using Pari/GP. Math. surveys and monographs 254, American Math. Soc. (2021).

    Google Scholar 

  19. K. Belabas, F. Diaz y Diaz and E. Friedman. Small generators of the ideal class group. Math. Comp.77, 1185–1197 (2008).

    Google Scholar 

  20. K. Belabas and B. Perrin-Riou. Overconvergent modular symbols and p-adic L-functions. arXiv:2101.06960 (2021).

    Google Scholar 

  21. K. Belabas, M. Van Hoeij, J. Klüners and A. Steel. Factoring polynomials over global fields. J. Théor. Nombres Bordeaux21, 15–39 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Bhargava. Higher composition laws I, II, III, and IV. Ann. Math.159, 217–250, 865–886, 1329–1360 (2004) and 172, 1559–1591 (2010).

    Google Scholar 

  23. M. Bhargava. The density of discriminants of quartic rings and fields. Ann. Math.162, 1031–1063 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Bhargava. The density of discriminants of quintic rings and fields. Ann. Math.172, 1559–1591 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman and Y. Zhao. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves. J. Amer. Math. Soc.33, 1087–1099 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  26. M Bhargava, A. Shankar, and X. Wang, An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree, arXiv:2204.01331 (2022).

    Google Scholar 

  27. Y. Bilu and G. Hanrot. Solving Thue equations of high degree. J. Number Theory60, 373–392 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Birch and P. Swinnerton-Dyer. Notes on elliptic curves I and II. J. reine angew. Math.212, 7–25 (1963), 218, 79–108 (1965).

    Google Scholar 

  29. B. Birch and W. Kuyk (eds). Modular forms in one variable IV. Lecture Notes in Math. 476, Springer-Verlag (1975).

    Google Scholar 

  30. A. Booker. Artin’s conjecture, Turing’s method, and the Riemann hypothesis. Experiment. Math.15, 385–407 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Booker, A. Strömbergsson and A. Venkatesh. Effective computation of Maass cusp forms. Int. Math. Res. Not., Art. ID 71281, 34 (2006).

    Google Scholar 

  32. W. Bosma, J. Cannon and C. Playoust. The Magma algebra system I. The user language. J. Symbolic Comput.24, 235–265 (1997).

    Google Scholar 

  33. J. Buchmann. On the computation of units and class numbers by a generalization of Lagrange’s algorithm. J. Number Theory26, 8–30(1987).

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Buchmann, C. Thiel and H. Williams. Short representations of quadratic integers. In: Math. and its applications325, Kluwer, 159–185 (1995).

    Google Scholar 

  35. J. Buhler. Icosahedral Galois representations. Lecture Notes in Math. 654, Springer–Verlag (1978).

    Google Scholar 

  36. A. Brumer and K. Kramer. Paramodular abelian varieties of odd conductor. Trans. Amer. Math. Soc.366, 2463–2516 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  37. H. Cohen. A Course in Computational Algebraic Number Theory (fourth corrected printing). Graduate Texts in Math. 138, Springer-Verlag (2000).

    Google Scholar 

  38. H. Cohen. Advanced Topics in Computational Number Theory. Graduate Texts in Math. 193, Springer-Verlag (2000).

    Google Scholar 

  39. H. Cohen. Number Theory I, Tools and Diophantine Equations. Graduate Texts in Math. 239, Springer-Verlag (2007).

    Google Scholar 

  40. H. Cohen. Number Theory II, Analytic and Modern Tools. Graduate Texts in Math. 240, Springer-Verlag (2007).

    Google Scholar 

  41. H. Cohen. Computing L-functions: A survey. J. Th. Nombres Bordeaux27, 699–726 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  42. H. Cohen. Computational number theory in relation with L-functions. In: I. Inam and E. Büyükaşik (eds.), International Autumn School on Computational Number Theory, 171–266, Birkhäuser (2019).

    Google Scholar 

  43. H. Cohen, F. Diaz y Diaz and M. Olivier. Computing ray class groups, conductors, and discriminants., Math. Comp.67, 773–795 (1998).

    Google Scholar 

  44. H. Cohen, F. Diaz y Diaz and M. Olivier. A table of totally complex number fields of small discriminant. In: Proceedings ANTS XIII, Lecture Notes in Computer Science 1423, Springer-Verlag, 381–391 (1998).

    Google Scholar 

  45. H. Cohen, F. Diaz y Diaz and M. Olivier. Enumerating quartic dihedral extensions of \({\mathbb Q}\). Compositio Math.133, 65–93 (2002).

    Google Scholar 

  46. H. Cohen, F. Diaz y Diaz and M. Olivier. Constructing complete tables of quartic fields using Kummer theory. Math. Comp.72, 941–951 (2003).

    Google Scholar 

  47. H. Cohen and G. Frey (eds). Handbook of elliptic and elliptic and hyperelliptic curve cryptography. CRC Press (2006).

    Google Scholar 

  48. H. Cohen and H.W. Lenstra, Jr. Primality testing and Jacobi sums. Math. Comp.42, 297–330 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Cohen and A.K. Lenstra. Implementation of a new primality test. Math. Comp.48, 103–121 and S1–S4 (1987).

    Google Scholar 

  50. H. Cohen and H.W. Lenstra, Jr. Heuristics on class groups of number fields. Lecture Notes in Math. 1068, 33–62, Springer-Verlag (1984).

    Google Scholar 

  51. H. Cohen and J. Martinet. Étude heuristique des groupes de classes des corps de nombres. J. für die reine und angew. Math.404, 39–76 (1990).

    MathSciNet  MATH  Google Scholar 

  52. H. Cohn. The density of abelian cubic fields. Proc. Amer. Math. Soc.5, 476–477 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  53. D. Corwin. From Chabauty’s method to Kim’s non-abelian Chabauty’s method. Unpublished manuscript, 41 pp, (2021).

    Google Scholar 

  54. J.-M. Couveignes. Enumerating number fields. Ann. Math.192, 487–497 (2020) and arXiv:1907.13617 (2019).

    Google Scholar 

  55. J. Cremona. Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. Compositio Math.51, 275–324 (1984).

    MathSciNet  MATH  Google Scholar 

  56. J. Cremona. Algorithms for modular elliptic curves (2nd ed.) Cambridge Univ. Press (1997).

    Google Scholar 

  57. J. Cremona. The L-functions and modular forms database project. Found. Comp. Math.16, 1541–1553 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  58. H. Davenport and H. Heilbronn. On the density of discriminants of cubic fields I. Bull. London Math. Soc.1, 345–348 (1969), and II, Proc. Royal Soc. A322, 405–420 (1971).

    Google Scholar 

  59. L. Dembélé and J. Voight. Explicit methods for Hilbert modular forms. In: Elliptic curves, Hilbert modular forms, and Galois deformations, 135–198, Birkhäuser (2013).

    Google Scholar 

  60. J.-M. Deshouillers, F. Hennecart and B. Landreau. Waring’s problem for sixteen biquadrates - Numerical results. J. Th. Nombres Bordeaux12, 411–422 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  61. T. Dokchitser. Computing special values of motivic L-functions. Exp. Math.13, 137–149 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  62. B. Edixhoven and J.-M. Couveignes (eds.) Computational aspects of modular forms and Galois representations. Annals of Math. Studies 176, Princeton Univ. Press (2011).

    Google Scholar 

  63. N. Elkies. Elliptic and modular curves over finite fields and related computational issues. In: Computational perspectives in number theory7, 21–76, American Math. Soc. (1998).

    Google Scholar 

  64. N. Elkies. Three lectures on elliptic surfaces and curves of high rank. arXiv:0709.2908 (2007).

    Google Scholar 

  65. J. Ellenberg and A. Venkatesh. Reflection principles and bounds for class group torsion. Int. Math. Res. Not.1 (2007).

    Google Scholar 

  66. A. Enge. Computing modular polynomials in quasi-linear time. Math. Comp.78, 1809–1824 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  67. A. Enge and F. Johansson. paritwine 0.1. INRIA (2019). https://www.multiprecision.org/paritwine/.

  68. A. Enge and A. Sutherland. Class invariants by the CRT method. In: Proceedings ANTS IX, Lecture Notes in Comp. Science 6197, 142–156 (2010).

    Google Scholar 

  69. D. Farmer, S. Koutsioliotas and S. Lemurell. Maass forms on \( \operatorname {\mathrm {GL}}(3)\) and \( \operatorname {\mathrm {GL}}(4)\)., Int. research notices22 and arXiv:1212.4544 (2012).

    Google Scholar 

  70. K. Fischer. The Zetafast algorithm for computing zeta functions. arXiv:1703.01414v7 (2017).

    Google Scholar 

  71. F. Fité, K. Kedlaya and A. Sutherland. Sato–Tate groups of abelian threefolds: a preview of the classification. Contemp. Math.770, 103–129 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  72. E. Fouvry and J. Klüners. On the 4-rank of class groups of quadratic number fields. Invent. Math.167, 455–513 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  73. F. Gerth. The 4-class ranks of quadratic fields. Invent. Math.77, 489–515 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  74. D. Goldfeld. Gauss’ class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc.13, 23–37 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  75. X. Gourdon. Algorithmique du théorème fondamental de l’algèbre. Rapport de recherche 1852 INRIA (1993).

    Google Scholar 

  76. L. Grenié and G. Molteni. Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH. Math. Comp.85, 889–906 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Hafner and K. McCurley. A rigorous subexponential algorithm for computation of class groups. J. American Math. Soc.2, 837–850 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  78. D. Harvey. Counting points on hyperelliptic curves in average polynomial time. Ann. Math.179, 783–803 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  79. D. Harvey. Computing zeta functions of arithmetic schemes. Proc. London Math. Soc.111, 1379–1401 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  80. D. Harvey and J. van der Hoeven. Integer multiplication in time \(O(n\log (n))\). Annals of Math.193, 563–617 (2021).

    Google Scholar 

  81. D. Hejhal. The Selberg trace formula for\( \operatorname {\mathrm {PSL}}_2({\mathbb R})\). I and II, Lecture Notes in Math. 548 and 1001, Springer-Verlag (1976 and 1983).

    Google Scholar 

  82. H. Helfgott. The ternary Goldbach problem. arXiv:1501.05438v2 (2015).

    Google Scholar 

  83. F. Johansson. Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Trans. on Computers66, 1281–1292 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  84. C. Jeannerod, C. Pernet and A. Storjohann. Rank-profile revealing Gaussian elimination and the CUP matrix decomposition. J. Symbolic Comput.56, 46–68 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  85. K. Kedlaya. Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology. J. Ramanujan Math. Soc.16, 318–330 (2001) and 18, 417–418 (2003).

    Google Scholar 

  86. K. Khuri-Makdisi. Asymptotically fast group operations on Jacobians of general curves. Math. Comp.76, 2213–2239 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  87. D.H. Lehmer. Factorization of certain cyclotomic functions. Ann. Math. (2)34, 461–479 (1933).

    Article  MathSciNet  MATH  Google Scholar 

  88. R. Lemke Oliver and F. Thorne. Upper bounds on number fields of given degree and bounded discriminant. arXiv:2005.14110v2 (2020).

    Google Scholar 

  89. H.W. Lenstra, Jr. Factoring integers with elliptic curves. Annals of Math.126, 649–673 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  90. A.K. Lenstra and H.W. Lenstra, Jr. (eds.) The development of the number field sieve. Lecture Notes in Math. 1554, Springer-Verlag (1993).

    Google Scholar 

  91. A.K. Lenstra, H.W. Lenstra, Jr. and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann.261, 515–534 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  92. H.W. Lenstra and R. Tijdeman (eds). Computational methods in number theory. Math. Center Tracts 154/155, Math. Centrum Amsterdam (1982).

    Google Scholar 

  93. The LMFDB collaboration. The L-function and modular form database. http://www.lmfdb.org and http://beta.lmfdb.org.

  94. Q. Liu. Modèles minimaux des courbes de genre deux. J. für die reine und angew. Math.453, 137–164 (1994).

    MathSciNet  MATH  Google Scholar 

  95. W. McCallum and B. Poonen. The method of Chabauty and Coleman. In: Panoramas et synthèses36, Soc. Math. de France, 99–117 (2012).

    Google Scholar 

  96. S. Mäki. The conductor density of abelian number fields. J. London Math. Soc. (2) 47, 18–30 (1993).

    Google Scholar 

  97. G. Malle. On the distribution of Galois groups I., J. Number Theory92, 315–322 (2002), and II., Exp. Math.13, 129–135 (2004).

    Google Scholar 

  98. J.-F. Mestre. Unpublished, but see Section 17.3.2.b in [47].

  99. P. Molin and A. Page. Computing groups of Hecke characters. arXiv:2210.02716.

    Google Scholar 

  100. P. Nguyen and D. Stehlé. Floating-point LLL revisited. In: Proceedings Eurocrypt ’2005, Springer-Verlag (2005).

    Google Scholar 

  101. P. Nguyen and B. Vallée (eds). The LLL algorithm: Survey and applications. Information Security and Cryptography, Springer (2010).

    Google Scholar 

  102. A. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators, and zeros of zeta functions: a survey of recent results. J. Th. Nombres Bordeaux2, 114–141 (1990).

    MathSciNet  MATH  Google Scholar 

  103. A. Odlyzko and A. Schönhage. Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Amer. Math. Soc.309, 797–809 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  104. The PARI Group. PARI/GPversion 2.15.1. Univ. Bordeaux (2022). http://pari.math.u-bordeaux.fr/.

  105. W. Plesken and B. Souvignier. Computing isometries of lattices. J. Symbolic Comp.24, 327–334 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  106. M. Pohst and H. Zassenhaus. Algorithmic algebraic number theory (3rd ed.) Cambridge Univ. Press (1993).

    Google Scholar 

  107. R. Pollack and G. Stevens. Overconvergent modular symbols and p-adic L-functions. Ann. Sci. ENS44, 1–42 (2011).

    MathSciNet  MATH  Google Scholar 

  108. C. Pomerance. The quadratic sieve factoring algorithm. In: EUROCRYPT, Lecture Notes in Comp. Science 209, 169–182, Springer-Verlag (1984).

    Google Scholar 

  109. A. Pethö, M. Pohst, H. Williams and H. Zimmer (eds). Computational number theory, Walter de Gruyter (1991).

    Google Scholar 

  110. B. Poonen. Heuristics for the arithmetic of elliptic curves. arXiv:1711.10112v2 (2017).

    Google Scholar 

  111. B. Poonen and E. Rains. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc.25, 245–269 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  112. C. Poor and D. Yuen. Paramodular cusp forms. Math. Comp.84, 1401–1438 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  113. D. Roberts and F. Rodriguez-Villegas. Hypergeometric motives. arXiv:2109.00027 (2021).

    Google Scholar 

  114. The Sage Developers. SageMath, the Sage Mathematics Software System version 9.7 (2022). https://www.sagemath.org.

  115. T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc.15, 247–270 (2000).

    MathSciNet  MATH  Google Scholar 

  116. R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp.44, 483–494 (1985).

    Google Scholar 

  117. D. Shanks. Class number, a theory of factorization, and genera. In: Proc. Sympos. Pure Math.20, 415–440, American Math. Soc. (1971).

    Google Scholar 

  118. P. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. arXiv:quant-ph/9508027v2 (1996).

    Google Scholar 

  119. C.-L. Siegel. Contributions to the theory of the Dirichlet L-series and the Epstein zeta-functions. Ann. of Math. (2)44, 143–172 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  120. A. Silvester, M. Jacobson and H. Williams. Shorter compact representations in real quadratic fields. In: Number Theory and Cryptography, Lecture Notes in Computer Science 8260, 50–72 (2013).

    Article  MATH  Google Scholar 

  121. N. Smart. The algorithmic resolution of Diophantine equations. London Math. Soc. student texts 41 (1998).

    Google Scholar 

  122. A. Smith. 2∞-Selmer groups, 2∞-class groups, and Goldfeld’s conjecture. arXiv:1702.02325v2 (2017).

    Google Scholar 

  123. C. Smyth. The Mahler measure of algebraic numbers: a survey. In: J. McKee and C. Smyth (eds.), Number theory and polynomials, 322–349, Cambridge Univ. Press (2008).

    Google Scholar 

  124. H. Stark. L-functions at s = 1 I, II, III, IV. Adv. Math.7, 301–343 (1971), 17, 60–92 (1975), 22, 64–84 (1976), 35, 197–235 (1980).

    Google Scholar 

  125. H. Stark. Fourier coefficients of Maass waveforms. In: Modular forms, R. Rankin ed., 263–269, Ellis Horwood (1984).

    Google Scholar 

  126. W. Stein. Modular forms, a computational approach. Graduate Studies in Math. 79, American Math. Soc. (2007).

    Google Scholar 

  127. M. Stoll, Documentation for the ratpoints program, arXiv:0803.3165v5 (2022).

    Google Scholar 

  128. A. Sutherland. Computing Hilbert class polynomials with the Chinese remainder theorem. Math. Comp.80, 501–538 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  129. A. Sutherland. On the evaluation of modular polynomials. In: Proceedings ANTS X, 531–555, Open Book Series 1, msp (2013).

    Google Scholar 

  130. A. Sutherland. Sato–Tate distributions. Contemp. Math.740, 197–248 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  131. A. Sutherland. Counting points on superelliptic curves in average polynomial time. In: Proceedings ANTS XIV, 403–422, Open Book Series 4, msp (2020), or arXiv:2004.10189v4.

    Google Scholar 

  132. D. Tingley. Elliptic curves uniformized by modular functions. Ph.D. thesis, Univ. of Oxford (1975).

    Google Scholar 

  133. A. Turing. Some calculations of the Riemann zeta-function. Proc. London Math. Soc.3, 99–117 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  134. W. Wang and M. Matchett Wood. Moments and interpretations of the Cohen–Lenstra–Martinet heuristics. arXiv:1907.11201v2 (2020).

    Google Scholar 

  135. M. Watkins. Some remarks on Heegner point computations. In: Panoramas et synthèses36, 81–97, Soc. Math. de France (2012).

    Google Scholar 

  136. M. Watkins. A discursus on 21 as a bound for ranks of elliptic curves over \({\mathbb Q}\), and sundry related topics. http://magma.maths.usyd.edu.au/~watkins/papers/DISCURSUS.pdf (2015).

  137. M. Watkins. Class numbers of imaginary quadratic fields. Math. Comp.73, 907–938 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  138. A.-E. Wilke. Thesis, in preparation.

    Google Scholar 

  139. S. Wong. Densities of quartic fields with even Galois groups. Proc. Amer. Math. Soc.133, 2873–2881 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  140. H. Zimmer. Computational problems, methods, and results in algebraic number theory. Lecture Notes in Math. 262, Springer–Verlag (1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Henri Cohen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Cohen, H. (2023). Computational Number Theory, Past, Present, and Future. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_38

Download citation

Publish with us

Policies and ethics