Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results

Odlyzko, A. M.

Séminaire de théorie des nombres de Bordeaux, Série 2, Tome 2 (1990) no. 1, pp. 119-141.

Nous présentons une bibliographie d'articles récents sur les bornes inférieures des discriminants de corps de nombres et sur des sujets voisins. Nous discutons quelques unes des principales méthodes, et nous donnons les résultats principaux et des problèmes ouverts.

A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.

1 W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN-Polish Scientific Publishers Warsaw (1974). MR 50# 268. (Very complete references for work before 1973.) | MR | Zbl

2 L.C. Washington, Introduction to Cyclotomic Fields, Springer. 1982. MR 85g: 11001. | MR | Zbl

3 J. Martinet, Méthodes géométriques dans la recherche des petits discriminants. pp. 147-179, Séminaire Théorie des Nombres, Paris, 1983-84, C. Goldstein éd., Birkhäuser Boston, 1985. MR 88h:11083. | MR | Zbl

1 H.M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. math. 23 (1974), 135-152. MR 49 # 7218. | MR | Zbl

2 H.M. Stark, The analytic theory of algebraic numbers, Bull. Am. Math. Soc. 81 (1975), 961-972. MR 56 # 2961. | MR | Zbl

3 A.M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. math. 29 (1975), 275-286. MR 51 # 12788. | MR | Zbl

4 A.M. Odlyzko, Lower bounds. for discriminants of number fields, Acta Arith. 29 (1976), 275-297. MR 53 # 5531. | MR | Zbl

5 A.M. Odplyzko, Lower bounds for discriminants of number fields. II. MR 56 # 309. | Zbl

6 A.M. Odlyzko, On conductors and discriminants. pp. 377-407, Algebraic Number Fields, (Proc. 1975 Durham Symp.), A. Fröhlich, ed., Academic Press 1977. MR 56 #11961. | MR | Zbl

7 J.-P. Serre, Minorations de discriminants. note of October 1975, published on pp. 240-243 in vol. 3 of Jean-Pierre Serre, Collected Papers, Springer 1986.

8 A.M. Odlyzko, Discriminant bounds. tables dated Nov. 29, 1976 (unpublished). Some of these bounds are included in Ref. B12.

9 G. Poitou, Minorations de discriminants (d'après A.M. Odlyzko), Séminaire Bourbaki. Vol. 1975/76 28ème année, Exp. No. 479, pp. 136-153, Lecture Notes in Math. #567, Springer 1977. MR 55 #7995. | Numdam | MR | Zbl

10 G. Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou. 18e année: (1976/77), Théorie des nombres, Fasc. 1, Exp. No. 6, 18pp., Secrétariat Math., Paris, 1977. MR 81i:12007. | Numdam | MR | Zbl

11 F. Diaz Y. Diaz, Tables minorant la racine n-ième du discriminant d'un corps de degré n. Publications Mathématiques d'Orsay 80.06. Université de Paris-Sud, Département de Mathématique, Orsay, (1980). 59 pp. MR 82i:12007. (Some of these bounds are included in Ref. B12.) | MR | Zbl

12 J. Martinet, Petits discriminants des corps de nombres. pp. 151-193, Journées Arithmétiques 1980, J.V. Armitage, ed., Cambridge Univ. Press 1982. MR 84g:12009. | MR | Zbl

1 H.W. Lenstra, Euclidean number fields of large degree, Invent. math. 38 (1977), 237-254. MR 55 # 2836. | MR | Zbl

2 J. Martinet, Tours de corps de classes et estimations de discriminants, Invent. math. 44 (1978), 65-73.. MR 57 # 275. | MR | Zbl

3 J. Martinet, Petits discriminants, Ann. Inst. Fourier (Grenoble) 29, no 1 (1979), 159-170. MR 81h:12006. | Numdam | MR | Zbl

4 R. Schoof, Infinite class field towers of quadratic fields, J. reine angew. Math. 372 (1986), 209-220. MR 88a:11121. | MR | Zbl

1 J.M. Masley, Odlyzko bounds and class number problems. pp. 465-474, Algebraic Number Fields (Proc. Durham Symp., 1975), A. Fröhlich, ed., Academic Press 1977. MR 56 # 5493. | MR | Zbl

2 J.M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), 297-319. MR 80e:12005. | Numdam | MR | Zbl

3 J.M. Masley, Where are number fields with small class numbers ?. pp. 221-242, Number Theory, Carbondale 1979, Lecture Notes in Math. #751, Springer, 1979. MR 81f:12004. | MR | Zbl

4 J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. math. 55 (1979), 37-47. MR 80k:12019. | MR | Zbl

5 J.M. Masley, Class groups of abelian number fields. pp. 475-497, Proc. Queen's Number Theory Conf. 1979, P. Ribenboim ed., Queen's Papers in Pure and Applied Mathematics no. 54, Queen's Univ., 1980, MR 83f:12007. | MR | Zbl

6 J. Martinet, Sur la constante de Lenstra des corps de nombres, Sém. Theorie des Nombres de Bordeaux (1979-1980). Exp. #17, 21 pp., Univ. Bordeaux 1980. MR 83b:12007. | MR | Zbl

7 F.J. Van Der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), 693-707. MR 84e:12005. | MR | Zbl

8 A. Leutbecher and J. Martinet, Lenstra's constant and Euclidean number fields, Astérisque 94 (1982), 87-131. MR 85b:11090. | Numdam | MR | Zbl

9 A. Leutbecher, Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier (Grenoble) 35. no.2 (1985), 83-106. MR 86j:11107. | Numdam | MR | Zbl

10 J. Hoffstein and N. Jochnowitz, On Artin's conjecture and the class number of certain CM fields, Duke Math. J. 59 (1989), 553-563. | MR | Zbl

11 J. Hoffstein and N. Jochnowitz, On Artin's conjecture and the class number of certain CM fields-II, Duke Math. J. 59 (1989), 565-584. | MR | Zbl

1 M. Pohst, Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory 9 (1977), 459-492. MR 57 # 268. | MR | Zbl

2 G. Gras and M.-N. Gras, Calcul du nombre de classes et des unités des extensions abéliennes réelles de Q, Bull. Sci. Math. 101 (2) (1977), 97-129. MR 58 # 586. | MR | Zbl

3 M. Pohst, Eine Regulatorabschätzung, Abh. Math. Sem. Univ. Hamburg 47 (1978), 95-106. MR 58 # 16596. | MR | Zbl

4 R. Zimmert, Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. math. 62 (1981), 367-380. MR 83g:12008. | MR | Zbl

5 G. Poitou, Le théorème des classes jumelles de R. Zimmert, Sém. de Théorie des Nombres de Bordeaux (1983-1984). Exp. # 5, 4 pp., Univ. Bordeaux 1984, (Listed in MR 86b:11003.) | MR | Zbl

6 J. Oesterlé, Le théorème des classes jumelles de Zimmert et les formules explicites de Weil. pp. 181-197, Sém. Théorie des Nombres, Paris 1983-84, C. Goldstein ed., Birkhäuser Boston, 1985. | MR | Zbl

7 J. Silverman, An inequality connecting the regulator and the discriminant of a number field, J. Number Theory 19 (1984), 437-442. MR 86c:11094. | MR | Zbl

8 T.W. Cusick, Lower bounds for regulators. pp. 63-73 in Number Theory, Noordwijkerhout 1983, H. Jager ed., Lecture Notes in Math. # 1068, Springer 1984. MR 85k:11052. | MR | Zbl

9 A.-M. Bergé and J. Martinet, Sur les minorations géométriques des régulateurs. pp. 23-50, Séminaire Théorie des Nombres, Paris 1987- 88, C. Goldstein ed., Birkhäuser Boston, 1990. | MR | Zbl

10 E. Friedman, Analytic formulas for regulators of number fields, Invent. math. 98 (1989), 599-622. | MR | Zbl

11 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press. (1989). | MR | Zbl

12 R. Schoof and L.C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. | MR | Zbl

13 A.-M. Bergé and J. Martinet, Notions relatives de régulateurs et de hauteurs, Acta Arith. 54 (1989), 155-170. | MR | Zbl

14 A.-M. Bergé and J. Martinet, Minorations de hauteurs et petits régulateurs relatifs, Sém. Théorie des Nombres Bordeaux (1987-88). Exp.#11, Univ. Bordeaux 1988. | Zbl

15 A. Costa and E. Friedman, Ratios of regulators in totally real extensions of number fields. to be published.

16 E. Friedman and N.-P. Skoruppa, Explicit formulas for regulators and ratios of regulators of number fields. manuscript in preparation.

17 T.W. Cusick, The: egulator spectrum of totally real cubic fields. to appear. | MR | Zbl

1 P. Cartier and Y. Roy, On the enumeration of quintic fields with small discriminants, J. reine angew. Math 268/269 (1974), 213-215. MR 50 # 2119. | MR | Zbl

2 M. Pohst, Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper, J. reine angew. Math. 278/279 (1975), 278-300. MR 52 # 8085. | MR | Zbl

3 M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields. pp. 235-240, Number theory and algebra, H. Zassenhaus ed., Academic Press 1977. MR 57 # 5952. | MR | Zbl

4 J. Liang and H. Zassenhaus, The minimum discriminant of sixth degree totally complex algebraic number field, J. Number Theory 9 (1977), 16-35. MR 55 # 305. | MR | Zbl

5 M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117. MR 83g:12009. | MR | Zbl

6 M. Pohst, P. Weiler, and H. Zassenhaus, On effective computation of fundamental units, Math. Comp. 38 (1982), 293-329. MR 83e:12005b. | MR | Zbl

7 D.G. Rish, On algebraic number fields of degree five, Vestnik Moskov. Univ. Ser. I Mat. Mekh. no 2, (1982), 76-80. English translation in Moscow Univ. Math. Bull. 37 (1982), no. 99-103. MR 83g:12006. | MR | Zbl

8 F. Diaz Y. Diaz, Valeurs minima du discriminant des corps de degré 7 ayant une seule place réelle, C.R. Acad. Sc. Paris 296 (1983), 137-139. MR 84i:12004. | MR | Zbl

9 F. Diaz Y. Diaz, Valeurs minima du discriminant pour certains types de corps de degré 7, Ann. Inst. Fourier (Grenoble) 34, no 3 (1984), 29-38. MR 86d:11091. | Numdam | MR | Zbl

10 K. Takeuchi, Totally real algebraic number fields of degree 5 and 6 with small discriminant, Saitama Math. J. 2 (1984), 21-32. MR 86i:11060. | MR | Zbl

11 H.J. Godwin, On quartic fields of signature one with small discriminant. II, Math. Comp. 42 (1984). 707-711. Corrigendum, Math. Comp. 43 (1984), 621. MR 85i:11092a, 11092b. | MR | Zbl

12 F. Diaz Y. Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory 25 (1987), 34-52. | MR | Zbl

13 S.-H. Kwon and J. Martinet, Sur les corps résolubles de degré premier, J. reine angew. Math. 375/376 (1987), 12-23. MR 88g:11080. | MR | Zbl

14 F. Diaz Y. Diaz, Discriminants minima et petits discriminants des corps de nombres de degré 7 avec cinq places réelles, J. London Math. Soc. 38 (2) (1988), 33-46.. | MR | Zbl

15 J. Buchmann and D. Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), 161-174. | MR | Zbl

16 S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens. preprint 1987.

17 P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 107, Math. Comp. 50 (1988), 581-594. | MR | Zbl

18 J. Buchmann, M. Pohst and J. V. Schmettow, On the computation of unit groups and class groups of totally real quartic fields, Math. Comp. 53 (1989), 387-397. | MR | Zbl

19 A.-M. Bergé, J. Martinet and M. Olivier, The computation of sextic fields with a quadratic subfield. in Math. Comp. to appear. | MR | Zbl

20 F. Diaz Y. Diaz, Table de corps quintiques totalement réels. Université de Paris-Sud, Département de Mathématiques, Orsay, Report no. 89-14, 35 pp., 1989.

21 M. Pohst, J. Martinet and F. Diaz Y. Diaz, The minimum discriminant of totally real octic fields, J. Number Theory. to appear. | Zbl

1 J. Hoffstein, Some results related to minimal discriminants. pp. 185-194, Number Theory, Carbondale 1979, Lecture Notes in Math. # 751, Springer 1979, MR 81d:12005. | MR | Zbl

2 A. Neugebauer, On zeros of zeta functions in low rectangles in the critical strip. (in Polish), Ph.D. Thesis, A. Mickiewicz University, Poznan, Poland, 1985.

3 A. Neugebauer, On the zeros of the Dedekind zeta-function near the real axis, Funct. Approx. Comment. Math. 16 (1988), 165-167. | MR | Zbl

4 A. Neugebauer, Every Dedekind zeta-function has a zero in the rectangle 1/2 ≤ σ ≤ 1, 0 < t < 60,, Discuss. Math.. to appear. | Zbl

5 A.M. Odlyzko, Low zeros of Dedekind zeta function. manuscript in preparation.

1 J.-F. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), 209-232. MR 87j:11059. | Numdam | MR | Zbl

2 E. Friedman, The zero near 1 of an ideal class zeta function, J. London Math. Soc. 35 (2) (1987). 1-17. MR 88g:11087. | MR | Zbl

3 E. Friedman, Hecke's integral formula, Sém. Théorie des Nombres de Bordeaux (1987-88). Exp. #5, 23 pp., Univ. Bordeaux 1988. | Zbl

1 E. Landau, Zur Theorie der Heckeschen Zetafunktionen, welche komplexen Charakteren entsprechen, Math. Zeit. 4 (1919), 152-162.. Reprinted on pp. 176-186 of vol. 7, Edmund Landau: Collected Works, P.T. Bateman, et al., eds., Thales Verlag. | JFM | MR

2 E. Landau, Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, 2nd ed., Göttingen (1927). Reprinted by Chelsea, 1949. | MR | Zbl

3 R. Remak, Über die Abschätzung des absoluten Betrages des Regulators eines algebraischen Zahlkörpers nach unten, J. reine angew. Math. 167. (1932), 360-378. | JFM | Zbl

4 R.P. Boas and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189-206. MR 6-265. | MR | Zbl

5. A.P. Guinand, A summation formula in the theory of prime numbers, Proc. London Math. Soc. 50 (2) (1948), 107-119. MR 10, 104g. | MR | Zbl

6 A.P. Guinand, Fourier reciprocities and the Riemann zeta- function, Proc. London Math. Soc. 51 (2) (1949), 401-414. MR 11, 162d. | MR | Zbl

7 A. Weil, Sur les "formules explicites" de la théorie des nombres premiers, Comm. Sem. Math. Univ. Lund, tome supplémentaire (1952), 252-265. MR 14, 727e. | MR | Zbl

8 R. Remak, Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compos. Math. 10 (1952), 245-285. MR 14, 952d. | Numdam | MR | Zbl

9 R. Remak, Über algebraische Zahlkörper mit schwachem Einheitsdefekt, Compos. Math. 12 (1954), 35-80. MR 16, 116a. | Numdam | MR | Zbl

10 A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972). 3-18. MR 52 # 345. Reprinted in A. Weil, Oeuvres Scientifiques, vol. 3, pp. 249-264, Springer 1979. | MR | Zbl

11 H.-J. Besenfelder, Die Weilsche "Explizite Formel" und temperierte Distributionen, J. reine angew. Math. 293/294 (1977), 228-257. MR 57 # 254. | MR | Zbl

12 J.-P. Serre. note on p. 710 in vol. 3 of Jean-Pierre Serre, Collected Papers, Springer 1986.

13 H. Cohen and H.W. Lenstra Jr., Heuristics on class groups of number fields. pp. 33-62 in Number Theory, Noordwijkerhout 1983, H. Jager ed., Lecture Notes in Math. # 1068, Springer 1984. MR 85j:11144. | MR | Zbl

14 J.-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C.R. Acad. Sci. Paris 296 (1983). sér. I, 397-402. MR 85b:14027. Reprinted on pp. 658-663 in vol. 3 of Jean-Pierre Serre, Collected Papers, Springer 1986. | MR | Zbl

15 A.M. Odlyzko and H.J.J. Te Riele, Disproof of the Mertens conjecture, J. reine angew. Math. 357 (1985), 138-160. MR 86m:11070. | MR | Zbl

16 J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z, Invent. math. 81 (1985), 515-538. MR 87g:11073. | MR | Zbl

17 H. Cohen and J. Martinet, Etude heuristique des groupes de classes des corps de nombres, J. reine angew. Math. 404 (1990), 39-76. | MR | Zbl

18 A. Borel and G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ: Math. I.H.E.S. 69 (1989), 119-171. | Numdam | MR | Zbl