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Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results. Journal de théorie des nombres de Bordeaux, 2 no. 1 (1990), p. 119-141
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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 347767 | Zbl 0276.12002 2 L.C. Washington, Introduction to Cyclotomic Fields, Springer. 1982. MR 85g: 11001. MR 718674 | Zbl 0484.12001 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 902831 | Zbl 0567.12009 1 H.M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. math. 23 (1974), 135-152. MR 49 # 7218. MR 342472 | Zbl 0278.12005 2 H.M. Stark, The analytic theory of algebraic numbers, Bull. Am. Math. Soc. 81 (1975), 961-972. MR 56 # 2961.
Article | MR 444611 | Zbl 0329.12010 3 A.M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. math. 29 (1975), 275-286. MR 51 # 12788. MR 376613 | Zbl 0306.12005 4 A.M. Odlyzko, Lower bounds. for discriminants of number fields, Acta Arith. 29 (1976), 275-297. MR 53 # 5531.
Article | MR 401704 | Zbl 0286.12006 5 A.M. Odplyzko, Lower bounds for discriminants of number fields. II. MR 56 # 309. Zbl 0362.12005 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 453701 | Zbl 0362.12006 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 tables dated Nov. 29, 1976 (unpublished).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 435033 | Zbl 0359.12010 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 551335 | Zbl 0393.12010 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.)
Article | MR 607864 | Zbl 0482.12003 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 697261 | Zbl 0491.12005 1 H.W. Lenstra, Euclidean number fields of large degree, Invent. math. 38 (1977), 237-254. MR 55 # 2836. MR 429826 | Zbl 0328.12007 2 J. Martinet, Tours de corps de classes et estimations de discriminants, Invent. math. 44 (1978), 65-73.. MR 57 # 275. MR 460281 | Zbl 0369.12007 3 J. Martinet, Petits discriminants, Ann. Inst. Fourier (Grenoble) 29, no 1 (1979), 159-170. MR 81h:12006.
Numdam | MR 526782 | Zbl 0387.12006 4 R. Schoof, Infinite class field towers of quadratic fields, J. reine angew. Math. 372 (1986), 209-220. MR 88a:11121.
Article | MR 863524 | Zbl 0589.12011 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 447178 | Zbl 0365.12007 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 511747 | Zbl 0428.12003 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 564932 | Zbl 0421.12005 4 J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. math. 55 (1979), 37-47. MR 80k:12019. MR 553994 | Zbl 0474.12009 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 634704 | Zbl 0471.12007 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.
Article | MR 604214 | Zbl 0457.12003 7 F.J. van der LINDEN, Class number computations of real abelian number fields, Math. Comp. 39 (1982), 693-707. MR 84e:12005. MR 669662 | Zbl 0505.12010 8 A. Leutbecher and J. Martinet, Lenstra's constant and Euclidean number fields, Astérisque 94 (1982), 87-131. MR 85b:11090. MR 702368 | Zbl 0499.12013 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 786536 | Zbl 0546.12005 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.
Article | MR 1016903 | Zbl 0711.11042 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 1016903 | Zbl 0711.11042 1 M. Pohst, Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory 9 (1977), 459-492. MR 57 # 268. MR 460274 | Zbl 0366.12011 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 480423 | Zbl 0359.12007 3 M. Pohst, Eine Regulatorabschätzung, Abh. Math. Sem. Univ. Hamburg 47 (1978), 95-106. MR 58 # 16596. MR 498487 | Zbl 0381.12006 4 R. Zimmert, Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. math. 62 (1981), 367-380. MR 83g:12008. MR 604833 | Zbl 0456.12003 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.)
Article | MR 784057 | Zbl 0543.12006 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 902832 | Zbl 0569.12004 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 769793 | Zbl 0552.12003 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 756083 | Zbl 0549.12003 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 1042763 | Zbl 0699.12014 10 E. Friedman, Analytic formulas for regulators of number fields, Invent. math. 98 (1989), 599-622. MR 1022309 | Zbl 0694.12006 11 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press. (1989). MR 1033013 | Zbl 0685.12001 12 R. Schoof and L.C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 929552 | Zbl 0649.12007 13 A.-M. Bergé and J. Martinet, Notions relatives de régulateurs et de hauteurs, Acta Arith. 54 (1989), 155-170. MR 1024424 | Zbl 0642.12011 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.
Article | Zbl 0699.12013 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 1139098 | Zbl 0736.11063 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.
Article | MR 349626 | Zbl 0285.12001 2 M. Pohst, Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper, J. reine angew. Math. 278/279 (1975), 278-300. MR 52 # 8085. MR 387242 | Zbl 0314.12014 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 466069 | Zbl 0373.12006 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 427270 | Zbl 0345.12005 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 644904 | Zbl 0478.12005 6 M. Pohst, P. Weiler, and H. Zassenhaus, On effective computation of fundamental units, Math. Comp. 38 (1982), 293-329. MR 83e:12005b. MR 637308 | Zbl 0493.12005 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 655408 | Zbl 0512.12009 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 693185 | Zbl 0527.12007 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 762692 | Zbl 0546.12004 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 769447 | Zbl 0581.12002 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 736462 | Zbl 0535.12003 12 F. Diaz Y. Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory 25 (1987), 34-52. MR 871167 | Zbl 0606.12005 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 882288 | Zbl 0601.12013 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 949079 | Zbl 0653.12003 15 J. Buchmann and D. Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), 161-174. MR 946599 | Zbl 0668.12001 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 929555 | Zbl 0651.12001 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 970698 | Zbl 0714.11002 19 A.-M. Bergé, J. Martinet and M. Olivier, The computation of sextic fields with a quadratic subfield. in Math. Comp. to appear. MR 1011438 | Zbl 0709.11056 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 0719.11079 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 564928 | Zbl 0418.12002 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 965377 | Zbl 0677.12005 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 0592.12010 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 844410 | Zbl 0607.14012 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 871761 | Zbl 0583.12010 3 E. Friedman, Hecke's integral formula, Sém. Théorie des Nombres de Bordeaux (1987-88). Exp. #5, 23 pp., Univ. Bordeaux 1988.
Article | Zbl 0697.12010 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.
Article | MR 1544358 | JFM 47.0164.01 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 31002 | Zbl 0041.01103 3 R. Remak, Über die Abschätzung des absoluten Betrages des Regulators eines algebraischen Zahlkörpers nach unten, J. reine angew. Math. 167. (1932), 360-378.
Article | Zbl 0003.19801 | JFM 58.0173.04 4 R.P. Boas and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189-206. MR 6-265.
Article | MR 12152 | Zbl 0060.25602 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 26086 | Zbl 0031.11003 6 A.P. Guinand, Fourier reciprocities and the Riemann zeta- function, Proc. London Math. Soc. 51 (2) (1949), 401-414. MR 11, 162d. MR 31513 | Zbl 0039.11503 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 53152 | Zbl 0049.03205 8 R. Remak, Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compos. Math. 10 (1952), 245-285. MR 14, 952d.
Numdam | MR 54641 | Zbl 0047.27202 9 R. Remak, Über algebraische Zahlkörper mit schwachem Einheitsdefekt, Compos. Math. 12 (1954), 35-80. MR 16, 116a.
Numdam | MR 63403 | Zbl 0055.26805 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 379440 | Zbl 0245.12010 11 H.-J. Besenfelder, Die Weilsche "Explizite Formel" und temperierte Distributionen, J. reine angew. Math. 293/294 (1977), 228-257. MR 57 # 254. MR 460260 | Zbl 0354.12007 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 756082 | Zbl 0558.12002 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 703906 | Zbl 0538.14015 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 783538 | Zbl 0544.10047 16 J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z, Invent. math. 81 (1985), 515-538. MR 87g:11073. MR 807070 | Zbl 0612.14043 17 H. Cohen and J. Martinet, Etude heuristique des groupes de classes des corps de nombres, J. reine angew. Math. 404 (1990), 39-76.
Article | MR 1037430 | Zbl 0699.12016 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 1019963 | Zbl 0707.11032 |