Bulletin of the American Mathematical Society

Author(s): Barry Mazur
Journal: Bull. Amer. Math. Soc. 45 (2008), 185-228.
MSC (2000): Primary 11-02, 11F03, 11F80, 11G05, 11G40
Posted: February 6, 2008
Retrieve article in: PDF

1.: Arthur, J., Clozel, L.: Simple algebras, base change and the advanced theory of the trace formula, Annals of Math. Studies 120, Princeton University Press (1989). MR 1007299 (90m:22041)
2.: Akiyama, S., Tanigawa, Y.: Calculation of values of -functions associated to elliptic curves, Mathematics of Computation 68, No. 227 (1999) 1201-1231. MR 1627842 (99m:11070)
3.: Clozel, L., Harris, M., Taylor, R.: Automorphy for some $\ell$ -adic lifts of automorphic mod $\ell$ representations (preprint), http://www.math.harvard.edu/rtaylor/.
4.: Cogdell, J., Piatetski-Shapiro, I.: Remarks on Rankin-Selberg convolutions, in Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Press, Baltimore, MD (2004). MR 2058610 (2005d:11075)
5.: Cremona, J. E.: Algorithms for Modular Elliptic Curves, Cambridge University Press (1992). MR 1201151 (93m:11053)
6.: Davenport, H.: Multiplicative Number Theory, Third Edition, Graduate Texts in Mathematics, Springer (2000). MR 1790423 (2001f:11001)
7.: Deligne, P.: La conjecture de Weil: I, I.H.E.S. Publ. Math. 43 (1974) 273-307. MR 0340258 (49:5013)
8.: Deligne, P.: La conjecture de Weil II, I.H.E.S. Publ. Math. 52 (1981) 313-428. MR 601520 (83c:14017)
9.: Diamond, F., Shurman, J.: A First Course in Modular Forms, Graduate Texts in Mathematics, Springer (2005). MR 2112196 (2006f:11045)
10.: Dieulefait, L.: The level case of Serre's conjecture revisited, preprint, ArXiv:0705.0457v1 (2007).
11.: Elkies, N.: The existence of infinitely many supersingular primes for every elliptic curve over $\mathbf{Q}$ , Invent. Math. 89 (1987) 561-568. MR 903384 (88i:11034)
12.: Elkies, N.: Distribution of supersingular primes, Journées Arithmétiques, 1989 (Luminy, 1989), Astérisque 198-200 (1991); 127-132 (1992). MR 1144318 (93b:11070)
13.: Fontaine, J.-M., Mazur, B.: Geometric Galois representations, in Elliptic Curves, Modular Forms, & Fermat's Last Theorem, J. Coates and S. T. Yau, editors, vol. 1 of Series in Number Theory, International Press, Cambridge, MA (1995). MR 1363495 (96h:11049)
14.: Gelbart, S., Jacquet, H.: A relation between automorphic representations of and , Ann. Sci. École Norm. Sup. (4) 11 (1978) 471-552. MR 533066 (81e:10025)
15.: Granville, A., Martin, G.: Prime number races, American Mathematical Monthly 113 (2006) 1-33. MR 2202918 (2006h:11112)
16.: Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, Oxford University Press, fifth edition (1979). MR 568909 (81i:10002)
17.: Harris, M.: Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applications. To appear in Algebra, Arithmetic, and Geometry--Manin Festschrift, Birkhäuser (in press.)
18.: Harris, M., Shepherd-Barron, N., Taylor, R.: Ihara's lemma and potential automorphy (preprint), http://www.math.harvard.edu/rtaylor/.
19.: Hasse, H.: Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F.K. Schmidtschen Kongruenzzetafunktionen in gewissen zyklischen Fällen. Vorläufige Mitteilung. Nachr. Ges. Wiss. Göttingen I. Math.-Phys. Kl. Fachgr. I Math. Nr. 42 (1933) 253-262. See also: Helmut Hasse Mathematische Abhandlungen, Band 2, de Gruyter (1975) 85-94. MR 0465757 (57:5648b)
20.: Jacquet, H., Piatetskii-Shapiro, I., Shalika, J.: Rankin-Selberg Convolutions, American Journal of Mathematics 105 (1983) 367-464. MR 701565 (85g:11044)
21.: Katz, N.: An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields, A.M.S. Proc. Symp. Pure Math 28 (1976) 275-305. MR 0424822 (54:12780)
22.: Katz, N., Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy, AMS Colloquium Publications, 45, AMS (1999). MR 1659828 (2000b:11070)
23.: Khare, C.: Serre's modularity conjecture: The level one case, Duke Math. J. 134 (2006) 557-589. MR 2254626 (2007e:11060)
24.: Khare, C., Wintenberger, J-P.: On Serre's reciprocity conjecture for -dimensional mod representations of the Galois group of $\mathbf{Q}$ , preprint (2004); available at: www.arxiv.org.
25.: Kim, H.: Functoriality for the exterior square of ${\rm GL}_4$ and the symmetric fourth power of ${\rm GL}_2$ , J. Amer. Math. Soc. 16 (2003) 139-183. With Appendix 1 by Dinakar Ramakrihnan and Appendix 2 by Kim and Peter Sarnak. MR 1937203 (2003k:11083)
26.: Kim, H., Shahidi, F.: Cuspidality of symmetric powers with applications, Duke Math. J. 112, no. 1 (2002) 177-197. MR 1890650 (2003a:11057)
27.: Kim, H., Shahidi, F.: Functorial products for $\mathrm{GL}(2)\times \mathrm{GL}(3)$ and the symmetric cube for $\mathrm{GL}(2)$ , Annals of Math. 155 (2002) 837-893. MR 1923967 (2003m:11075)
28.: Kisin, M.: Moduli of finite flat group schemes and modularity, preprint (2004).
29.: Kisin, M.: Modularity of -adic Barsotti-Tate representations, preprint (2006).
30.: Kisin, M.: The Fontaine-Mazur conjecture for ${\rm GL}_2$ , preprint (2006).
31.: Korevaar, J.: Tauberian Theory: A century of developments, Grundlehren der mathematischen Wissenshaften, 329, Springer (2004). MR 2073637 (2006e:11139)
32.: Lang, S.: Algebraic Number Theory, Second Edition, Springer (1994). MR 1282723 (95f:11085)
33.: Lang, S.: Math Talks for Undergraduates, Springer (1999). MR 1697559 (2000g:00004)
34.: Lang, S., Trotter, H.: Frobenius distributions in $\mathit{GL}_2$ -extensions, Lecture Notes in Mathematics, 504, Springer (1976). MR 0568299 (58:27900)
35.: Langlands, R.: Euler Products, Yale University Press (1971). MR 0419366 (54:7387)
36.: Langlands, R.: On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, 544, Springer (1976). MR 0579181 (58:28319)
37.: Lagarias, J., Odlyzko, A.: Effective versions of the Cebotarev density theorem. In: A. Frolich (ed.), Algebraic Number Fields, Proceedings of the 1975 Durham Symposium, Academic Press, London and New York (1977). MR 0447191 (56:5506)
38.: Mazur, B.: Controlling Our Errors, Nature 443, 7 (2006) 38-40.
39.: Ogg, A.: A remark on the Sato-Tate conjecture, Invent. Math. 9 (1970) 198-200. MR 0258835 (41:3481)
40.: Riemann, G. F. B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsber. Königl. Preuss. Akad. Wiss. Berlin (1859) 671-680.
41.: Rubinstein, M., Sarnak, P.: Chebyshev's Bias, Experimental Mathematics 3 (1994) 174-197. MR 1329368 (96d:11099)
42.: Serre, J-P.: Abelian $\ell$ -adic Representations, Benjamin (1968). MR 0263823 (41:8422)
43.: Serre, J-P.: Quelques applications du théorème de densité de Cebotarev, I.H.E.S Publ. Math. 54 (1981); also: pp. 563-641 in Jean-Pierre Serre. Oeuvres. Collected Papers. Volume III, 1972-1984, Springer (1986). MR 0644559 (83k:12011), MR 0926691 (89h:01109c)
44.: Serre, J-P.: Letter to F. Shahidi, January 24, 1992; Appendix (pp. 175-180) in reference 48.
45.: Serre, J-P.: Propriété conjecturales des groups de Galois motiviques et des représentations $\ell$ -adiques, pp. 325-348 in Jean-Pierre Serre. Oeuvres. Collected Papers. Volume IV, 1985-1998, Springer (2000). MR 1265537 (95m:11059), MR 1730973 (2001e:01037)
46.: Shahidi, F.: On certain -functions, Amer. J. Math. 103 (1981) 297-355. MR 610479 (82i:10030)
47.: Shahidi, F.: On the Ramanujan Conjecture and finiteness of poles of certain -functions, Annals of Math. 127 (1988) 547-584. MR 942520 (89h:11021)
48.: Shahidi, F.: A proof of Langlands' conjecture on Plancherel measures; complementary series for -adic groups. Ann. of Math. (2) 132 (1990), no. 2, 273-330 (Reviewer: Stephen Gelbart). MR 1070599 (91m:11095)
49.: Shahidi, F.: Symmetric power -functions for $\mathrm{GL}(2)$ , pp. 159-182 in Elliptic Curves and Related Topics, Volume 4 of CRM Proceedings and Lecture Notes (Eds.: H. Kisilevsky, M.R. Murty), American Mathematical Society (1994). MR 1260961 (95c:11066)
50.: Shahidi, F.: Twists of a general class of -functions by highly ramified characters. Canad. Math. Bull. 43 (2000), no. 3, 380-384. MR 1776066 (2001g:22030)
51.: Shahidi, F.: Langlands-Shahidi method, pp. 299-330 in Automorphic Forms and Applications (Eds: P. Sarnak, F. Shahidi), IAS/Park City Mathematics Series 12, AMS/IAS (2007). MR 2331347
52.: Silverman, J.: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Princeton University Press (1992). MR 1329092 (95m:11054)
53.: Tate, J.: Algebraic Cycles and Poles of Zeta Functions, pp. 93-110 in Arithmetic Algebraic Geometry, Proceedings of a conference held in Purdue, Dec. 5-7, 1963, Harpers (1965). MR 0225778 (37:1371)
54.: Taylor, R.: Automorphy for some $\ell$ -adic lifts of automorphic mod $\ell$ representations. II (preprint). http://www.math.harvard.edu/rtaylor/.
55.: von Koch, H.: Sur la distribution des nombres premiers, Acta Mathematica 24 (1) (1901) 159-182. MR 1554926
56.: Weil, A. Sur les courbes algébriques et les variétés qui s'en déduisent, Hermann (Paris), 1948. MR 0029522 (10:621d)

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11-02, 11F03, 11F80, 11G05, 11G40

Barry Mazur
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138-2901

DOI: 10.1090/S0273-0979-08-01207-X
PII: S 0273-0979(08)01207-X
Received by editor(s): September 9, 2007
Posted: February 6, 2008
Dedicated: In memory of Serge Lang
Additional Notes: Part I of this article was presented in the {\it Current Events Bulletin} section of the winter meeting of the AMS on January 7, 2007, in New Orleans. The title of the talk was ``The structure of error terms in number theory and an introduction to the Sato-Tate Conjecture''. Part I and some of Part II were published in the {\it Current Events Bulletin} of the AMS that was distributed at the meeting.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS