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1 : ( ) 9 10 : 1. 9/6( ) 10:20 12:40 GL(2) Hecke ( ) 12:50 13:50 14:00 14:50 15:00 16:30 Selberg ( ) 16:45 18:15 GL(2) I ( ) 18:45 20:00 20:15 21:45 Selberg ( ) 9/7( ) 7:00 9:00 9:15 10:30 GL(2) II ( ) 10:45 12:00 GL(2) III ( ) 12:15 13:15 13:30 14:30 Eichler-Selberg ( ) 14:45 16:15 Jacquet-Langlands I ( ) 16:30 18:00 Jacquet-Langlands II ( ) 18:30 19:45 20:15 21:15 degree 2 p Siegel-Eisenstein p- Euler Abel Fitting 9/8( ) 7:00 9:00 9:15 10:30 GL(3) I ( ) 10:45 12:00 GL(3) II ( ) 12:15 13:15 13:30 15:30 ( ) 16:00 18:00 ( ) 18:30 19:45 20:15 21:15 1
2 9/9( ) 7:00 9:00 9:15 10:30 ( ) 10:45 12:00 ( ) 12:15 13:15 13:30 14:30 ( ) 14:45 16:15 I ( ) 16:30 18:00 II ( ) 19:00 21:00 9/10( ) 7:00 9:00 9:15 10:45 III ( ) 11:00 12:30 IV ( ) 12:45 13:45 2
3 : GL(2) Hecke ( ) 140. (i) Lie : SL(2, R) Haar Casimir (ii) SL(2, R) : SL(2, R) ( ) SL(2, R) (Gel fand-graev- Piatetski-Shapiro ) (iii) adele GL(2, A) Hecke : adele GL(2, A) Hecke GL(2, Q p ) Hecke Hecek-eigen form Eisenstein [1] A. Borel, Automorphic forms on SL 2 (R), Cambridge Tracts in Mathematics 130, Cambridge, Cambridge University Press, [2] D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics 55, Cambridge, Cambridge University Press, [3] H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics 17, Providence, RI: American Mathematical Society (AMS), Selberg ( ) 90. R Z. G Γ G f L 2 (Γ\G) r Γ (f),. SL(2, Z) G = SL(2, R) Γ = SL(2, Z). r Γ (f) L 2 (Γ\G), L 2 dis (Γ\G).. [1], Selberg Trace Formula. [2] A.W. Knapp, Theoretical aspects of the trace formula for GL(2), Representation theory and automorphic forms, American Mathematical Society, Proc. Symp. Pure Math. 61 (1997), [3] A. Selberg, Harmonic analysis, Collected Papers, Vol. I, , Springer- Verlag,
4 [4] G. Warner, Selberg s trace formula for nonuniform lattices: The R-rank one case, Studies in Algebra and Number Theory, Adv. Math., Suppl. Stud. 6 (1979), GL(2) 90. GL(2) Arthur-Selberg L 2 - GL(1) Poisson GL(2) L 2 -space GL 2 (A) φ R(φ) Eisenstein Eisenstein : Weil, A., Basic number theory, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen Band 144, Springer-Verlag New York Heiderberg Berlin Tate, J. Fourier analysis in number fields and Hecke s zeta functions, Algebraic Number Theory, Academic Press (1990), Ramakrishnan, D., Valenza, R.J., Fourier analysis on number fields, Graduate Texts in Mathematics vol. 186, Springer (1998). Knapp, A., Theoretical aspects of the trace formula for GL(2), Proceedings of Symposia in Pure Mathematics, 61 (1997), Gelbert, S., Jacquet, H., Forms of GL(2) from the analytic point of view, Proceedings of Symposia in Pure Mathematrics Vol. 33 (1979), part 1, Bump, D., Automorphic forms and representations, Cambridge studies in Advanced Mathematics: 55, Cambridge University Press (1998). Borel,. A.,Automorphic forms on SL 2 (R), Cambridge tactics in Mathematics, Cambridge University Press (1997). 4
5 Iwaniec, H., Spectral methods of automorphic forms (second edition), Graduate Studies in Mathematics 53, American Mathematical Society Providence, Rhode Islamd (2002) Selberg ( ) 90. G = SL(2, R) K,.,. G = SL(2, Z),. [1] D. A. Hejhal, The Selberg trace formula for PSL(2, R) Vol. I, Lecture Notes in Mathematics 548, Berlin- Heidelberg-New York: Springer-Verlag, [2] D. A. Hejhal, The Selberg trace formula for PSL(2, R) Vol. II, Lecture Notes in Mathematics 1001, Berlin- Heidelberg-New York: Springer-Verlag, [3] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), no. 2, [4] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), GL(2) II, III ( ) 150. GL(2) I. L 2 (GL(2, Q)\GL(2, A) 1 ) Arthur-Selberg Arthur., Eichler-Selberg Jacquet-Langlands.. Arthur s modified kernel Simple trace formula GL(2, R) [1] H. Jacquet, R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Berlin-Heidelberg-New York: Springer-Verlag, [2] S. Gelbart, H. Jacquet, Forms of GL(2) from the analytic point of view, Proceedings of Symposia in Pure Mathematics 33 Part 1 (1979), [3] S. Gelbart, Lectures on the Arthur-Selberg trace formula, University Lecture Series 9, Providence, RI: American Mathematical Society, [4] A.W. Knapp, Theoretical aspects of the trace formula for GL(2), Proc. Symp. Pure Math. 61 (1997),
6 2.6. Eichler-Selberg ( ) 60. Gottschling Sp(2, Z) Sp(2, Z) Hasse (cf. Hashimoto and Ibukiyama ), SL 2 (Z) Sp(2) Miyake [5] local-global [1] M. Eichler, On the class number of imaginary quadratic fields and the sums of divisors of natural numbers, J. Indian Math. Soc. 19 (1955), [2] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), [3] K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (I), J. Fac. Sci. Univ. Tokyo Sect IA. Math. Vol. 27 No. 3 (1980), [4] A. Knightly, C. Li, Traces of Hecke operators, Mathematical Surveys and Monographs 133, Providence, RI: American Mathematical Society (AMS), [5] T. Miyake, Modular forms, Springer-Verlag, Berlin, [6] D. Zagier, The Eichler-Selberg trace formula on SL 2 (Z), Introduction to modular forms by S. Lang, Appendix, pp , Springer, Berlin, Jacquet-Langlands ( 90 ). Arthur-Selberg lifting) Jacquet-Langlands GL(2) inner forms Jacquet-Langlands GL(2) smooth ) GL(2) inner forms 6
7 GL(2) smooth Jacquet-Langlands L Jacquet-Langlands II ( 90 ). Arthur-Selberg Jacquet-Langlands Hecke maching Arthur-Selberg : Jacquet, H., Langlands, L.P.,Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, Springer-Verlag, Berline, Heiderberg, New York (1970). Knapp, A. W, Rogawski, J. D, Applications of the trace formula, Proc. Symp. Pure Math. 61 (1997), Rogawski, Jonathan D, Representations of GL(n) and division algebras over a p-adic field, Duke Math. J. 50 (1983), no. 1, Deligne, P., Kazhdan, D., Vigneras, M.-F. Representations des algebres centrales simples p-adiques, , Travaux en Cours, Hermann, Paris, 1984 Gelbert, Stephen S., Automorphic forms on adele grous, Annals of Mathematics Studies, Princeton University Press (1975) GL(3) ( ) 150. GL(3). GL(3). (invariant distribution),... GL(3) (G, M)-family distribution GL(3) GL(3) 7
8 [1] J. Arthur, A trace formula for reductive groups. I. Terms associated to classes in G(Q), Duke Math. J. 45 (1978), no. 4, [2] J. Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, [3] J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), [4] J. Arthur, The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), no. 2, [5] J. Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Soc. 1 (1988), no. 3, [6] J. Arthur, An introduction to the trace formula, Clay Mathematics Proceedings 4 (2005), ( ) ( ) 120. Arthur-Selberg Langlands L Langlands Lefschetz Arthur Tate Serre, J.-P. Local fields Springer-Verlag, 1979, 67, viii+241. Milne, J. Arithmetic duality theorems Booksurge Publishing, 2006, 339+viii. Hida, H. Modular forms and Galois cohomology Cambridge University Press, 2000, 69, x+343. Langlands Labesse, J.-P. Cohomologie, L-groupes et fonctorialit? Compositio Math., 1985, 55, Langlands, R. P. Les débuts d une formule des traces stable Université de Paris VII U.E.R. de Mathématiques, 1983, v+18. Kottwitz, R. E. Rational conjugacy classes in reductive groups Duke Math. J., 1982, 49,
9 Kottwitz, R. E. Stable trace formula: elliptic singular terms Math. Ann., 1986, 275, ( ) 75. ( ) Langlands-Shelstad Hecke Hecke Langlands, R. P. & Shelstad, D. On the definition of transfer factors Math. Ann., 1987, 278, Harris, M. Arithmetic applications of the Langlands program Jpn. J. Math., 2010, 5, ( ) 75. Arthur Langlands, R. P. Les débuts d une formule des traces stable Université de Paris VII U.E.R. de Mathématiques, 1983, v+18. Kottwitz, R. E. Stable trace formula: elliptic singular terms Math. Ann., 1986, 275, ( ) 60. Arthur Clozel-Labesse Kottwitz, 1963, 15, Kottwitz, R. E. Tamagawa numbers Ann. of Math. (2), 1988, 127,
10 Clozel, L. Nombres de Tamagawa des groupes semi-simples (d après Kottwitz) Astérisque, 1989, Exp. No. 702, ( ) 360. trace formula elliptic term stabilization 3 U(3) trace formula stabilization trace formula GL twisted trace formula 1. Langlands A-packet 2. GL 3. multiplicity formula [1] J. D. Rogawski, The multiplicity formula for A-packets. The zeta functions of Picard modular surfaces, , Univ. Montréal, Montreal, QC, [2] J. D. Rogawski, Automorphic representations of unitary groups in three variables. Annals of Mathematics Studies, 123. Princeton University Press, Princeton, NJ, xii+259 pp. [3] J. Arthur, An introduction to the trace formula. Harmonic analysis, the trace formula, and Shimura varieties, 1 263, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI,
11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))
![11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))](/thumbs/91/105488540.jpg "11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))")
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