1057:
1593:
779:
3188:
2642:
1407:
655:
2923:
3353:
1239:
3048:
1787:
2461:
507:
2252:
2815:
1052:{\displaystyle \sum _{n=0}^{\infty }h(r_{n})={\frac {\mu (X)}{4\pi }}\int _{-\infty }^{\infty }r\,h(r)\tanh(\pi r)\,dr+\sum _{\{T\}}{\frac {\log N(T_{0})}{N(T)^{\frac {1}{2}}-N(T)^{-{\frac {1}{2}}}}}g(\log N(T)).}
2820:
3253:
1588:{\displaystyle \sum _{\gamma \in \{\Gamma \}}a_{\Gamma }^{G}(\gamma )\int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx=\sum _{\pi \in {\widehat {G}}}a_{\Gamma }^{G}(\pi )\operatorname {tr} \pi (\phi )}
1338:
474:
729:
243:. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the
2366:
3659:. The orbital integrals and the traces in irreducible summands can then be computed further and in particular one can recover the case of the trace formula for hyperbolic surfaces in this way.
351:
2456:
1827:
1143:
2138:
3525:
2408:
2052:
3443:
1963:
3657:
2290:
2013:
2668:
1706:
169:
to prime numbers, with the zeta zeros corresponding to eigenvalues of the
Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the
2996:
1668:
768:
3023:
2949:
2081:
1619:
3382:
1873:
3228:
2733:
2697:
1711:
3631:
3043:
2969:
1847:
1639:
1374:
3607:
1398:
3463:
1987:
1902:
4256:
3573:
3553:
3483:
3402:
3248:
2165:
1922:
3183:{\displaystyle \operatorname {tr} R(\phi )=\sum _{\gamma \in \{\Gamma \}}a_{\Gamma }^{G}(\gamma )\int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx.}
3678:
The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup
392:
The
Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group
2637:{\displaystyle (R(\phi )f)(x)=\int _{G}\phi (y)f(xy)\,dy=\int _{\Gamma \setminus G}\left(\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma y)\right)f(y)\,dy}
161:
in terms of geometric data involving the lengths of geodesics on the
Riemann surface. In this case the Selberg trace formula is formally similar to the
1255:
414:
264:
4164:
4130:
4027:
3994:
2182:
662:
3792:
2738:
650:{\displaystyle {\begin{cases}u(\gamma z)=u(z),\qquad \forall \gamma \in \Gamma \\y^{2}\left(u_{xx}+u_{yy}\right)+\mu _{n}u=0.\end{cases}}}
2295:
325:
4197:
3955:
3926:
3892:
3865:
386:
4223:
3847:
3768:
1065:, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes
143:
4295:(1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series",
3672:
162:
123:
400:
4189:
4052:
2918:{\displaystyle \operatorname {tr} R(\phi )=\int _{\Gamma \setminus G}\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma x)\,dx.}
201:
3348:{\displaystyle \operatorname {tr} R(\phi )=\sum _{\pi \in {\hat {G}}}a_{\Gamma }^{G}(\pi )\operatorname {tr} \pi (\phi )}
2413:
734:
4348:
3857:
1879:
1792:
54:
268:
4077:
3780:
109:
2086:
244:
3488:
2371:
2018:
182:
3407:
1927:
4231:
1966:
1882:
85:
81:
80:, when the representation breaks up into discrete summands. Here the trace formula is an extension of the
77:
32:
3636:
2269:
1992:
3729:
2647:
1685:
299:
197:
170:
166:
2974:
1234:{\displaystyle \vert h(r)\vert \leq M\left(1+\left|\operatorname {Re} (r)\right|\right)^{-2-\delta }.}
3776:
303:
280:
196:
differential operator and its powers. The traces of powers of a
Laplacian can be used to define the
3783:). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.
3633:
can be described in geometric terms using the compact
Riemannian manifold (more generally orbifold)
1644:
516:
408:
256:
236:
3001:
2928:
2057:
1782:{\displaystyle a_{\Gamma }^{G}(\gamma )={\text{volume}}(\Gamma ^{\gamma }\setminus G^{\gamma }).}
1598:
284:
3358:
1852:
4273:
4219:
4193:
4160:
4126:
4094:
4023:
3990:
3951:
3888:
3861:
3713:
3668:
3204:
2709:
2673:
127:
3682:
into an algebraic group over a field which is technically easier to work with. The case of SL
3616:
3028:
2954:
1832:
1624:
1359:
4315:
4265:
4152:
4118:
4117:, Lecture Notes in Mathematics, Vol. 548, vol. 548, Berlin, New York: Springer-Verlag,
4086:
3982:
3911:
3758:
2255:
372:
288:
4308:
4285:
4243:
4207:
4174:
4140:
4106:
4060:
4037:
4004:
3965:
3934:
3902:
3875:
3578:
2703:
and the trace formula is the result of computing its trace in two ways as explained below.
1383:
4304:
4281:
4239:
4203:
4170:
4136:
4102:
4056:
4033:
4000:
3978:
3961:
3947:
3930:
3898:
3871:
3610:
3448:
1972:
1887:
307:
209:
173:
of a
Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
154:
157:, the Selberg trace formula describes the spectrum of differential operators such as the
4333:
4181:
4044:
4019:
4011:
3717:
3558:
3538:
3468:
3387:
3233:
2150:
1907:
228:
220:
62:
4342:
4068:
272:
260:
240:
3744:)); they also handle the adelic case in characteristic 0, combining all completions
279:'s methods by-passed the analysis involved in the trace formula. The development of
200:. The interest of this case was the analogy between the formula obtained, and the
4292:
4251:
4151:, Lecture Notes in Mathematics, vol. 1001, Berlin, New York: Springer-Verlag,
3843:
2144:
1671:
276:
205:
189:
4090:
4072:
2700:
20:
407:
is discrete and real, since the
Laplace operator is self adjoint with compact
4277:
4098:
4215:
292:
232:
193:
158:
36:
4269:
4254:(1972), "Selberg's trace formula as applied to a compact Riemann surface",
4156:
4122:
3986:
3883:
Chavel, Isaac; Randol, Burton (1984). "XI. The
Selberg Trace Formula".
2247:{\displaystyle \int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx}
3975:
An approach to the
Selberg trace formula via the Selberg zeta-function
3944:
Groups acting on hyperbolic space: Harmonic analysis and number theory
2810:{\displaystyle K(x,y)=\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma y)}
181:
Cases of particular interest include those for which the space is a
4188:. Graduate Studies in Mathematics. Vol. 53 (Second ed.).
3691:
295:
characteristic of non-compact
Riemann surfaces and modular curves.
3977:, Lecture Notes in Mathematics, vol. 1253, Berlin, New York:
2266:
Define the following operator on compactly supported functions on
1061:
The right hand side is a sum over conjugacy classes of the group
396:
has no parabolic or elliptic elements (other than the identity).
3045:. Then the above integral can, after manipulation, be written
1404:. The trace formula in this setting is the following equality:
3887:. Pure and Applied Mathematics. Vol. 115. Academic Press.
2951:
denote a collection of representatives of conjugacy classes in
1333:{\displaystyle h(r)=\int _{-\infty }^{\infty }g(u)e^{iru}\,du.}
469:{\displaystyle 0=\mu _{0}<\mu _{1}\leq \mu _{2}\leq \cdots }
310:
is an isospectral invariant, essentially by the trace formula.
65:. The character is given by the trace of certain functions on
3912:"Die Selbergsche Spurformel für kompakte Riemannsche Flächen"
3671:
was largely motivated by the requirement to separate out the
3643:
2276:
2038:
1999:
332:
643:
3555:
is a semisimple Lie group with a maximal compact subgroup
4073:"The Selberg trace formula and the Riemann zeta function"
3695:
3230:
using the decomposition of the regular representation of
1343:
The general Selberg trace formula for cocompact quotients
3813:
Arthur (1989). "The trace formula and Hecke operators".
2143:
all integrals and volumes are taken with respect to the
4322:, Proc. ICM-90 Kyoto, Springer-Verlag, pp. 577–585
3201:
of the trace formula comes from computing the trace of
724:{\displaystyle \mu =s(1-s),\qquad s={\tfrac {1}{2}}+ir}
302:
applications. For instance, by a result of Buser, the
3531:
The case of semisimple Lie groups and symmetric spaces
1073:(which are all hyperbolic in this case). The function
701:
4047:(1966). "The decomposition of L(G/Γ) for Γ=SL(2,Z)".
3639:
3619:
3581:
3561:
3541:
3491:
3471:
3451:
3410:
3390:
3384:
is the set of irreducible unitary representations of
3361:
3256:
3236:
3207:
3051:
3031:
3004:
2977:
2957:
2931:
2823:
2741:
2712:
2676:
2650:
2464:
2416:
2374:
2298:
2272:
2185:
2153:
2089:
2060:
2021:
1995:
1975:
1930:
1910:
1890:
1855:
1835:
1795:
1714:
1688:
1647:
1627:
1601:
1410:
1386:
1362:
1258:
1146:
782:
737:
665:
510:
417:
328:
314:
Selberg trace formula for compact hyperbolic surfaces
3775:, and the many studies of the trace formula in the
3675:, which is characteristic of the non-compact case.
2361:{\displaystyle R(\phi )=\int _{G}\phi (x)R(x)\,dx,}
231:, for a Hecke operator acting on a vector space of
4018:, Generalized Functions, vol. 6, Boston, MA:
4014:; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1990),
3942:Elstrodt, J.; Grunewald, F.; Mennicke, J. (1998).
3856:. Cambridge Tracts in Mathematics. Vol. 130.
3651:
3625:
3601:
3567:
3547:
3519:
3477:
3457:
3437:
3396:
3376:
3347:
3242:
3222:
3182:
3037:
3017:
2990:
2963:
2943:
2917:
2809:
2727:
2691:
2662:
2636:
2450:
2402:
2360:
2284:
2246:
2159:
2132:
2075:
2046:
2007:
1981:
1957:
1916:
1896:
1867:
1841:
1821:
1781:
1700:
1662:
1633:
1613:
1587:
1392:
1368:
1332:
1233:
1051:
762:
723:
649:
468:
345:
3815:Number theory, trace formulas and discrete groups
2171:The left-hand side of the formula is called the
4149:The Selberg trace formula for PSL(2,R). Vol. 2
4016:Representation theory and automorphic functions
3692:Gel'fand, Graev & Pyatetskii-Shapiro (1990)
2735:can be expressed as the integral of the kernel
346:{\displaystyle \Gamma \backslash \mathbf {H} ,}
287:) provided a purely algebraic setting based on
130:. Selberg worked out the non-compact case when
108:, the Selberg trace formula is essentially the
4257:Communications on Pure and Applied Mathematics
4115:The Selberg trace formula for PSL(2,R). Vol. I
3767:Contemporary successors of the theory are the
2451:{\displaystyle f\in L^{2}(\Gamma \setminus G)}
4224:"Scattering theory for automorphic functions"
1822:{\displaystyle G_{\gamma },\Gamma _{\gamma }}
1400:a compactly supported continuous function on
142:; the extension to higher rank groups is the
122:is not compact is harder, because there is a
8:
4049:Algebraic Groups and Discontinuous Subgroups
3712:is a locally compact topological field with
3090:
3084:
2938:
2932:
1608:
1602:
1428:
1422:
1162:
1147:
925:
919:
219:At the same time, interest in the traces of
31:, is an expression for the character of the
3827:
3771:applying to the case of general semisimple
1356:be a unimodular locally compact group, and
2133:{\displaystyle \int _{G}\phi (g)\pi (g)dg}
263:. For instance, using the trace theorem,
3696:Elstrodt, Grunewald & Mennicke (1998)
3638:
3618:
3591:
3580:
3560:
3540:
3520:{\displaystyle L^{2}(\Gamma \setminus G)}
3496:
3490:
3470:
3450:
3420:
3415:
3409:
3389:
3363:
3362:
3360:
3312:
3307:
3290:
3289:
3282:
3255:
3235:
3206:
3170:
3152:
3128:
3123:
3104:
3099:
3077:
3050:
3030:
3009:
3003:
2982:
2976:
2956:
2930:
2905:
2887:
2865:
2849:
2822:
2789:
2767:
2740:
2711:
2675:
2649:
2627:
2592:
2570:
2549:
2535:
2502:
2463:
2427:
2415:
2403:{\displaystyle L^{2}(\Gamma \setminus G)}
2379:
2373:
2348:
2318:
2297:
2271:
2237:
2219:
2195:
2190:
2184:
2152:
2094:
2088:
2059:
2047:{\displaystyle L^{2}(\Gamma \backslash G}
2026:
2020:
1994:
1974:
1940:
1935:
1929:
1909:
1889:
1854:
1834:
1813:
1800:
1794:
1767:
1754:
1742:
1724:
1719:
1713:
1687:
1649:
1648:
1646:
1626:
1600:
1552:
1547:
1530:
1529:
1522:
1508:
1490:
1466:
1461:
1442:
1437:
1415:
1409:
1385:
1361:
1320:
1308:
1286:
1278:
1257:
1213:
1145:
1004:
1000:
973:
949:
930:
918:
904:
873:
864:
856:
826:
814:
798:
787:
781:
742:
736:
700:
664:
625:
604:
588:
573:
511:
509:
454:
441:
428:
416:
335:
327:
4051:. Proc. Sympos. Pure Math. Providence:
3804:
3779:(dealing with technical issues such as
3508:
3134:
2853:
2654:
2553:
2439:
2391:
2201:
1760:
1472:
28:
3946:. Springer Monographs in Mathematics.
3250:into its irreducible components. Thus
2644:after a change of variables. Assuming
322:can be written as the space of orbits
255:The trace formula has applications to
4186:Spectral methods of automorphic forms
3438:{\displaystyle a_{\Gamma }^{G}(\pi )}
1958:{\displaystyle a_{\Gamma }^{G}(\pi )}
188:. The initial publication in 1956 of
7:
4334:Selberg trace formula resource page
4320:Trace formulae in spectral geometry
3652:{\displaystyle \Gamma \backslash X}
2285:{\displaystyle \Gamma \backslash G}
2008:{\displaystyle \Gamma \backslash G}
1621:is the set of conjugacy classes in
3885:Eigenvalues in Riemannian geometry
3640:
3620:
3505:
3416:
3404:(recall that the positive integer
3308:
3100:
3087:
2979:
2958:
2935:
2872:
2850:
2774:
2663:{\displaystyle \Gamma \setminus G}
2651:
2577:
2550:
2436:
2388:
2273:
2035:
1996:
1936:
1862:
1810:
1751:
1720:
1701:{\displaystyle \gamma \in \Gamma }
1695:
1628:
1605:
1548:
1438:
1425:
1363:
1287:
1282:
865:
860:
799:
562:
553:
329:
298:The trace formula also has purely
14:
3927:Deutsche Mathematiker-Vereinigung
3919:Jahresber. Deutsch. Math.-Verein.
2991:{\displaystyle \Gamma ^{\gamma }}
1376:a discrete cocompact subgroup of
504:of the Laplacian; in other words
387:linear fractional transformations
3793:Jacquet–Langlands correspondence
659:Using the variable substitution
336:
216:play the role of prime numbers.
3716:, so a finite extension of the
3025:the respective centralizers of
1989:in the right-representation on
1132:there exist positive constants
693:
552:
235:of a given weight, for a given
3698:. Gel'fand et al also treat SL
3514:
3502:
3465:in the unitary representation
3432:
3426:
3368:
3342:
3336:
3324:
3318:
3295:
3272:
3266:
3217:
3211:
3167:
3145:
3116:
3110:
3067:
3061:
2902:
2880:
2839:
2833:
2804:
2782:
2757:
2745:
2722:
2716:
2686:
2680:
2624:
2618:
2607:
2585:
2532:
2523:
2517:
2511:
2492:
2486:
2483:
2477:
2471:
2465:
2445:
2433:
2397:
2385:
2345:
2339:
2333:
2327:
2308:
2302:
2234:
2212:
2121:
2115:
2109:
2103:
2070:
2064:
2032:
1952:
1946:
1773:
1747:
1736:
1730:
1663:{\displaystyle {\widehat {G}}}
1582:
1576:
1564:
1558:
1505:
1483:
1454:
1448:
1301:
1295:
1268:
1262:
1200:
1194:
1159:
1153:
1077:has to satisfy the following:
1043:
1040:
1034:
1022:
997:
990:
970:
963:
955:
942:
901:
892:
883:
877:
838:
832:
820:
807:
763:{\displaystyle r_{n},n\geq 0.}
687:
675:
546:
540:
531:
522:
153:is the fundamental group of a
1:
4190:American Mathematical Society
4091:10.1215/S0012-7094-76-04338-6
4053:American Mathematical Society
2817:along the diagonal, that is:
318:A compact hyperbolic surface
225:Eichler–Selberg trace formula
3769:Arthur–Selberg trace formula
2175:and the right-hand side the
1248:is the Fourier transform of
731:the eigenvalues are labeled
144:Arthur–Selberg trace formula
3018:{\displaystyle G^{\gamma }}
2944:{\displaystyle \{\Gamma \}}
2368:It extends continuously to
2076:{\displaystyle \pi (\phi )}
1614:{\displaystyle \{\Gamma \}}
55:square-integrable functions
4365:
4147:Hejhal, Dennis A. (1983),
4113:Hejhal, Dennis A. (1976),
3858:Cambridge University Press
3811:This presentation is from
3377:{\displaystyle {\hat {G}}}
489:-invariant eigenfunctions
399:Then the spectrum for the
192:dealt with this case, its
165:relating the zeros of the
92:is the cocompact subgroup
72:The simplest case is when
4078:Duke Mathematical Journal
3910:Elstrodt, Jürgen (1981).
3613:the conjugacy classes in
2670:is compact, the operator
1868:{\displaystyle G,\Gamma }
401:Laplace–Beltrami operator
110:Poisson summation formula
3973:Fischer, Jürgen (1987),
3223:{\displaystyle R(\phi )}
2728:{\displaystyle R(\phi )}
2692:{\displaystyle R(\phi )}
291:, taking account of the
84:for the character of an
3849:Automorphic forms on SL
3828:Lax & Phillips 1980
3626:{\displaystyle \Gamma }
3445:is the multiplicity of
3038:{\displaystyle \gamma }
2964:{\displaystyle \Gamma }
1842:{\displaystyle \gamma }
1634:{\displaystyle \Gamma }
1369:{\displaystyle \Gamma }
183:compact Riemann surface
88:of finite groups. When
4270:10.1002/cpa.3160250302
4232:Bull. Amer. Math. Soc.
3667:The general theory of
3653:
3627:
3603:
3569:
3549:
3521:
3479:
3459:
3439:
3398:
3378:
3349:
3244:
3224:
3194:of the trace formula.
3184:
3039:
3019:
2992:
2965:
2945:
2919:
2811:
2729:
2693:
2664:
2638:
2452:
2404:
2362:
2286:
2248:
2161:
2134:
2077:
2048:
2009:
1983:
1959:
1918:
1898:
1883:unitary representation
1869:
1843:
1823:
1783:
1702:
1664:
1635:
1615:
1589:
1394:
1370:
1334:
1235:
1053:
803:
764:
725:
651:
476:where the eigenvalues
470:
347:
300:differential-geometric
269:Hasse–Weil L-functions
86:induced representation
33:unitary representation
3730:formal Laurent series
3654:
3628:
3604:
3602:{\displaystyle X=G/K}
3570:
3550:
3522:
3480:
3460:
3440:
3399:
3379:
3350:
3245:
3225:
3185:
3040:
3020:
2993:
2966:
2946:
2920:
2812:
2730:
2694:
2665:
2639:
2453:
2405:
2363:
2287:
2249:
2162:
2135:
2078:
2049:
2010:
1984:
1960:
1919:
1899:
1870:
1844:
1824:
1784:
1703:
1665:
1636:
1616:
1590:
1395:
1393:{\displaystyle \phi }
1371:
1335:
1236:
1054:
783:
774:Selberg trace formula
765:
726:
652:
471:
348:
198:Selberg zeta function
171:Selberg zeta function
167:Riemann zeta function
25:Selberg trace formula
4297:J. Indian Math. Soc.
4055:. pp. 211–224.
3777:Langlands philosophy
3637:
3617:
3579:
3559:
3539:
3489:
3469:
3458:{\displaystyle \pi }
3449:
3408:
3388:
3359:
3254:
3234:
3205:
3049:
3029:
3002:
2975:
2955:
2929:
2821:
2739:
2710:
2674:
2648:
2462:
2414:
2372:
2296:
2270:
2183:
2151:
2087:
2058:
2019:
1993:
1982:{\displaystyle \pi }
1973:
1928:
1908:
1897:{\displaystyle \pi }
1888:
1853:
1833:
1829:the centralisers of
1793:
1712:
1686:
1645:
1625:
1599:
1408:
1384:
1360:
1256:
1144:
780:
735:
663:
508:
415:
326:
281:parabolic cohomology
245:Riemann–Roch theorem
98:of the real numbers
16:Mathematical theorem
3673:continuous spectrum
3425:
3317:
3109:
1945:
1729:
1557:
1447:
1291:
869:
265:Eichler and Shimura
257:arithmetic geometry
237:congruence subgroup
124:continuous spectrum
4220:Phillips, Ralph S.
4157:10.1007/BFb0061302
4123:10.1007/BFb0079608
3987:10.1007/BFb0077696
3690:) is discussed in
3649:
3623:
3609:is the associated
3599:
3565:
3545:
3517:
3475:
3455:
3435:
3411:
3394:
3374:
3345:
3303:
3302:
3240:
3220:
3180:
3095:
3094:
3035:
3015:
2988:
2961:
2941:
2915:
2876:
2807:
2778:
2725:
2689:
2660:
2634:
2581:
2448:
2400:
2358:
2282:
2244:
2157:
2130:
2073:
2044:
2005:
1979:
1955:
1931:
1914:
1894:
1865:
1839:
1819:
1779:
1715:
1698:
1660:
1631:
1611:
1585:
1543:
1542:
1433:
1432:
1390:
1366:
1330:
1274:
1231:
1049:
929:
852:
760:
721:
710:
647:
642:
466:
343:
285:Eichler cohomology
223:was linked to the
126:, described using
4349:Automorphic forms
4316:Sunada, Toshikazu
4166:978-3-540-12323-1
4132:978-3-540-07988-0
4069:Hejhal, Dennis A.
4029:978-0-12-279506-0
3996:978-3-540-15208-8
3817:. Academic Press.
3669:Eisenstein series
3568:{\displaystyle K}
3548:{\displaystyle G}
3478:{\displaystyle R}
3397:{\displaystyle G}
3371:
3298:
3278:
3243:{\displaystyle G}
3073:
2861:
2763:
2566:
2256:orbital integrals
2167:or its quotients.
2160:{\displaystyle G}
1917:{\displaystyle G}
1745:
1657:
1538:
1518:
1411:
1348:General statement
1017:
1012:
981:
914:
850:
709:
357:is a subgroup of
227:, of Selberg and
208:theory. Here the
202:explicit formulae
163:explicit formulas
128:Eisenstein series
82:Frobenius formula
27:, introduced by
4356:
4323:
4311:
4288:
4247:
4228:
4211:
4177:
4143:
4109:
4064:
4040:
4007:
3969:
3938:
3916:
3906:
3879:
3830:
3825:
3819:
3818:
3809:
3759:rational numbers
3714:ultrametric norm
3711:
3707:
3681:
3658:
3656:
3655:
3650:
3632:
3630:
3629:
3624:
3608:
3606:
3605:
3600:
3595:
3574:
3572:
3571:
3566:
3554:
3552:
3551:
3546:
3526:
3524:
3523:
3518:
3501:
3500:
3484:
3482:
3481:
3476:
3464:
3462:
3461:
3456:
3444:
3442:
3441:
3436:
3424:
3419:
3403:
3401:
3400:
3395:
3383:
3381:
3380:
3375:
3373:
3372:
3364:
3354:
3352:
3351:
3346:
3316:
3311:
3301:
3300:
3299:
3291:
3249:
3247:
3246:
3241:
3229:
3227:
3226:
3221:
3189:
3187:
3186:
3181:
3160:
3159:
3141:
3140:
3133:
3132:
3108:
3103:
3093:
3044:
3042:
3041:
3036:
3024:
3022:
3021:
3016:
3014:
3013:
2997:
2995:
2994:
2989:
2987:
2986:
2970:
2968:
2967:
2962:
2950:
2948:
2947:
2942:
2924:
2922:
2921:
2916:
2895:
2894:
2875:
2860:
2859:
2816:
2814:
2813:
2808:
2797:
2796:
2777:
2734:
2732:
2731:
2726:
2698:
2696:
2695:
2690:
2669:
2667:
2666:
2661:
2643:
2641:
2640:
2635:
2614:
2610:
2600:
2599:
2580:
2560:
2559:
2507:
2506:
2457:
2455:
2454:
2449:
2432:
2431:
2409:
2407:
2406:
2401:
2384:
2383:
2367:
2365:
2364:
2359:
2323:
2322:
2291:
2289:
2288:
2283:
2253:
2251:
2250:
2245:
2227:
2226:
2208:
2207:
2200:
2199:
2166:
2164:
2163:
2158:
2139:
2137:
2136:
2131:
2099:
2098:
2083:is the operator
2082:
2080:
2079:
2074:
2053:
2051:
2050:
2045:
2031:
2030:
2014:
2012:
2011:
2006:
1988:
1986:
1985:
1980:
1964:
1962:
1961:
1956:
1944:
1939:
1923:
1921:
1920:
1915:
1903:
1901:
1900:
1895:
1874:
1872:
1871:
1866:
1848:
1846:
1845:
1840:
1828:
1826:
1825:
1820:
1818:
1817:
1805:
1804:
1788:
1786:
1785:
1780:
1772:
1771:
1759:
1758:
1746:
1743:
1728:
1723:
1707:
1705:
1704:
1699:
1669:
1667:
1666:
1661:
1659:
1658:
1650:
1640:
1638:
1637:
1632:
1620:
1618:
1617:
1612:
1594:
1592:
1591:
1586:
1556:
1551:
1541:
1540:
1539:
1531:
1498:
1497:
1479:
1478:
1471:
1470:
1446:
1441:
1431:
1399:
1397:
1396:
1391:
1375:
1373:
1372:
1367:
1339:
1337:
1336:
1331:
1319:
1318:
1290:
1285:
1251:
1247:
1240:
1238:
1237:
1232:
1227:
1226:
1212:
1208:
1207:
1203:
1139:
1135:
1128:
1107:
1102:
1100:
1099:
1096:
1093:
1076:
1072:
1064:
1058:
1056:
1055:
1050:
1018:
1016:
1015:
1014:
1013:
1005:
983:
982:
974:
958:
954:
953:
931:
928:
868:
863:
851:
849:
841:
827:
819:
818:
802:
797:
769:
767:
766:
761:
747:
746:
730:
728:
727:
722:
711:
702:
656:
654:
653:
648:
646:
645:
630:
629:
617:
613:
612:
611:
596:
595:
578:
577:
503:
492:
488:
484:
475:
473:
472:
467:
459:
458:
446:
445:
433:
432:
406:
395:
384:
378:
373:upper half plane
370:
364:
356:
352:
350:
349:
344:
339:
321:
289:group cohomology
215:
210:closed geodesics
187:
152:
141:
133:
121:
107:
97:
91:
75:
68:
60:
52:
41:
4364:
4363:
4359:
4358:
4357:
4355:
4354:
4353:
4339:
4338:
4330:
4314:
4291:
4250:
4226:
4214:
4200:
4182:Iwaniec, Henryk
4180:
4167:
4146:
4133:
4112:
4067:
4045:Godement, Roger
4043:
4030:
4012:Gel'fand, I. M.
4010:
3997:
3979:Springer-Verlag
3972:
3958:
3948:Springer-Verlag
3941:
3914:
3909:
3895:
3882:
3868:
3852:
3842:
3839:
3834:
3833:
3826:
3822:
3812:
3810:
3806:
3801:
3789:
3756:
3739:
3727:
3709:
3703:
3701:
3685:
3679:
3665:
3635:
3634:
3615:
3614:
3611:symmetric space
3577:
3576:
3557:
3556:
3537:
3536:
3533:
3492:
3487:
3486:
3467:
3466:
3447:
3446:
3406:
3405:
3386:
3385:
3357:
3356:
3252:
3251:
3232:
3231:
3203:
3202:
3190:This gives the
3148:
3124:
3119:
3047:
3046:
3027:
3026:
3005:
3000:
2999:
2978:
2973:
2972:
2953:
2952:
2927:
2926:
2883:
2845:
2819:
2818:
2785:
2737:
2736:
2708:
2707:
2672:
2671:
2646:
2645:
2588:
2565:
2561:
2545:
2498:
2460:
2459:
2423:
2412:
2411:
2375:
2370:
2369:
2314:
2294:
2293:
2268:
2267:
2264:
2215:
2191:
2186:
2181:
2180:
2149:
2148:
2090:
2085:
2084:
2056:
2055:
2022:
2017:
2016:
1991:
1990:
1971:
1970:
1926:
1925:
1906:
1905:
1886:
1885:
1851:
1850:
1831:
1830:
1809:
1796:
1791:
1790:
1763:
1750:
1710:
1709:
1684:
1683:
1682:for an element
1643:
1642:
1623:
1622:
1597:
1596:
1486:
1462:
1457:
1406:
1405:
1382:
1381:
1358:
1357:
1350:
1345:
1304:
1254:
1253:
1249:
1245:
1187:
1183:
1176:
1172:
1171:
1142:
1141:
1137:
1133:
1111:
1097:
1094:
1091:
1090:
1088:
1082:
1081:be analytic on
1074:
1066:
1062:
996:
969:
959:
945:
932:
842:
828:
810:
778:
777:
738:
733:
732:
661:
660:
641:
640:
621:
600:
584:
583:
579:
569:
566:
565:
512:
506:
505:
494:
490:
486:
482:
477:
450:
437:
424:
413:
412:
404:
393:
380:
376:
366:
358:
354:
324:
323:
319:
316:
308:Riemann surface
304:length spectrum
267:calculated the
253:
221:Hecke operators
213:
185:
179:
155:Riemann surface
150:
135:
131:
116:
99:
93:
89:
73:
66:
58:
43:
39:
17:
12:
11:
5:
4362:
4360:
4352:
4351:
4341:
4340:
4337:
4336:
4329:
4328:External links
4326:
4325:
4324:
4312:
4299:, New Series,
4289:
4264:(3): 225–246,
4248:
4212:
4198:
4178:
4165:
4144:
4131:
4110:
4085:(3): 441–482,
4065:
4041:
4028:
4020:Academic Press
4008:
3995:
3970:
3956:
3939:
3907:
3893:
3880:
3866:
3850:
3838:
3835:
3832:
3831:
3820:
3803:
3802:
3800:
3797:
3796:
3795:
3788:
3785:
3752:
3735:
3723:
3718:p-adic numbers
3699:
3683:
3664:
3661:
3648:
3645:
3642:
3622:
3598:
3594:
3590:
3587:
3584:
3564:
3544:
3532:
3529:
3516:
3513:
3510:
3507:
3504:
3499:
3495:
3474:
3454:
3434:
3431:
3428:
3423:
3418:
3414:
3393:
3370:
3367:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3315:
3310:
3306:
3297:
3294:
3288:
3285:
3281:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3239:
3219:
3216:
3213:
3210:
3192:geometric side
3179:
3176:
3173:
3169:
3166:
3163:
3158:
3155:
3151:
3147:
3144:
3139:
3136:
3131:
3127:
3122:
3118:
3115:
3112:
3107:
3102:
3098:
3092:
3089:
3086:
3083:
3080:
3076:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3034:
3012:
3008:
2985:
2981:
2960:
2940:
2937:
2934:
2914:
2911:
2908:
2904:
2901:
2898:
2893:
2890:
2886:
2882:
2879:
2874:
2871:
2868:
2864:
2858:
2855:
2852:
2848:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2806:
2803:
2800:
2795:
2792:
2788:
2784:
2781:
2776:
2773:
2770:
2766:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2724:
2721:
2718:
2715:
2688:
2685:
2682:
2679:
2659:
2656:
2653:
2633:
2630:
2626:
2623:
2620:
2617:
2613:
2609:
2606:
2603:
2598:
2595:
2591:
2587:
2584:
2579:
2576:
2573:
2569:
2564:
2558:
2555:
2552:
2548:
2544:
2541:
2538:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2505:
2501:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2447:
2444:
2441:
2438:
2435:
2430:
2426:
2422:
2419:
2399:
2396:
2393:
2390:
2387:
2382:
2378:
2357:
2354:
2351:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2321:
2317:
2313:
2310:
2307:
2304:
2301:
2281:
2278:
2275:
2263:
2260:
2243:
2240:
2236:
2233:
2230:
2225:
2222:
2218:
2214:
2211:
2206:
2203:
2198:
2194:
2189:
2173:geometric side
2169:
2168:
2156:
2141:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2097:
2093:
2072:
2069:
2066:
2063:
2043:
2040:
2037:
2034:
2029:
2025:
2004:
2001:
1998:
1978:
1954:
1951:
1948:
1943:
1938:
1934:
1913:
1893:
1876:
1864:
1861:
1858:
1838:
1816:
1812:
1808:
1803:
1799:
1778:
1775:
1770:
1766:
1762:
1757:
1753:
1749:
1741:
1738:
1735:
1732:
1727:
1722:
1718:
1697:
1694:
1691:
1656:
1653:
1630:
1610:
1607:
1604:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1555:
1550:
1546:
1537:
1534:
1528:
1525:
1521:
1517:
1514:
1511:
1507:
1504:
1501:
1496:
1493:
1489:
1485:
1482:
1477:
1474:
1469:
1465:
1460:
1456:
1453:
1450:
1445:
1440:
1436:
1430:
1427:
1424:
1421:
1418:
1414:
1389:
1365:
1349:
1346:
1344:
1341:
1329:
1326:
1323:
1317:
1314:
1311:
1307:
1303:
1300:
1297:
1294:
1289:
1284:
1281:
1277:
1273:
1270:
1267:
1264:
1261:
1242:
1241:
1230:
1225:
1222:
1219:
1216:
1211:
1206:
1202:
1199:
1196:
1193:
1190:
1186:
1182:
1179:
1175:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1130:
1109:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1011:
1008:
1003:
999:
995:
992:
989:
986:
980:
977:
972:
968:
965:
962:
957:
952:
948:
944:
941:
938:
935:
927:
924:
921:
917:
913:
910:
907:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
872:
867:
862:
859:
855:
848:
845:
840:
837:
834:
831:
825:
822:
817:
813:
809:
806:
801:
796:
793:
790:
786:
759:
756:
753:
750:
745:
741:
720:
717:
714:
708:
705:
699:
696:
692:
689:
686:
683:
680:
677:
674:
671:
668:
644:
639:
636:
633:
628:
624:
620:
616:
610:
607:
603:
599:
594:
591:
587:
582:
576:
572:
568:
567:
564:
561:
558:
555:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
517:
515:
485:correspond to
480:
465:
462:
457:
453:
449:
444:
440:
436:
431:
427:
423:
420:
342:
338:
334:
331:
315:
312:
273:modular curves
271:associated to
252:
249:
229:Martin Eichler
178:
175:
115:The case when
63:discrete group
61:is a cofinite
29:Selberg (1956)
15:
13:
10:
9:
6:
4:
3:
2:
4361:
4350:
4347:
4346:
4344:
4335:
4332:
4331:
4327:
4321:
4317:
4313:
4310:
4306:
4302:
4298:
4294:
4293:Selberg, Atle
4290:
4287:
4283:
4279:
4275:
4271:
4267:
4263:
4259:
4258:
4253:
4252:McKean, H. P.
4249:
4245:
4241:
4237:
4234:
4233:
4225:
4221:
4217:
4216:Lax, Peter D.
4213:
4209:
4205:
4201:
4199:0-8218-3160-7
4195:
4191:
4187:
4183:
4179:
4176:
4172:
4168:
4162:
4158:
4154:
4150:
4145:
4142:
4138:
4134:
4128:
4124:
4120:
4116:
4111:
4108:
4104:
4100:
4096:
4092:
4088:
4084:
4080:
4079:
4074:
4070:
4066:
4062:
4058:
4054:
4050:
4046:
4042:
4039:
4035:
4031:
4025:
4021:
4017:
4013:
4009:
4006:
4002:
3998:
3992:
3988:
3984:
3980:
3976:
3971:
3967:
3963:
3959:
3957:3-540-62745-6
3953:
3949:
3945:
3940:
3936:
3932:
3928:
3925:. Bielefeld:
3924:
3921:(in German).
3920:
3913:
3908:
3904:
3900:
3896:
3894:0-12-170640-0
3890:
3886:
3881:
3877:
3873:
3869:
3867:0-521-58049-8
3863:
3859:
3855:
3854:
3845:
3844:Borel, Armand
3841:
3840:
3836:
3829:
3824:
3821:
3816:
3808:
3805:
3798:
3794:
3791:
3790:
3786:
3784:
3782:
3778:
3774:
3770:
3765:
3763:
3760:
3755:
3751:
3747:
3743:
3738:
3734:
3731:
3726:
3722:
3719:
3715:
3706:
3697:
3693:
3689:
3676:
3674:
3670:
3662:
3660:
3646:
3612:
3596:
3592:
3588:
3585:
3582:
3562:
3542:
3530:
3528:
3511:
3497:
3493:
3472:
3452:
3429:
3421:
3412:
3391:
3365:
3339:
3333:
3330:
3327:
3321:
3313:
3304:
3292:
3286:
3283:
3279:
3275:
3269:
3263:
3260:
3257:
3237:
3214:
3208:
3200:
3199:spectral side
3195:
3193:
3177:
3174:
3171:
3164:
3161:
3156:
3153:
3149:
3142:
3137:
3129:
3125:
3120:
3113:
3105:
3096:
3081:
3078:
3074:
3070:
3064:
3058:
3055:
3052:
3032:
3010:
3006:
2983:
2912:
2909:
2906:
2899:
2896:
2891:
2888:
2884:
2877:
2869:
2866:
2862:
2856:
2846:
2842:
2836:
2830:
2827:
2824:
2801:
2798:
2793:
2790:
2786:
2779:
2771:
2768:
2764:
2760:
2754:
2751:
2748:
2742:
2719:
2713:
2706:The trace of
2704:
2702:
2683:
2677:
2657:
2631:
2628:
2621:
2615:
2611:
2604:
2601:
2596:
2593:
2589:
2582:
2574:
2571:
2567:
2562:
2556:
2546:
2542:
2539:
2536:
2529:
2526:
2520:
2514:
2508:
2503:
2499:
2495:
2489:
2480:
2474:
2468:
2442:
2428:
2424:
2420:
2417:
2394:
2380:
2376:
2355:
2352:
2349:
2342:
2336:
2330:
2324:
2319:
2315:
2311:
2305:
2299:
2279:
2261:
2259:
2257:
2241:
2238:
2231:
2228:
2223:
2220:
2216:
2209:
2204:
2196:
2192:
2187:
2178:
2177:spectral side
2174:
2154:
2146:
2142:
2127:
2124:
2118:
2112:
2106:
2100:
2095:
2091:
2067:
2061:
2041:
2027:
2023:
2002:
1976:
1968:
1949:
1941:
1932:
1911:
1891:
1884:
1881:
1877:
1875:respectively;
1859:
1856:
1836:
1814:
1806:
1801:
1797:
1776:
1768:
1764:
1755:
1739:
1733:
1725:
1716:
1692:
1689:
1681:
1680:
1679:
1677:
1673:
1654:
1651:
1579:
1573:
1570:
1567:
1561:
1553:
1544:
1535:
1532:
1526:
1523:
1519:
1515:
1512:
1509:
1502:
1499:
1494:
1491:
1487:
1480:
1475:
1467:
1463:
1458:
1451:
1443:
1434:
1419:
1416:
1412:
1403:
1387:
1379:
1355:
1347:
1342:
1340:
1327:
1324:
1321:
1315:
1312:
1309:
1305:
1298:
1292:
1279:
1275:
1271:
1265:
1259:
1244:The function
1228:
1223:
1220:
1217:
1214:
1209:
1204:
1197:
1191:
1188:
1184:
1180:
1177:
1173:
1168:
1165:
1156:
1150:
1131:
1126:
1122:
1118:
1114:
1110:
1106:
1086:
1080:
1079:
1078:
1070:
1059:
1046:
1037:
1031:
1028:
1025:
1019:
1009:
1006:
1001:
993:
987:
984:
978:
975:
966:
960:
950:
946:
939:
936:
933:
922:
915:
911:
908:
905:
898:
895:
889:
886:
880:
874:
870:
857:
853:
846:
843:
835:
829:
823:
815:
811:
804:
794:
791:
788:
784:
775:
770:
757:
754:
751:
748:
743:
739:
718:
715:
712:
706:
703:
697:
694:
690:
684:
681:
678:
672:
669:
666:
657:
637:
634:
631:
626:
622:
618:
614:
608:
605:
601:
597:
592:
589:
585:
580:
574:
570:
559:
556:
549:
543:
537:
534:
528:
525:
519:
513:
501:
497:
483:
463:
460:
455:
451:
447:
442:
438:
434:
429:
425:
421:
418:
410:
402:
397:
390:
388:
383:
374:
369:
362:
340:
313:
311:
309:
305:
301:
296:
294:
290:
286:
282:
278:
274:
270:
266:
262:
261:number theory
258:
250:
248:
246:
242:
241:modular group
238:
234:
230:
226:
222:
217:
211:
207:
203:
199:
195:
191:
184:
177:Early history
176:
174:
172:
168:
164:
160:
156:
147:
145:
139:
134:is the group
129:
125:
120:
113:
111:
106:
102:
96:
87:
83:
79:
70:
64:
56:
50:
46:
42:on the space
38:
34:
30:
26:
22:
4319:
4300:
4296:
4261:
4255:
4235:
4230:
4185:
4148:
4114:
4082:
4076:
4048:
4015:
3974:
3943:
3922:
3918:
3884:
3848:
3823:
3814:
3807:
3772:
3766:
3761:
3753:
3749:
3745:
3741:
3736:
3732:
3724:
3720:
3704:
3687:
3677:
3666:
3534:
3198:
3196:
3191:
2705:
2265:
2179:. The terms
2176:
2172:
2170:
2145:Haar measure
1967:multiplicity
1675:
1672:unitary dual
1401:
1377:
1353:
1351:
1243:
1124:
1120:
1116:
1112:
1104:
1084:
1068:
1060:
776:is given by
773:
771:
658:
499:
495:
478:
398:
391:
381:
367:
360:
317:
297:
277:Goro Shimura
254:
251:Applications
224:
218:
206:prime number
190:Atle Selberg
180:
148:
137:
118:
114:
104:
100:
94:
71:
48:
44:
24:
18:
4238:: 261–295.
2701:trace-class
1880:irreducible
1252:, that is,
1140:such that:
21:mathematics
3837:References
3728:or of the
3663:Later work
411:; that is
233:cusp forms
4303:: 47–87,
4278:0010-3640
4099:0012-7094
3929:: 45–77.
3781:endoscopy
3644:∖
3641:Γ
3621:Γ
3509:∖
3506:Γ
3453:π
3430:π
3417:Γ
3369:^
3340:ϕ
3334:π
3331:
3322:π
3309:Γ
3296:^
3287:∈
3284:π
3280:∑
3270:ϕ
3261:
3215:ϕ
3162:γ
3154:−
3143:ϕ
3135:∖
3130:γ
3121:∫
3114:γ
3101:Γ
3088:Γ
3082:∈
3079:γ
3075:∑
3065:ϕ
3056:
3033:γ
3011:γ
2984:γ
2980:Γ
2959:Γ
2936:Γ
2897:γ
2889:−
2878:ϕ
2873:Γ
2870:∈
2867:γ
2863:∑
2854:∖
2851:Γ
2847:∫
2837:ϕ
2828:
2799:γ
2791:−
2780:ϕ
2775:Γ
2772:∈
2769:γ
2765:∑
2720:ϕ
2684:ϕ
2655:∖
2652:Γ
2602:γ
2594:−
2583:ϕ
2578:Γ
2575:∈
2572:γ
2568:∑
2554:∖
2551:Γ
2547:∫
2509:ϕ
2500:∫
2475:ϕ
2458:we have:
2440:∖
2437:Γ
2421:∈
2392:∖
2389:Γ
2325:ϕ
2316:∫
2306:ϕ
2277:∖
2274:Γ
2229:γ
2221:−
2210:ϕ
2202:∖
2197:γ
2188:∫
2113:π
2101:ϕ
2092:∫
2068:ϕ
2062:π
2039:∖
2036:Γ
2000:∖
1997:Γ
1977:π
1950:π
1937:Γ
1892:π
1863:Γ
1837:γ
1815:γ
1811:Γ
1802:γ
1769:γ
1761:∖
1756:γ
1752:Γ
1734:γ
1721:Γ
1696:Γ
1693:∈
1690:γ
1655:^
1629:Γ
1606:Γ
1580:ϕ
1574:π
1571:
1562:π
1549:Γ
1536:^
1527:∈
1524:π
1520:∑
1500:γ
1492:−
1481:ϕ
1473:∖
1468:γ
1459:∫
1452:γ
1439:Γ
1426:Γ
1420:∈
1417:γ
1413:∑
1388:ϕ
1364:Γ
1288:∞
1283:∞
1280:−
1276:∫
1224:δ
1221:−
1215:−
1192:
1166:≤
1071: }
1029:
1002:−
985:−
937:
916:∑
896:π
890:
866:∞
861:∞
858:−
854:∫
847:π
830:μ
800:∞
785:∑
772:Then the
755:≥
682:−
667:μ
623:μ
563:Γ
560:∈
557:γ
554:∀
526:γ
464:⋯
461:≤
452:μ
448:≤
439:μ
426:μ
409:resolvent
333:∖
330:Γ
194:Laplacian
159:Laplacian
78:cocompact
37:Lie group
4343:Category
4318:(1991),
4222:(1980).
4184:(2002).
4071:(1976),
3846:(1997).
3787:See also
3708:) where
2410:and for
379:acts on
57:, where
4309:0088511
4286:0473166
4244:0555264
4208:1942691
4175:0711197
4141:0439755
4107:0414490
4061:0210827
4038:1071179
4005:0892317
3966:1483315
3935:0612411
3903:0768584
3876:1482800
3757:of the
2054:), and
1965:is the
1878:for an
1670:is the
1101:
1089:
371:is the
359:PSL(2,
239:of the
4307:
4284:
4276:
4242:
4206:
4196:
4173:
4163:
4139:
4129:
4105:
4097:
4059:
4036:
4026:
4003:
3993:
3964:
3954:
3933:
3901:
3891:
3874:
3864:
3355:where
2971:, and
1744:volume
1595:where
375:, and
365:, and
353:where
283:(from
136:SL(2,
23:, the
4227:(PDF)
3915:(PDF)
3799:Notes
3535:When
2262:Proof
1789:with
1678:and:
1087:)| ≤
306:of a
293:cusps
149:When
35:of a
4274:ISSN
4194:ISBN
4161:ISBN
4127:ISBN
4095:ISSN
4024:ISBN
3991:ISBN
3952:ISBN
3889:ISBN
3862:ISBN
3748:and
3694:and
3575:and
3527:).
3197:The
2998:and
2925:Let
2254:are
1380:and
1352:Let
1136:and
1119:) =
1083:|Im(
887:tanh
435:<
259:and
4266:doi
4153:doi
4119:doi
4087:doi
3983:doi
3853:(R)
3485:on
2699:is
2292::
2258:.
2147:on
2015:in
1969:of
1904:of
1849:in
1674:of
1026:log
934:log
493:in
403:on
385:by
212:on
204:of
76:is
53:of
47:(Γ\
19:In
4345::
4305:MR
4301:20
4282:MR
4280:,
4272:,
4262:25
4260:,
4240:MR
4229:.
4218:;
4204:MR
4202:.
4192:.
4171:MR
4169:,
4159:,
4137:MR
4135:,
4125:,
4103:MR
4101:,
4093:,
4083:43
4081:,
4075:,
4057:MR
4034:MR
4032:,
4022:,
4001:MR
3999:,
3989:,
3981:,
3962:MR
3960:.
3950:.
3931:MR
3923:83
3917:.
3899:MR
3897:.
3872:MR
3870:.
3860:.
3764:.
3740:((
3328:tr
3258:tr
3053:tr
2825:tr
1924:,
1708:,
1641:,
1568:tr
1189:Re
1115:(−
1103:+
758:0.
638:0.
389:.
275:;
247:.
146:.
117:Γ\
112:.
103:=
69:.
4268::
4246:.
4236:2
4210:.
4155::
4121::
4089::
4063:.
3985::
3968:.
3937:.
3905:.
3878:.
3851:2
3773:G
3762:Q
3754:p
3750:Q
3746:R
3742:T
3737:q
3733:F
3725:p
3721:Q
3710:F
3705:F
3702:(
3700:2
3688:C
3686:(
3684:2
3680:Γ
3647:X
3597:K
3593:/
3589:G
3586:=
3583:X
3563:K
3543:G
3515:)
3512:G
3503:(
3498:2
3494:L
3473:R
3433:)
3427:(
3422:G
3413:a
3392:G
3366:G
3343:)
3337:(
3325:)
3319:(
3314:G
3305:a
3293:G
3276:=
3273:)
3267:(
3264:R
3238:G
3218:)
3212:(
3209:R
3178:.
3175:x
3172:d
3168:)
3165:x
3157:1
3150:x
3146:(
3138:G
3126:G
3117:)
3111:(
3106:G
3097:a
3091:}
3085:{
3071:=
3068:)
3062:(
3059:R
3007:G
2939:}
2933:{
2913:.
2910:x
2907:d
2903:)
2900:x
2892:1
2885:x
2881:(
2857:G
2843:=
2840:)
2834:(
2831:R
2805:)
2802:y
2794:1
2787:x
2783:(
2761:=
2758:)
2755:y
2752:,
2749:x
2746:(
2743:K
2723:)
2717:(
2714:R
2687:)
2681:(
2678:R
2658:G
2632:y
2629:d
2625:)
2622:y
2619:(
2616:f
2612:)
2608:)
2605:y
2597:1
2590:x
2586:(
2563:(
2557:G
2543:=
2540:y
2537:d
2533:)
2530:y
2527:x
2524:(
2521:f
2518:)
2515:y
2512:(
2504:G
2496:=
2493:)
2490:x
2487:(
2484:)
2481:f
2478:)
2472:(
2469:R
2466:(
2446:)
2443:G
2434:(
2429:2
2425:L
2418:f
2398:)
2395:G
2386:(
2381:2
2377:L
2356:,
2353:x
2350:d
2346:)
2343:x
2340:(
2337:R
2334:)
2331:x
2328:(
2320:G
2312:=
2309:)
2303:(
2300:R
2280:G
2242:x
2239:d
2235:)
2232:x
2224:1
2217:x
2213:(
2205:G
2193:G
2155:G
2140:;
2128:g
2125:d
2122:)
2119:g
2116:(
2110:)
2107:g
2104:(
2096:G
2071:)
2065:(
2042:G
2033:(
2028:2
2024:L
2003:G
1953:)
1947:(
1942:G
1933:a
1912:G
1860:,
1857:G
1807:,
1798:G
1777:.
1774:)
1765:G
1748:(
1740:=
1737:)
1731:(
1726:G
1717:a
1676:G
1652:G
1609:}
1603:{
1583:)
1577:(
1565:)
1559:(
1554:G
1545:a
1533:G
1516:=
1513:x
1510:d
1506:)
1503:x
1495:1
1488:x
1484:(
1476:G
1464:G
1455:)
1449:(
1444:G
1435:a
1429:}
1423:{
1402:G
1378:G
1354:G
1328:.
1325:u
1322:d
1316:u
1313:r
1310:i
1306:e
1302:)
1299:u
1296:(
1293:g
1272:=
1269:)
1266:r
1263:(
1260:h
1250:h
1246:g
1229:.
1218:2
1210:)
1205:|
1201:)
1198:r
1195:(
1185:|
1181:+
1178:1
1174:(
1169:M
1163:|
1160:)
1157:r
1154:(
1151:h
1148:|
1138:M
1134:δ
1129:;
1127:)
1125:r
1123:(
1121:h
1117:r
1113:h
1108:;
1105:δ
1098:2
1095:/
1092:1
1085:r
1075:h
1069:T
1067:{
1063:Γ
1047:.
1044:)
1041:)
1038:T
1035:(
1032:N
1023:(
1020:g
1010:2
1007:1
998:)
994:T
991:(
988:N
979:2
976:1
971:)
967:T
964:(
961:N
956:)
951:0
947:T
943:(
940:N
926:}
923:T
920:{
912:+
909:r
906:d
902:)
899:r
893:(
884:)
881:r
878:(
875:h
871:r
844:4
839:)
836:X
833:(
824:=
821:)
816:n
812:r
808:(
805:h
795:0
792:=
789:n
752:n
749:,
744:n
740:r
719:r
716:i
713:+
707:2
704:1
698:=
695:s
691:,
688:)
685:s
679:1
676:(
673:s
670:=
635:=
632:u
627:n
619:+
615:)
609:y
606:y
602:u
598:+
593:x
590:x
586:u
581:(
575:2
571:y
550:,
547:)
544:z
541:(
538:u
535:=
532:)
529:z
523:(
520:u
514:{
502:)
500:H
498:(
496:C
491:u
487:Γ
481:n
479:μ
456:2
443:1
430:0
422:=
419:0
405:X
394:Γ
382:H
377:Γ
368:H
363:)
361:R
355:Γ
341:,
337:H
320:X
214:S
186:S
151:Γ
140:)
138:R
132:G
119:G
105:R
101:G
95:Z
90:Γ
74:Γ
67:G
59:Γ
51:)
49:G
45:L
40:G
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