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Eichler–Shimura congruence relation

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Eichler, Martin (1954), "Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion",
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Shimura, Goro (1958), "Correspondances modulaires et les fonctions ζ de courbes algébriques",
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Expresses the local L-function of a modular curve at a prime in terms of Hecke operators
392: 343: 320: 88:). Roughly speaking, it says that the correspondence on the modular curve inducing the 396: 376: 324: 304: 188: 366: 296: 196: 384: 312: 184: 89: 73: 69: 416: 380: 308: 105: 54: 38: 371: 402: 200: 195:
of a modular curve or a more general modular variety, with the product of
81: 58: 338:. Lecture Notes in Mathematics. Vol. 349. Antwerp. pp. 317–360. 300: 65: 183:
The Eichler–Shimura congruence relation and its generalizations to
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Introduction to the arithmetic theory of automorphic functions
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Archiv für mathematische Logik und Grundlagenforschung
330: 414: 242:but its sources remain unclear because it lacks 359:Journal of the Mathematical Society of Japan 334:(1972). "Zeta functions of modular curves". 348:: CS1 maint: location missing publisher ( 370: 273:Learn how and when to remove this message 409:, Publ. of Math. Soc. of Japan, 11, 1971 356: 286: 85: 77: 14: 415: 203:or a product of analogous automorphic 336:Modular functions of one variable II 214: 43:Eichler–Shimura congruence relation 24: 25: 444: 138:as endomorphisms of the Jacobian 219: 191:, by identifying a part of the 13: 1: 210: 7: 187:play a pivotal role in the 32:Eichler–Shimura isomorphism 10: 449: 29: 433:Theorems in number theory 228:This article includes a 193:Hasse–Weil zeta function 171:) over the finite field 30:Not to be confused with 18:Eichler-Shimura relation 332:Piatetski-Shapiro, Ilya 257:more precise citations. 72:. It was introduced by 372:10.2969/jmsj/01010001 160:of the modular curve 80:) and generalized by 428:Zeta and L-functions 301:10.1007/BF01898377 230:list of references 115:. In other words, 111:and its transpose 104:to the sum of the 283: 282: 275: 197:Mellin transforms 189:Langlands program 185:Shimura varieties 100:is congruent mod 16:(Redirected from 440: 399: 374: 353: 347: 339: 327: 295:(4–6): 355–366, 278: 271: 267: 264: 258: 253:this article by 244:inline citations 223: 222: 215: 64:in terms of the 21: 448: 447: 443: 442: 441: 439: 438: 437: 413: 412: 341: 340: 279: 268: 262: 259: 248: 234:related reading 224: 220: 213: 179: 166: 159: 158: 144: 126: 99: 70:Hecke operators 35: 28: 23: 22: 15: 12: 11: 5: 446: 436: 435: 430: 425: 411: 410: 400: 354: 328: 281: 280: 238:external links 227: 225: 218: 212: 209: 175: 164: 154: 150: 142: 136: 135: 122: 95: 90:Hecke operator 45:expresses the 26: 9: 6: 4: 3: 2: 445: 434: 431: 429: 426: 424: 423:Modular forms 421: 420: 418: 408: 404: 401: 398: 394: 390: 386: 382: 378: 373: 368: 364: 360: 355: 351: 345: 337: 333: 329: 326: 322: 318: 314: 310: 306: 302: 298: 294: 290: 285: 284: 277: 274: 266: 256: 252: 246: 245: 239: 235: 231: 226: 217: 216: 208: 206: 202: 201:modular forms 198: 194: 190: 186: 181: 178: 174: 170: 163: 157: 153: 148: 141: 134: 130: 125: 121: 118: 117: 116: 114: 110: 107: 106:Frobenius map 103: 98: 94: 91: 87: 83: 79: 75: 71: 67: 63: 60: 56: 55:modular curve 52: 50: 44: 40: 39:number theory 33: 19: 406: 403:Goro Shimura 362: 358: 335: 292: 288: 269: 260: 249:Please help 241: 207:-functions. 204: 199:of weight 2 182: 176: 172: 168: 161: 155: 151: 146: 139: 137: 132: 128: 123: 119: 112: 108: 101: 96: 92: 61: 48: 42: 36: 255:introducing 66:eigenvalues 417:Categories 263:March 2016 211:References 397:119360118 381:0025-5645 344:cite book 325:119801181 309:0003-9268 51:-function 365:: 1–28, 389:0095173 317:0063406 251:improve 84: ( 82:Shimura 76: ( 74:Eichler 395:  387:  379:  323:  315:  307:  47:local 41:, the 393:S2CID 321:S2CID 236:, or 59:prime 57:at a 53:of a 377:ISSN 350:link 305:ISSN 129:Frob 109:Frob 86:1958 78:1954 367:doi 297:doi 133:Ver 113:Ver 68:of 37:In 419:: 405:, 391:, 385:MR 383:, 375:, 363:10 361:, 346:}} 342:{{ 319:, 313:MR 311:, 303:, 291:, 240:, 232:, 180:. 131:+ 127:= 369:: 352:) 299:: 293:5 276:) 270:( 265:) 261:( 247:. 205:L 177:p 173:F 169:N 167:( 165:0 162:X 156:p 152:F 149:) 147:N 145:( 143:0 140:J 124:p 120:T 102:p 97:p 93:T 62:p 49:L 34:. 20:)

Index

Eichler-Shimura relation
Eichler–Shimura isomorphism
number theory
local L-function
modular curve
prime
eigenvalues
Hecke operators
Eichler
1954
Shimura
1958
Hecke operator
Frobenius map
Shimura varieties
Langlands program
Hasse–Weil zeta function
Mellin transforms
modular forms
list of references
related reading
external links
inline citations
improve
introducing
Learn how and when to remove this message
doi
10.1007/BF01898377
ISSN
0003-9268

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