LEADER 08170nam 2201429 a 450 001 9910789850303321 005 20210518030658.0 010 $a1-283-05180-X 010 $a9786613051806 010 $a1-4008-3900-9 024 7 $a10.1515/9781400839001 035 $a(CKB)2670000000079377 035 $a(EBL)670341 035 $a(OCoLC)729386470 035 $a(SSID)ssj0000483479 035 $a(PQKBManifestationID)11337977 035 $a(PQKBTitleCode)TC0000483479 035 $a(PQKBWorkID)10528800 035 $a(PQKB)11285546 035 $a(MiAaPQ)EBC670341 035 $a(StDuBDS)EDZ0000514998 035 $a(WaSeSS)Ind00023282 035 $a(DE-B1597)446741 035 $a(OCoLC)979742221 035 $a(DE-B1597)9781400839001 035 $a(Au-PeEL)EBL670341 035 $a(CaPaEBR)ebr10456327 035 $a(CaONFJC)MIL305180 035 $z(PPN)199244936 035 $a(PPN)187957568 035 $a(EXLCZ)992670000000079377 100 $a20110124d2011 uy 0 101 0 $aeng 135 $aurun#---|uu|u 181 $ctxt 182 $cc 183 $acr 200 00$aComputational aspects of modular forms and Galois representations$b[electronic resource] $ehow one can compute in polynomial time the value of Ramanujan's tau at a prime /$fedited by Jean-Marc Couveignes and Bas Edixhoven 205 $aCourse Book 210 $aPrinceton, N.J. $cPrinceton University Press$dc2011 215 $a1 online resource (438 p.) 225 1 $aAnnals of mathematics studies ;$v176 300 $aDescription based upon print version of record. 311 0 $a0-691-14201-7 311 0 $a0-691-14202-5 320 $aIncludes bibliographical references and index. 327 $tFront matter --$tContents --$tPreface --$tAcknowledgments --$tAuthor information --$tDependencies between the chapters --$tChapter 1. Introduction, main results, context /$rEdixhoven, Bas --$tChapter 2. Modular curves, modular forms, lattices, Galois representations /$rEdixhoven, Bas --$tChapter 3. First description of the algorithms /$rCouveignes, Jean-Marc / Edixhoven, Bas --$tChapter 4. Short introduction to heights and Arakelov theory /$rEdixhoven, Bas / de Jong, Robin --$tChapter 5. Computing complex zeros of polynomials and power series /$rCouveignes, Jean-Marc --$tChapter 6. Computations with modular forms and Galois representations /$rBosman, Johan --$tChapter 7. Polynomials for projective representations of level one forms /$rBosman, Johan --$tChapter 8. Description of X1(5l) /$rEdixhoven, Bas --$tChapter 9. Applying Arakelov theory /$rEdixhoven, Bas / de Jong, Robin --$tChapter 10. An upper bound for Green functions on Riemann surfaces /$rMerkl, Franz --$tChapter 11. Bounds for Arakelov invariants of modular curves /$rEdixhoven, B. / de Jong, R. --$tChapter 12. Approximating Vf over the complex numbers /$rCouveignes, Jean-Marc --$tChapter 13. Computing Vf modulo p /$rCouveignes, Jean-Marc --$tChapter 14. Computing the residual Galois representations /$rEdixhoven, Bas --$tChapter 15. Computing coefficients of modular forms /$rEdixhoven, Bas --$tEpilogue --$tBibliography --$tIndex 330 $a"Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number P can be computed in time bounded by a fixed power of the logarithm of P. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations"--$cProvided by publisher. 330 $a"This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"--$cProvided by publisher. 410 0$aAnnals of mathematics studies ;$vno. 176. 606 $aGalois modules (Algebra) 606 $aClass field theory 610 $aArakelov invariants. 610 $aArakelov theory. 610 $aFourier coefficients. 610 $aGalois representation. 610 $aGalois representations. 610 $aGreen functions. 610 $aHecke operators. 610 $aJacobians. 610 $aLanglands program. 610 $aLas Vegas algorithm. 610 $aLehmer. 610 $aPeter Bruin. 610 $aRamanujan's tau function. 610 $aRamanujan's tau-function. 610 $aRamanujan's tau. 610 $aRiemann surfaces. 610 $aSchoof's algorithm. 610 $aTuring machines. 610 $aalgorithms. 610 $aarithmetic geometry. 610 $aarithmetic surfaces. 610 $abounding heights. 610 $abounds. 610 $acoefficients. 610 $acomplex roots. 610 $acomputation. 610 $acomputing algorithms. 610 $acomputing coefficients. 610 $acusp forms. 610 $acuspidal divisor. 610 $aeigenforms. 610 $afinite fields. 610 $aheight functions. 610 $ainequality. 610 $alattices. 610 $aminimal polynomial. 610 $amodular curves. 610 $amodular forms. 610 $amodular representation. 610 $amodular representations. 610 $amodular symbols. 610 $anonvanishing conjecture. 610 $ap-adic methods. 610 $aplane curves. 610 $apolynomial time algorithm. 610 $apolynomial time algoriths. 610 $apolynomial time. 610 $apolynomials. 610 $apower series. 610 $aprobabilistic polynomial time. 610 $arandom divisors. 610 $aresidual representation. 610 $asquare root. 610 $asquare-free levels. 610 $atale cohomology. 610 $atorsion divisors. 610 $atorsion. 615 0$aGalois modules (Algebra) 615 0$aClass field theory. 676 $a512/.32 686 $aMAT001000$aMAT012010$2bisacsh 701 $aEdixhoven$b B$g(Bas),$f1962-$060658 701 $aCouveignes$b Jean-Marc$01580823 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910789850303321 996 $aComputational aspects of modular forms and Galois representations$93862029 997 $aUNINA