


ID:
744875.0,
MPI für Mathematik / Export Zagier 
Report 32/2005: Explicit methods in number theory (July 17th  July 23rd, 2005). 
Editors:  Cohen, Henri; Lenstra, Hendrik W.; Zagier, Don B.  Date of Publication (YYYYMMDD):  2005  Title of Journal:  Oberwolfach Reports  Volume:  2  Issue / Number:  3  Start Page:  1799  End Page:  1866  Audience:  Not Specified  Intended Educational Use:  No  Abstract / Description:  From the text: These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics included modular forms, varieties over finite fields, rational and integral points on varieties, class groups, and integer factorization.\par The workshop `Explicit Methods in Number Theory' was organised by Henri Cohen (Talence), Hendrik W. Lenstra (Leiden), and Don B. Zagier (Bonn) and was held July 1723, 2005. Three previous workshops on the topic had been held in 1999, 2001, and 2003. The goal of this meeting was to present new methods and results on concrete aspects of number theory. In many cases, this included computational and experimental work, but the primary emphasis was placed on the implications for number theory rather than on the computational methods employed. \par There was a ``miniseries'' of five 1hour morning talks given by Bas Edixhoven, Johan Bosman, Robin de Jong, and JeanMarc Couveignes on the topic of computing the coefficients of modular forms. Let Δ = q \prodn≥1 (1q^n)^24 = \sumn≥1 τ(n)q^n be Ramanujan's tau function, a newform of weight 12 for \textSL2(\mathbb Z). The speakers exhibited a method to compute τ(p) for p prime in time polynomial in \log p.\par Some of the other main themes included:\par \bullet Modular forms, qexpansions, and Arakelov geometry \par \bullet Rational and integral points on curves and higherdimensional varieties \par \bullet Integer factorization \par \bullet Counting points on varieties over finite fields\par \bullet Class groups of quadratic and cubic fields and their relationship to geometry, analysis, and arithmetic.\par Contributions:\par  Bas Edixhoven (joint with JeanMarc Couveignes, Robin de Jong), On the computation of the coefficients of a modular form, I: introduction (p. 1803) \par  Bjorn Poonen, Characterizing characteristic 0 function fields (p. 1805) \par  Kiran S. Kedlaya, Computing zeta functions of surfaces (p. 1808)\par  E. Victor Flynn (joint with Nils Bruin), Annihilation of Sha on Jacobians (p. 1810) \par  John Voight, Computing maximal orders of quaternion algebras (p. 1812) \par  Fernando RodriguezVillegas, Ratios of factorial and algebraic hypergeometric functions (p. 1813) \par  Johan Bosman (joint with Bas Edixhoven), On the computation of the coefficients of a modular form, II: explicit calculations (p. 1816)\par  Thorsten Kleinjung, Polynomial Selection for NFS I (p. 1818) \par  Daniel Bernstein, Polynomial Selection for NFS II (p. 1819) \par  Frank Calegari (joint with Nathan Dunfield), Automorphic forms and rational homology spheres (p. 1820) \par  Jürgen Klüners (joint with Étienne Fouvry), CohenLenstra heuristics for 4ranks of class groups of quadratic number fields (p. 1821)\par  Dongho Byeon, Class numbers, elliptic curves, and hyperelliptic curves (p. 1823) \par  Michael Stoll, Finite coverings and rational points (p. 1824)\par  Robin de Jong (joint with JeanMarc Couveignes, Bas Edixhoven), On the computation of the coefficients of a modular form, III: Application of Arakelov intersection theory (p. 1827)\par  Neeraj Kayal, Solvability of polynomial equations over finite fields (p. 1828) \par  Mark Watkins, Random matrix theory and Heegner points (p. 1829)\par  Bas Edixhoven (joint with JeanMarc Couveignes, Robin de Jong), On the computation of the coefficients of a modular form, IV: the Arakelov contribution (p. 1830)\par  Samir Siksek (joint with Martin Bright), Functions, reciprocity and the obstruction to divisors on curves (p. 1832) \par  Michael A. Bennett (joint with P.G. Walsh), Integral points on congruent number curves (p. 1835)\par  Bart de Smit (joint with Lara Thomas), Local Galois module structure for ArtinSchreier extensions of degree p (p. 1837) \par  Nicole Raulf, Hecke operators and class numbers (p. 1838) \par  Reinier Bröker, Class invariants in a nonarchimedean setting (p. 1839) \par  Ronald van Luijk, Explicit computations on the Manin conjectures (p. 1841) \par  JeanMarc Couveignes, On the computation of the coefficients of a modular form, V: computational aspects (p. 1842)\par  Tim Dokchitser (joint with Vladimir Dokchitser), Computations in noncommutative Iwasawa theory of elliptic curves (p. 1846) \par  William A. Stein (joint with G. Grigorov and A. Jorza and S. Patrikis and C. Tarni\ctaP\uatraşcu), Computational verification of the Birch and SwinnertonDyer conjecture for individual elliptic curves (p. 1848) \par  H.M. Stark, The BrauerSiegel theorem (p. 1850)\par  Robert Carls, Theta null points of canonical lifts (p. 1853)\par  Bas Jansen, Mersenne primes and class field theory (p. 1855)\par  Mark van Hoeij (joint with Jürgen Klüners), Generating Subfields (p. 1858)  External Publication Status:  published  Document Type:  Article 
Communicated by:  nn  Affiliations:  MPI für Mathematik
 Identifiers:  LOCALID:111 ISSN:16608933  


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