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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 17--23, 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 ``mini-series'' of five 1-hour morning talks given by Bas Edixhoven, Johan Bosman, Robin de Jong, and Jean-Marc Couveignes on the topic of computing the coefficients of modular forms. Let Δ = q \prodn≥1 (1-q^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, q-expansions, and Arakelov geometry \par \bullet Rational and integral points on curves and higher-dimensional 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 Jean-Marc 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 Rodriguez-Villegas, 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), Cohen-Lenstra heuristics for 4-ranks 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 Jean-Marc 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 Jean-Marc 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 Artin-Schreier extensions of degree p (p. 1837) \par -- Nicole Raulf, Hecke operators and class numbers (p. 1838) \par -- Reinier Bröker, Class invariants in a non-archimedean setting (p. 1839) \par -- Ronald van Luijk, Explicit computations on the Manin conjectures (p. 1841) \par -- Jean-Marc 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 non-commutative Iwasawa theory of elliptic curves (p. 1846) \par -- William A. Stein (joint with G. Grigorov and A. Jorza and S. Patrikis and C. Tarni\cta-P\uatraşcu), Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves (p. 1848) \par -- H.M. Stark, The Brauer-Siegel 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)
