Mahler's Measure and Special Values of L-functions

REFERENCES

• Batut , C. , Bernardi , D. , Cohen , H. and Olivier , M. 1995 . User's Guide to Pari-GP 1.39 Université de Bordeaux. . [Batut, Bernardi, Cohen et Olivier 1995], See ftp://megrez.math.u-bordeaux.fr/pub/pari
   Google Scholar
• Beauville , A. 1982 . “Les families stables de courbes elliptiques sur P1 admettant quatre fibres singuliéres” . C. R. Acad. Sci. Paris Sér. I Math. , 294 ( 19 ) : 657 – 660 . [Beauville 1982]
   Google Scholar
• Beilinson , A. A. 1980 . “Higher regulators and values of L-functions of curves” . Funktsional. Anal, i Prilozhen. , 14 ( 2 ) : 46 – 47 . [Beilinson 1980], In Russian; translation in Functional Anal. Appl. 14(1980), 116–118
   Google Scholar
• Belinson , A. A. 1984 . “Higher regulators and values of L-functions”. ” . In [Current problems in mathematics] Edited by: Gamkre-lidze , R. V. and Nauki i Tekhniki , Itogi . vol. 24 , 181 – 238 . Moscow : Akad. Nauk SSSR Vs-esoyuz. Inst. Nauchn. i Tekhn. Inform. . [Beilinson 1984], In Russian; translation in J. Soviet Math. 30(1985), 2036–2070
   Google Scholar
• Bloch , S. and Grayson , D. 1983 . “K2 and L-functions of elliptic curves: computer calculations”. ” . In Applications of algebraic K-theory to algebraic geometry and number theory 79 – 88 . Boulder , CO [Bloch and Grayson 1986]
   Google Scholar
• Bloch , S. and Kato , K. 1990 . “L-functions and Tamagawa numbers of motives”. ” . In The Grothendieck Festschrift Edited by: Cartier , R . vol. 1 , 333 – 400 . Boston : Birkhäuser. . [Bloch and Kato 1990], Progr. Math. 86
   Google Scholar
• Boyd , D. W. 1977 . “Small Salem numbers” . Duke Math. J. , 44 ( 2 ) : 315 – 328 . [Boyd 1977]
   Web of Science ®Google Scholar
• Boyd , D. W. 1980 . “Reciprocal polynomials having small measure” . Math. Comp. , 35 ( 152 ) : 1361 – 1377 . [Boyd 1980]
   Web of Science ®Google Scholar
• Boyd , D. W. 1981 . “Kronecker's theorem and Lehmer's problem for polynomials in several variables” . J. Number Theory , 13 ( 1 ) : 116 – 121 . [Boyd 1981a]
   Web of Science ®Google Scholar
• Boyd , D. W. 1981 . “Speculations concerning the range of Mahler's measure” . Canad. Math. Bull. , 24 ( 4 ) : 453 – 469 . [Boyd 1981b]
   Web of Science ®Google Scholar
• Boyd , D. W. 1989 . “Reciprocal polynomials having small measure . W, Math. Comp. , 53 : 187 355 – 357 . S1 – S5 . [Boyd 1989]
   Google Scholar
• Boyd , D. W. 1992 . “Two sharp inequalities for the norm of a factor of a polynomial” . Mathematika , 39 ( 2 ) : 341 – 349 . [Boyd 1992]
   Google Scholar
• Browkin , J. 1989 . “Conjectures on the dilogarithm” . K-Theory , 3 ( 1 ) : 29 – 56 . [Browkin 1989]
   Google Scholar
• Buhler , J. P. , Gross , B. H. and Zagier , D. B. 1985 . “On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3” . Math. Comp. , 44 ( 170 ) : 473 – 481 . [Buhler et al. 1985]
   Web of Science ®Google Scholar
• Cassels , J. W. S. 1991 . Lectures on elliptic curves Cambridge : Cambridge University Press. . [Cassels 1991], London Math. Soc. Student Texts 24
   Google Scholar
• Cassels , J. W. S. and Flynn , E. V. 1996 . Prolegomena to a middlebrow arithmetic of curves of genus Vol. 2 , Cambridge : Cambridge University Press. . [Cassels and Flynn 1996], London Mathematical Society Lecture Note Series 230
   Google Scholar
• Cohen , H. 1993 . A course in computational algebraic number theory Berlin : Springer-Verlag. . [Cohen 1993], Graduate Texts in Mathematics 138
   Google Scholar
• Cooper , D. , Culler , M. , Gillet , H. , Long , D. D. and Shalen , P. B. 1994 . “Plane curves associated to character varieties of 3-manifolds” . Invent. Math. , 118 ( 1 ) : 47 – 84 . [Cooper et al. 1994]
   Web of Science ®Google Scholar
• Cremona , J. E. 1992 . Algorithms for modular elliptic curves Cambridge : Cambridge University Press. . [Cremona 1992], Additional tables available at ftp://euclid.ex.ac.uk/pub/cremona/data
   Google Scholar
• Deligne , P. 1977 . “Valeurs de fonctions L et périodes d'intégrales”. ” . In Automorphic forms, representations and L-functions 313 – 346 . Corvallis , OR [Deligne 1979]
   Google Scholar
• Deninger , C. 1997 . “Deligne periods of mixed motives, K-theory and the entropy of certain Zn-actions” . J. Amer. Math. Soc. , 10 ( 2 ) : 259 – 281 . [Deninger 1997]
   Web of Science ®Google Scholar
• Deninger , C. and Scholl , A. J. 1989 . “The Belinson conjectures”. ” . In L-functions and arithmetic 173 – 209 . Durham [Deninger and Scholl 1991], Coates, J. Taylor, M. J., London Math. Soc. Lecture Note Ser. 153, Cambridge Univ. Press, Cambridge, 1991
   Google Scholar
• Deninger , C. and Wingberg , K. 1988 . “On the Belinson conjectures for elliptic curves with complex multiplication”. ” . In Betlinson's conjectures on special values of L-functions Edited by: Rapoport , M. 249 – 272 . Boston , MA : Academic Press. . [Deninger and Wingberg 1988], Perspect. Math. 4
   Google Scholar
• Diamond , F. 1996 . “On deformation rings and Hecke rings” . Ann. of Math. (2) , 144 ( 1 ) : 137 – 166 . [Diamond 1996]
   Web of Science ®Google Scholar
• ‘Fisher , M. E. 1961 . “Statistical mechanics of dimers on a plane lattice” . Phys. Rev. (2) , 124 : 1664 – 672 . [Fisher 1961]
   Google Scholar
• Grayson , D. R. 1980 . “Dilogarithm computations for K3“. ” . In Algebraic K-theory 168 – 178 . Evanston [Grayson 1981], Friedlander, E.M., Stein M.R., Lecture Notes in Math. 854, Springer, Berlin, 1981
   Google Scholar
• Griffiths , P. and Harris , J. 1978 . Principles of algebraic geometry New York : Pure and Applied Mathematics, Wiley-Interscience. . [Griffiths and Harris 1978], Reprinted 1995
   Google Scholar
• Hamming , R. W. 1962 . Numerical methods for scientists and engineers New York : McGraw-Hill. . [Hamming 1962], International Series Pure Appl. Math., Reprinted by Dover, New York, 1986
   Google Scholar
• Hilbert , D. 1993 . Theory of algebraic invariants Cambridge : Cambridge University Press. . [Hilbert 1993], A translation by Laubenbacher, R.C. of the 1897 edition
   Google Scholar
• Kasteleyn , P. W. 1961 . “The statistics of dimers on a lattice” . Physica , 27 : 1209 – 1225 . [Kasteleyn 1961]
   Google Scholar
• Lawton , W. M. 1983 . “A problem of Boyd concerning geometric means of polynomials” . J. Number Theory , 16 ( 3 ) : 356 – 362 . [Lawton 1983]
   Web of Science ®Google Scholar
• Lehmer , D. H. 1933 . “Factorization of certain cyclotomic functions” . Ann. of Math. (2) , 34 : 461 – 479 . [Lehmer 1933]
   Google Scholar
• Lewin , L. 1981 . Polylogarithms and associated functions New York : North-Holland Publishing Co. . [Lewin 1981]
   Google Scholar
• Ligozat , G. 1975 . Courbes modulaires de genre 1 Paris : Soc. math. France. . [Ligozat 1975], Mémoires Soc. Math. France 43
   Google Scholar
• Lind , D. , Schmidt , K. and Ward , T. 1990 . “Mahler measure and entropy for commuting automorphisms of compact groups” . Invent. Math. , 101 ( 3 ) : 593 – 629 . [Lind et al. 1990]
   Web of Science ®Google Scholar
• Mahler , K. 1962 . “On some inequalities for polynomials in several variables” . J. London Math. Soc. , 37 : 341 – 344 . [Mahler 1962]
   Google Scholar
• Maillot , V. January 1997 . “Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables” January , [Maillot 1997], preprint, See http://www.dmi.ens.fr/users/vmaillot/torique.ps
   Google Scholar
• Mestre , J.-F. and Schappacher , N. 1989 . “Séries de Kronecker et fonctions L des puissances symétriques de courbes elliptiques sur Q”. ” . In Arithmetic algebraic geometry 209 – 245 . Texel [Mestre and Schappacher 1991], G. van der Geer et al., Progr. Math. 89, Birkhäuser, Boston, 1991
   Google Scholar
• Mossinghoff , M. J. 1995 . Algorithms for the determination of polynomials with small Mahler measure , Ph.D. thesis Austin : University of Texas. . [Mossinghoff 1995], See http://www.mathsci.appstate.edu/u/math/mjm/docs/mmthesis.ps.gz
   Google Scholar
• Mossinghoff , M. J. “Polynomials with small Mahler measure” [Mossinghoff ≥ 1998], See http://www.mathsci.appstate.edu/u/math/mjm/docs/smallmeas.dvi. To appear in Math. Comp.
   Google Scholar
• Nakada , H. 1990 . “The metrical theory of complex continued fractions” . Acta Arith. , 56 ( 4 ) : 279 – 289 . [Nakada 1990]
   Web of Science ®Google Scholar
• Nekov , J. 1991 . “Beilinson's conjectures”. ” . In Motives 537 – 570 . Seattle , WA [Nekovář 1994], Jannsen, U. et al., Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994
   Google Scholar
• Poonen , B. 1996 . “Computational aspects of curves of genus at least 2”. ” . In Algorithmic number theory 283 – 306 . Talence [Poonen 1996], Cohen, H., Lecture Notes in Comput. Sci. 1122, Springer, Berlin, 1996
   Google Scholar
• Ray , G. A. 1987 . “Relations between Mahler's measure and values of L-series” . Canad. J. Math. , 39 ( 3 ) : 694 – 732 . [Ray 1987]
   Web of Science ®Google Scholar
• Rodriguez Villegas , F. December 1996 . “Modular Mahler measures” December , [Rodriguez Villegas 1996], preprint, See http://www.ma.utexas/users/villegas/mahler-ll.dvi
   Google Scholar
• Rodríguez Villegas , F. July 1997 . “Modular Mahler measures” July , Pennsylvania State University. . [Rodríguez Villegas 1997], Slides of a talk given at the conference “Topics in Number Theory”, See http://www.ma.utexas/users/villegas/talk.dvi
   Google Scholar
• Rolshausen , K. 1996 . Eléments explicites dansK₂d'une courbe elliptique , Ph.D. thesis Strasbourg : Institut de Recherche Mathématique Avancée, Université Louis Pasteur. . [Rolshausen 1996], See http://galois.ustrasbg.fr/irma/publications/996/96004.dvi.gz
   Google Scholar
• Salmon , G. 1876 . Lessons introductory to the modern higher algebra, , 3rd ed. Cambridge , , UK : Hodges, Foster and Co. . [Salmon 1876]
   Google Scholar
• Sarnak , P. 1982 . “Spectral behavior of quasiperiodic potentials” . Comm. Math. Phys. , 84 ( 3 ) : 377 – 401 . [Sarnak 1982]
   Web of Science ®Google Scholar
• Schmidt , A. L. 1991 . “Ergodic theory of complex continued fractions”. ” . In Number theory with an emphasis on the Markoff spectrum 215 – 226 . Provo , UT [Schmidt 1993], Lecture Notes in Pure and Appl. Math. 147, Dekker, New York, 1993
   Google Scholar
• Schmidt , K. 1995 . Dynamical systems of algebraic origin Basel : Birkhäuser. . [Schmidt 1995], Progr. Math. 128
   Google Scholar
• Scholl , A. J. 1991 . “Height pairings and special values of L-functions”. ” . In Motives 571 – 598 . Seattle , WA [Scholl 1994], Jannsen, U. et al., Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994
   Google Scholar
• Silverman , J. H. 1986 . The arithmetic of elliptic curves New York : Springer. . [Silverman 1986], Graduate Texts in Math.
   Google Scholar
• Smyth , C. J. 1971 . “On the product of the conjugates outside the unit circle of an algebraic integer” . Bull. London Math. Soc. , 3 : 169 – 175 . [Smyth 1971]
   Google Scholar
• Smyth , C. J. 1981 . “A Kronecker-type theorem for complex polynomials in several variables” . Canad. Math. Bull. , 24 ( 4 ) : 447 – 452 . [Smyth 1981a], Addenda and errata in 25 (1982), 504
   Web of Science ®Google Scholar
• Smyth , C. J. 1981 . “On measures of polynomials in several variables” . Bull. Austral. Math. Soc. , 23 ( 1 ) : 49 – 63 . [Smyth 1981b]
   Web of Science ®Google Scholar
• Taylor , R. and Wiles , A. 1995 . “Ring-theoretic properties of certain Hecke algebras” . Ann. of Math. (2) , 141 ( 3 ) : 553 – 572 . [Taylor and Wiles 1995]
   Web of Science ®Google Scholar
• Thouless , D. J. 1990 . “Scaling for the discrete Mathieu equation” . Comm. Math. Phys. , 127 ( 1 ) : 187 – 193 . [Thouless 1990]
   Web of Science ®Google Scholar
• Wiles , A. 1995 . “Modular elliptic curves and Fermat's last theorem” . Ann. of Math. (2) , 141 ( 3 ) : 443 – 551 . [Wiles 1995]
   Web of Science ®Google Scholar
• Zagier , D. 1991 . “Special values and functional equations of polylogarithms”. ” . In Structural properties of polylogarithms Edited by: Lewin , L. 377 – 400 . Providence , RI : Amer. Math. Soc. . [Zagier 1991], Math. Surveys Monogr. 37
   Google Scholar