Elliptic Curves

These are the notes for Math 679, University of Michigan, Winter 1996, exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at jmilne@umich.edu using \Math679" as the subject.

[1]  David Harari,et al.  Cohomologie galoisienne , 2020 .

[2]  G. Ballew,et al.  The Arithmetic of Elliptic Curves , 2020, Elliptic Curves.

[3]  Daniel Kříž.  Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture , 2020, 2002.04767.

[4]  Andrew V. Sutherland,et al.  Computing classical modular forms , 2020, 2002.04717.

[5]  C. Khare,et al.  Jean-Pierre Wintenberger , 2020 .

[6]  Jingbo Liu.  Legendre's theorem, Hasse invariant and Jacobi symbol , 2019 .

[7]  B. Poonen,et al.  A heuristic for boundedness of ranks of elliptic curves , 2016, Journal of the European Mathematical Society.

[8]  P. Roquette.  The Riemann Hypothesis in Characteristic p in Historical Perspective , 2018 .

[9]  David Schwein,et al.  Étale Cohomology , 2018, Translations of Mathematical Monographs.

[10]  A formula for the Jacobian of a genus one curve of arbitrary degree , 2015, Algebra & Number Theory.

[11]  J. Kujawa,et al.  Algebraic groups , 2019, Introduction to Arithmetic Groups.

[12]  J. Coates.  The Oldest Problem , 2017 .

[13]  H. Darmon.  Andrew Wiles’s Marvelous Proof , 2017 .

[14]  H. Lenstra,et al.  HEURISTICS FOR THE ARITHMETIC OF ELLIPTIC CURVES , 2017 .

[15]  J. Coates,et al.  The Conjecture of Birch and Swinnerton-Dyer , 1977, Open Problems in Mathematics.

[16]  F. Oort,et al.  Early history of the Riemann Hypothesis in positive characteristic , 2016 .

[17]  Anja Walter,et al.  Introduction To Elliptic Curves And Modular Forms , 2016 .

[18]  J. Milne.  The Riemann Hypothesis over Finite Fields: From Weil to the Present Day , 2015, 1509.00797.

[19]  George C. Ţurcaş.  The conjecture of Birch and Swinnerton-Dyer , 2015 .

[20]  A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture , 2014, 1407.1826.

[21]  J. Coates.  Lectures on the Birch-Swinnerton-Dyer Conjecture , 2013 .

[22]  Ezra Brown,et al.  Why Ellipses Are Not Elliptic Curves , 2012 .

[23]  Kristin E. Lauter,et al.  Modular polynomials via isogeny volcanoes , 2010, Math. Comput..

[24]  J. Milne.  Motives — Grothendieck’s Dream , 2012 .

[25]  Chandrashekhar Khare,et al.  Serre's Modularity Conjecture , 2011 .

[26]  Richard Taylor,et al.  A family of Calabi-Yau varieties and potential automorphy , 2010 .

[27]  J. Tate,et al.  DUALITY THEOREMS IN GALOIS COHOMOLOGY OVER NUMBER FIELDS , 2010 .

[28]  Chandrashekhar Khare,et al.  Serre’s modularity conjecture (II) , 2009 .

[29]  Fred Diamond,et al.  A First Course in Modular Forms , 2008 .

[30]  Jorma Jormakka,et al.  On the rank of elliptic curves , 2008, 0809.4091.

[31]  T. Browning,et al.  Local Fields , 2008 .

[32]  C. Khare.  Serre's modularity conjecture: The level one case , 2006 .

[33]  G. Everest,et al.  Rational points on elliptic curves , 2006, math/0606003.

[34]  K. Conrad,et al.  Partial Euler Products on the Critical Line , 2005, Canadian Journal of Mathematics.

[35]  M. Murty,et al.  On a Conjecture of Birch and Swinnerton-Dyer , 2005, Canadian Journal of Mathematics.

[36]  C. Wampler,et al.  Basic Algebraic Geometry , 2005 .

[37]  P. Sarnak.  Notes on the Generalized Ramanujan Conjectures , 2005 .

[38]  F. Murnaghan,et al.  LINEAR ALGEBRAIC GROUPS , 2005 .

[39]  B. Birch.  Heegner Points: the Beginnings , 2004 .

[40]  KANAYAMA Naoki,et al.  Jacobian Varieties , 2003 .

[41]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[42]  Charles J. Mozzochi.  The Fermat Diary , 2000 .

[43]  Henri Cohen,et al.  Advanced topics in computational number theory , 2000 .

[44]  J. Milne.  Descent for Shimura varieties. , 1997, alg-geom/9712031.

[45]  Gary Cornell,et al.  Modular Forms and Fermat's Last Theorem , 1997 .

[46]  Dino J. Lorenzini.  An Invitation to Arithmetic Geometry , 1996 .

[47]  N. Dummigan.  The Determinants of Certain Mordell-Weil Lattices , 1995 .

[48]  Beppo Levi and the arithmetic of elliptic curves. , 1995 .

[49]  J. Silverman.  Advanced Topics in the Arithmetic of Elliptic Curves , 1994 .

[50]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[51]  J. Cremona.  Algorithms for Modular Elliptic Curves , 1992 .

[52]  J. Cassels.  Lectures on elliptic curves , 1991 .

[53]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[54]  Joseph Oesterlé.  Empilements de sphères , 1990 .

[55]  J. Hoffstein,et al.  A nonvanishing theorem for derivatives of automorphic $L$-functions with applications to elliptic curves , 1989 .

[56]  V. Kolyvagin.  ON THE MORDELL-WEIL AND SHAFAREVICH-TATE GROUPS FOR WEIL ELLIPTIC CURVES , 1989 .

[57]  The work of kolyvagin on the arithmetic of elliptic curves , 1989 .

[58]  C. Hoffmann.  Algebraic curves , 1988 .

[59]  Karl Rubin,et al.  Tate-Shafarevich groups andL-functions of elliptic curves with complex multiplication , 1987 .

[60]  James S. Milne,et al.  Arithmetic Duality Theorems , 1987 .

[61]  N. Koblitz.  A Course in Number Theory and Cryptography , 1987 .

[62]  J. Cassels.  Mordell's finite basis theorem revisited , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[63]  Don Zagier,et al.  Heegner points and derivatives ofL-series , 1986 .

[64]  E. Grosswald.  Legendre’s Theorem , 1985 .

[65]  N. Koblitz.  Introduction to Elliptic Curves and Modular Forms , 1984 .

[66]  G. Faltings.  Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , 1983 .

[67]  M. Postnikov.  Lectures in algebraic topology , 1983 .

[68]  S. Lang.  Abelian varieties , 1983 .

[69]  J. Milne.  Comparison of the Brauer group with the Tate-Safarevic group , 1982 .

[70]  Barry Mazur,et al.  Modular curves and the eisenstein ideal , 1977 .

[71]  N. Koblitz.  p-adic Numbers, p-adic Analysis, and Zeta-Functions , 1977 .

[72]  J. Milne.  On a conjecture of Artin and Tate , 1975 .

[73]  N. Stephens,et al.  Congruence Properties of Congruent Numbers , 1975 .

[74]  H. Swinnerton-Dyer,et al.  Ellitpic curves and modular functions , 1975 .

[75]  J. Tate,et al.  Algorithm for determining the type of a singular fiber in an elliptic pencil , 1975 .

[76]  Roger North,et al.  The Mathematical Career of Pierre de Fermat , 1974, The Mathematical Gazette.

[77]  P. Deligne.  La conjecture de Weil. I , 1974 .

[78]  H. P. F. Swinnerton-Dyer,et al.  The Shafarevich-Tate conjecture for pencils of elliptic curves onK3 surfaces , 1973 .

[79]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[80]  Jean-Pierre Serre.  A Course in Arithmetic , 1973 .

[81]  B. Mazur.  Courbes Elliptiques et Symboles Modulaires , 1973 .

[82]  Jean-Pierre Serre.  Propriétés galoisiennes des points d'ordre fini des courbes elliptiques , 1971 .

[83]  Pierre Samuel,et al.  Algebraic theory of numbers , 1971 .

[84]  志村 五郎,et al.  Introduction to the arithmetic theory of automorphic functions , 1971 .

[85]  B. J. Birch,et al.  Weber's class invariants , 1969 .

[86]  J. Cassels,et al.  ABELIAN l -ADIC REPRESENTATIONS AND ELLIPTIC CURVES , 1969 .

[87]  W. Fulton,et al.  Algebraic Curves: An Introduction to Algebraic Geometry , 1969 .

[88]  James S. Milne.  The Tate-Šafarevič group of a constant abelian variety , 1968 .

[89]  André Weil,et al.  Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , 1967 .

[90]  J. Tate.  Endomorphisms of abelian varieties over finite fields , 1966 .

[91]  J. Cassels,et al.  Diophantine Equations with Special Reference To Elliptic Curves , 1966 .

[92]  J. Tate,et al.  On the conjectures of Birch and Swinnerton-Dyer and a geometric analog , 1966 .

[93]  J. Cassels,et al.  Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. , 1965 .

[94]  J. Tate,et al.  Algebraic cycles and poles of zeta functions , 1965 .

[95]  J. Cassels.  Arithmetic on curves of genus 1. VII. The dual exact sequence. , 1964 .

[96]  K. Kodaira.  On compact analytic surfaces II , 1963 .

[97]  H. Swinnerton-Dyer,et al.  Notes on elliptic curves. I. , 1963 .

[98]  J. W. S. Cassels,et al.  arithmetic on curves of genus 1 : iii. the tate‐Šafarevič and selmer groups , 1962 .

[99]  J. Cassels,et al.  Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermutung. , 1962 .

[100]  S. Lang,et al.  Principal Homogeneous Spaces Over Abelian Varieties , 1958 .

[101]  Par Goro Shimura.  Correspondances modulaires et les fonctions ζ de courbes algebriques , 1958 .

[102]  J. Tate,et al.  $WC$-groups over $p$-adic fields , 1958 .

[103]  Ernst S. Selmer,et al.  A conjecture concerning rational points on cubic curves , 1954 .

[104]  M. Eichler,et al.  Quaternäre quadratische Formen und die Riemannsche Vermutung fÜr die Kongruenzzetafunktion , 1954 .

[105]  Kurt Heegner.  Diophantische Analysis und Modulfunktionen , 1952 .

[106]  H. T. H. PIAGGIO,et al.  Foundations of Algebraic Geometry , 1948, Nature.

[107]  M. Deuring.  Die Typen der Multiplikatorenringe elliptischer Funktionenkörper , 1941 .

[108]  T. Nagell.  Sur les propriétés arithmétiques des cubiques planes du premier genre , 1929 .

[109]  A. Weil,et al.  L'arithmétique sur les courbes algébriques , 1929 .

[110]  L. Dickson.  History of the Theory of Numbers , 1924, Nature.

[111]  G. Fischer,et al.  Plane Algebraic Curves , 1921, Nature.

[112]  H. Weber,et al.  Theorie der algebraischen Functionen einer Veränderlichen. , 1882 .