Rational Points on Modular Elliptic Curves

Elliptic curves Modular forms Heegner points on $X_0(N)$ Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR points Integration on $\mathcal{H}_p\times\mathcal{H}$ Kolyvagin's theorem Bibliography.

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

[2]  Shou-wu Zhang.  Gross-Zagier Formula for GL(2), II , 2004 .

[3]  G. Shimura.  Algebraic Number Fields and Symplectic Discontinuous Groups , 1967 .

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

[5]  B. Mazur,et al.  -Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture , 1994 .

[6]  Ju. Manin,et al.  PARABOLIC POINTS AND ZETA-FUNCTIONS OF MODULAR CURVES , 1972 .

[7]  A. Atkin,et al.  Hecke operators on Γ0(m) , 1970 .

[8]  Fred Diamond,et al.  ON DEFORMATION RINGS AND HECKE RINGS , 1996 .

[9]  Jean-Pierre Serre.  Cohomologie des groupes discrets , 1971 .

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

[11]  Teitelbaum's exceptional zero conjecture in the anticyclotomic setting , 2002 .

[12]  Bas Edixhoven,et al.  On the Manin constants of modular elliptic curves , 1991 .

[13]  Henri Darmon,et al.  Heegner points on Mumford–Tate curves , 1996 .

[14]  Jean-François Mestre.  COURBES ELLIPTIQUES DE RANG >=12 SUR Q (T) , 1991 .

[15]  P. Green,et al.  Periods and Points Attached to Quadratic Algebras , 2004 .

[16]  H. Carayol.  Sur la mauvaise réduction des courbes de Shimura , 1986 .

[17]  Peter Green,et al.  Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence , 2002, Exp. Math..

[18]  Michael J. Razar,et al.  Modular forms for ₀() and Dirichlet series , 1977 .

[19]  A. Wiles,et al.  The Birch and Swinnerton-Dyer Conjecture , 2000 .

[20]  H. Darmon,et al.  A rigid analytic Gross-Zagier formula and arithmetic applications , 1997 .

[21]  Tanja Lange,et al.  Complex Multiplication , 2005, Handbook of Elliptic and Hyperelliptic Curve Cryptography.

[22]  Shou-wu Zhang.  GROSS-ZAGIER FORMULA FOR GL2 , 2001 .

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

[24]  J. Mennicke.  On Ihara's modular group , 1967 .

[25]  Fritz Grunewald,et al.  Groups Acting on Hyperbolic Space , 1998 .

[26]  M. Murty,et al.  Mean values of derivatives of modular L-series , 1991 .

[27]  David E. Rohrlich.  Galois theory, elliptic curves, and root numbers , 1996 .

[28]  Douglas Ulmer.  Elliptic curves with large rank over function fields , 2001, math/0109163.

[29]  Stephen S. Gelbart,et al.  Automorphic forms on Adele groups , 1975 .

[30]  A. Wiles,et al.  Ring-Theoretic Properties of Certain Hecke Algebras , 1995 .

[31]  伊原 康隆.  On congruence monodromy problems , 1968 .

[32]  E. Freitag.  Hilbert Modular Forms , 1990 .

[33]  Barry Mazur,et al.  Onp-adic analogues of the conjectures of Birch and Swinnerton-Dyer , 1986 .

[34]  Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives , 1990 .

[35]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[36]  H. Carayol,et al.  Uniformisation p-adique des courbes de Shimura : les théorèmes de Čerednik et de Drinfeld , 1991 .

[37]  L. Gerritzen,et al.  Schottky Groups and Mumford Curves , 1980 .

[38]  A. Wiles.  On ordinary λ-adic representations associated to modular forms , 1988 .

[39]  Fritz Grunewald,et al.  Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory , 1997 .

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

[41]  Henri Darmon,et al.  Integration on Hp × H and arithmetic applications , 2001 .

[42]  G. Shimura,et al.  On the Cohomology Groups Attached to Certain Vector Valued Differential Forms on the Product of the Upper Half Planes , 1963 .

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

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

[45]  H. Swinnerton-Dyer,et al.  Notes on elliptic curves. II. , 1963 .

[46]  Hervé Jacquet.  Automorphic Forms on GL(2) Part II , 1972 .

[47]  Ralph Greenberg,et al.  p-adicL-functions andp-adic periods of modular forms , 1993 .

[48]  Introduction to Abelian Varieties , 1993 .

[49]  David A. Cox.  Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication , 1989 .

[50]  織田 孝幸.  Periods of Hilbert modular surfaces , 1982 .

[51]  R. Greenberg.  On the Birch and Swinnerton-Dyer conjecture , 1983 .

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

[53]  Armand Brumer,et al.  The rank of elliptic curves , 1977 .

[54]  R. Taylor,et al.  On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises , 2001, Journal of the American Mathematical Society.

[55]  Henri Darmon,et al.  The p -Adic L -Functions of Modular Elliptic Curves , 2001 .

[56]  Shouwu Zhang.  Heights of Heegner points on Shimura curves , 2001 .

[57]  V. Kumar Murty,et al.  Non-vanishing of L-Functions and Applications , 1997 .

[58]  G. Shimura.  Construction of Class Fields and Zeta Functions of Algebraic Curves , 1967 .

[59]  Jean-Pierre Serre,et al.  Fermat ’ s Last Theorem , 2017 .

[60]  Fred Diamond,et al.  Modularity of Certain Potentially Barsotti-Tate Galois Representations , 1999 .

[61]  E. D. Shalit,et al.  p-adic periods and modular symbols of elliptic curves of prime conductor , 1995 .

[62]  D. Bump.  Automorphic Forms and Representations , 1998 .

[63]  Richard Taylor,et al.  l-adic representations associated to modular forms over imaginary quadratic fields , 1993 .

[64]  $p$-adic periods, $p$-adic $L$-functions, and the $p$-adic uniformization of Shimura curves , 1999 .

[65]  Don Zagier.  Modular points, modular curves, modular surfaces and modular forms , 1985 .

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

[67]  I V Čerednik,et al.  UNIFORMIZATION OF ALGEBRAIC CURVES BY DISCRETE ARITHMETIC SUBGROUPS OF $ PGL_2(k_w)$ WITH COMPACT QUOTIENTS , 1976 .

[68]  Jean-Pierre Serre,et al.  Le Probleme des Groupes de Congruence Pour SL 2 , 1970 .

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

[70]  H. Darmon,et al.  Periods of Hilbert modular forms and rational points on elliptic curves , 2003 .

[71]  Henri Darmon,et al.  Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization , 1998 .

[72]  Rigid-analytic L - transforms , 1984 .

[73]  C. Cornut.  Mazur’s conjecture on higher Heegner points , 2002 .

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

[75]  H. Carayol.  Formes modulaires et repr'esentations galoisiennes `a valeurs dans un anneau local complet , 1991 .

[76]  Vinayak Vatsal,et al.  Uniform distribution of Heegner points , 2002 .

[77]  A. Wiles.  Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?) , 1995 .

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

[79]  J. Waldspurger.  Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie , 1985 .

[80]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[81]  B. Mazur,et al.  Arithmetic of Weil curves , 1974 .

[82]  K. Ribet.  Fields of definition of abelian varieties with real multiplication , 1994 .

[83]  G. Shimura.  Class-fields and Automorphic Functions , 1964 .

[84]  J. Milne.  Elliptic Curves , 2020 .