Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute.

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

[2]  Michael Stoll,et al.  Implementing 2-descent for Jacobians of hyperelliptic curves , 2001 .

[3]  Qing Liu.  Conducteur et discriminant minimal de courbes de genre 2 , 1994 .

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

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

[6]  John Cremona,et al.  Visualizing Elements in the Shafarevich—Tate Group , 2000, Exp. Math..

[7]  Jacobiennes de certaines courbes de genre $2$ : torsion et simplicité , 1995 .

[8]  Bas Edixhoven.  L'action de l'algèbre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est «Eisenstein» , 1991 .

[9]  David R. Kohel,et al.  Component Groups of Quotients of J0(N) , 2000, ANTS.

[10]  Nigel P. Smart,et al.  Canonical heights on the jacobians of curves of genus 2 and the infinite descent , 1997 .

[11]  Gerhard Frey,et al.  Arithmetic of Modular Curves and Applications , 1997, Algorithmic Algebra and Number Theory.

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

[13]  P. Deligne,et al.  Groupes de monodromie en geometrie algebrique , 1972 .

[14]  Alain Robert,et al.  Introduction to modular forms , 1976 .

[15]  Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point , 1993 .

[16]  Siegfried Bosch,et al.  Rational points of the group of components¶of a Néron model , 1998, math/9804069.

[17]  J. L. Waldspurger,et al.  Sur les coefficients de Fourier des formes modulaires de poids demi-entier , 1981 .

[18]  Bjorn Poonen,et al.  Cycles of quadratic polynomials and rational points on a genus-$2$ curve , 1995 .

[19]  Xiangdong Wang.  2-dimensional simple factors ofJ0(N) , 1995 .

[20]  O. H. Lowry.  Academic press. , 1972, Analytical chemistry.

[21]  B. Mazur,et al.  Rational isogenies of prime degree , 1978 .

[22]  P. Steerenberg,et al.  Targeting pathophysiological rhythms: prednisone chronotherapy shows sustained efficacy in rheumatoid arthritis. , 2010, Annals of the rheumatic diseases.

[23]  Don Zagier,et al.  On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3 , 1985 .

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

[25]  Michael Stoll,et al.  On the height constant for curves of genus two, II , 1999 .

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

[27]  E. V. Flynn,et al.  Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Weddle's surface , 1996 .

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

[29]  Bjorn Poonen,et al.  The Cassels-Tate pairing on polarized abelian varieties , 1999 .

[30]  Bjorn Poonen,et al.  Explicit descent for Jacobians of cyclic coevers of the projective line. , 1997 .

[31]  K. Ueno,et al.  The complete classification of fibres in pencils of curves of genus two , 1973 .

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

[33]  Kenneth A. Ribet,et al.  On modular representations of $$(\bar Q/Q)$$ arising from modular forms , 1990 .

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

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

[36]  Some Abelian Varieties with Visible Shafarevich-Tate Groups , 1999 .

[37]  Edward F. Schaefer.  Computing a Selmer group of a Jacobian using functions on the curve , 1998 .

[38]  Ralph Duncan James,et al.  Proceedings of the International Congress of Mathematicians , 1975 .