Densities of Ranks of Quadratic Twists of Elliptic Curves: A Survey, Papers of Differential Equations

Download Densities of Ranks of Quadratic Twists of Elliptic Curves: A Survey and more Papers Differential Equations in PDF only on Docsity! 1 The distribution of ranks in families of quadratic twists of elliptic curves A. Silverberg 1 Department of Mathematics, University of California at Irvine, Irvine, CA 92697 USA This paper gives a very brief survey, in the form of a table, of some results and conjectures about densities of ranks of elliptic curves over Q in families of quadratic twists. The table summarizes some of the knowledge to date on the density of quadratic twists of rank r or ≥ r, for some small values of r. We also give a commentary on the table. For a more extensive survey of ranks of elliptic curves over Q, see [RS02]. The author thanks Roger Heath-Brown for helpful comments. We first fix notation. If E is an elliptic curve of the form y2 = f(x), let E(d) denote dy2 = f(x), the quadratic twist of E by d. If E is an elliptic curve over Q, it suffices to consider d that are squarefree integers. Let N∗(X) = #{squarefree d ∈ Z : |d| ≤ X, rank(E(d)(Q)) is ∗}, where ∗ can be “2”, “odd”, “≥ 3”, etc. The table below gives a summary of some of the known results (and conjectures) to date on the rate of growth of Nr(X) and N≥r(X). It is well-known (see Theorem 333 of [HW79]) that N≥0(X) = #{squarefree d ∈ Z : |d| ≤ X} ∼ 2X ∏ p (1− 1 p2 ) = 2 ζ(2) X = 12 π2 X. The first part of the Birch and Swinnerton-Dyer Conjecture [BSD63/5] says that the rank of an elliptic curve E over Q should be equal to the analytic rank (i.e., the order of vanishing at s = 1 of the L-function of E over Q). In particular, the Birch and Swinnerton-Dyer Conjecture implies the Parity Conjecture, which says that the rank has the same 1 Supported by NSA grant MDA904-03-1-0033. 1 2 A. Silverberg parity as the analytic rank. The parity of the analytic rank can be read off from the sign in the functional equation for the L-function. Using the way that the sign varies as one twists the curve, one can show that the Parity Conjecture implies that, as |d| grows, the ranks of the quadratic twists by squarefree d ∈ Z of a fixed elliptic curve over Q are even half the time and odd half the time. (See the Nodd, Neven table entry.) In 1979, Goldfeld [G79] conjectured that for every fixed elliptic curve, the average rank of its quadratic twists is 12 . Goldfeld Conjecture If E is an elliptic curve over Q, then lim X→∞ ∑ squarefree d∈Z,|d|≤X rank(E (d)(Q)) #{squarefree d ∈ Z : |d| ≤ X} = 1 2 . If one assumes both the Parity Conjecture and Goldfeld’s Conjecture, then for every fixed elliptic curve, the ranks of the quadratic twists should be zero half the time and one half the time. (See the last N0 and N1 table entries.) In the table, “w/PC” means that the result is conditional on the Parity Conjecture, and “w/PC & GC” means that the result is condi- tional on both the Parity Conjecture and Goldfeld’s Conjecture. Fur- ther, “w/RMTC” means this is a conjecture made in [CKRS02] (see Conjecture 1 and (7)), which is based on Random Matrix Theory. All “” and “” entries in the table, and in the discussion below, should be read as “there is a positive constant, depending on E but not on X, such that for all sufficiently large X, we have . . . ”. Ono and Skinner [OS98], using results of Waldspurger and of Friedberg and Hoffstein, show that N0(X)  X/ log X for all elliptic curves. It was known earlier that N0(X)  X/ log X for certain elliptic curves (see for example [R74] for y2 = x3 − x). Work of Monsky [Mo90]), Birch [B69, B70], and Heegner [H52] shows that certain elliptic curves E have rank(E(p)) ≥ 1 for all primes p in certain congruence classes (for example, for y2 = x3 − x and all primes p ≡ 5 or 7 (mod 8)), and thus N≥1(X)  X/ log X (in fact, N1(X)  X/ log X). For the elliptic curve y2 = x3 − x, Heath-Brown (see Theorem 2 of [HB94]) showed that N0(X) > (.279)6X/π2, and, subject to the Parity Conjecture, N1(X) > (.559)6X/π2. The methods for finding lower bounds for N≥r(X) when r ≥ 2 involve finding twists E(g(T )) of E over Q(T ) of rank ≥ r, and specializing T . By Theorem C of [Si83], for all but finitely many t ∈ Q one has Ranks in families of quadratic twists 5 twist by a 2+2 3 of (1) with n = a2−1 3 , and (2) for a = 0 is the quadratic twist by 2 of the curve y2 = x3 − x. References [B69] B. J. Birch, Diophantine analysis and modular functions, in Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), 35–42, Oxford Univ. Press, London, 1969. [B70] B. J. Birch, Elliptic curves and modular functions, in Symposia Mathe- matica, Vol. IV (INDAM, Rome, 1968/69), 27–32 Academic Press, London, 1970. [BSD63/5] B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, II, J. Reine Angew. Math. 212 (1963), 7–25; 218 (1965), 79–108. [CFMS04] J. B. Conrey, D. Farmer, F. Mezzadri, N. C. Snaith, workshop an- nouncement: Special Week on Ranks of Elliptic Curves and Random Matrix Theory, 9–13 February, 2004, http://www.newton.cam.ac.uk/programmes/RMA/rmaw01.html . [CKRS02] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular L-functions, in Number theory for the millennium, I (Urbana, IL, 2000), A. K. Peters, Natick, MA, 2002, 301–315. [G79] D. Goldfeld, Conjectures on elliptic curves over quadratic fields, in Num- ber theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illi- nois Univ., Carbondale, Ill., 1979), M. B. Nathanson, ed., Lect. Notes in Math. 751, Springer, Berlin, 1979, 108–118. [GM91] F. Gouvêa, B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), 1–23. [HW79] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, fifth edition, The Clarendon Press, Oxford University Press, New York, 1979. [HB94] D. R. Heath-Brown, The size of Selmer groups for the congruent num- ber problem. II, Invent. Math. 118 (1994), 331–370. [H52] K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253. [HLP00] E. Howe, F. Leprévost, B. Poonen, Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 (2000), 315– 364. [Me92] J-F. Mestre, Rang de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris 314 (1992), 919–922. [Me98] J-F. Mestre, Rang de certaines familles de courbes elliptiques d’invariant donné, C. R. Acad. Sci. Paris 327 (1998), 763–764. [Me00] J-F. 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