aa r X i v : . [ m a t h . N T ] S e p Conjecture: of elliptic surfaces over Q have rank zero Alex [email protected]
Abstract
Based on an equation for the rank of an elliptic surface over Q which appears in the work of Nagao, Rosen,and Silverman, we conjecture that of elliptic surfaces have rank when ordered by the size of thecoefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. Wethen discuss how it would follow from either understanding of certain L -functions, or from understanding ofthe local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finitefields, and highlight some experimental evidence supporting it. Let E be an elliptic surface over Q . Fix a Weierstrass equation E : y = x + A ( T ) x + B ( T ) with A ( T ) , B ( T ) ∈ Z [ T ] and A ( T ) + 27 B ( T ) . Given an integer t such that A ( t ) + 27 B ( t ) = 0 , let E t denote the elliptic curve over Q with Weierstrass equation E t : y = x + A ( t ) x + B ( t ) . Given an elliptic curve E/ Q and a prime p , define a p ( E ) to be the p th coefficient of the L -function attached to E .It was conjectured by Nagao [12] thatrank E ( Q ( T )) = − lim X →∞ X X p
For any fixed positive integers m and n , we have lim M →∞ S m,n ( M ) X E∈S m,n ( M ) rank E ( Q ( T )) = 0 . In section 4 we discuss the main obstacles for proving conjecture 1.1 using the analytic framework in [13]. Insection 5 we outline an approach to conjecture 1.1 based on investigating statistics of ranks of elliptic surfacesover finite fields.
We thank Noam Elkies, Bjorn Poonen, and Michael Snarski for helpful discussions. This work was supported bygrant 550031 from the Simons Foundation.
For a fixed prime p , Birch [2] gave all moments of the distribution of a p ( E ) when E is chosen by selecting a(possibly singular) Weierstrass equation with coefficients in F p uniformly at random. Let ( A p,t ) be a sequenceof independent random variables indexed by a prime p and a positive integer t , with the property that, for every t , the random variable A p,t has the same distribution as the values a p ( E ) for E a Weierstrass equation in F p chosen uniformly at random, as in Birch’s work. We highlight the following property of the sequence ( A p,t ) : Proposition 3.1.
For any ε > , the series X + ε X p For every prime ℓ , and every pair of integers m , n > , lim ℓ →∞ ρ ℓ ( m , n ) = lim m →∞ ρ ℓ ( m, n ) = lim n →∞ ρ ℓ ( m , n ) = 12 . This conjecture, beyond being interesting in its own right, provides an approach for proving conjecture 1.1. See[11] for experimental evidence towards conjecture 5.1, where ρ ℓ ( m, n ) is estimated computationally for ℓ = 7 , n = 6 , , , , , and m ≤ n/ .Let N be a squarefree positive integer. Let S ( N ) ( M ) denote the subset of S ( M ) for which gcd ( N, A ( T ) +27 B ( T ) ) = 1 . Then, for any m , n , and M , {E ∈ S ( N ) ( M ) : E / F ℓ has positive rank for all ℓ | N } S ( N ) ( M ) = Y ℓ | N ρ ℓ ( m, n ) + O ( M − ) (9)by the Chinese remainder theorem, where E / F ℓ denotes the reduction mod ℓ of E / Q .If E / Q has positive rank, then either the reduction E / F ℓ has positive rank, or the kernel of the reduction E ( Q ( T )) → E ( F ℓ ( T )) is of finite index in E ( Q ( T )) . If this kernel was never of finite index then conjecture 1.1would follow from conjecture 5.1 (as well as much weaker versions of this conjecture), via observation (9). Thekernel of the reduction E ( Q ( T )) → E ( F ℓ ( T )) is of finite index in E ( Q ( T )) occasionally, but presumably not nearlyenough for this approach to fail. However, proving as much seems difficult. The generators of E ( Q ( T )) will mapto the identity of E ( F ℓ ( T )) if their denominators are divisible by ℓ , so one is naturally lead to investigate thedependence of the height of the generators of E ( Q ( T )) on the size of the coefficients of the Weierstrass model of E / Q . References [1] Sandro Bettin, Chantal David, and Christophe Delaunay. Non-isotrivial elliptic surfaces with non-zeroaverage root number. J. Number Theory , 191:1–84, 2018.[2] B. J. Birch. How the number of points of an elliptic curve over a fixed prime field varies. J. London Math.Soc. , 43:57–60, 1968.[3] Keith Conrad. Partial Euler products on the critical line. Canad. J. Math. , 57(2):267–297, 2005.[4] Rick Durrett. Probability—theory and examples , volume 49 of Cambridge Series in Statistical and Proba-bilistic Mathematics . Cambridge University Press, Cambridge, 2019.[5] Noam D. Elkies. Three lectures on elliptic surfaces and curves of high rank. arxiv:0709.2908 , 2007.[6] Stéfane Fermigier. Étude expérimentale du rang de familles de courbes elliptiques sur q . Experiment. Math. ,5(2):119–130, 1996. 57] Dorian Goldfeld. Sur les produits partiels eulériens attachés aux courbes elliptiques. C. R. Acad. Sci. ParisSér. I Math. , 294(14):471–474, 1982.[8] D. R. Heath-Brown. The average analytic rank of elliptic curves. Duke Math. J. , 122(3):591–623, 2004.[9] Nicholas M. Katz and Peter Sarnak. Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. (N.S.) ,36(1):1–26, 1999.[10] Wentang Kuo and M. Ram Murty. On a conjecture of Birch and Swinnerton-Dyer. Canad. J. Math. ,57(2):328–337, 2005.[11] Alan G. B. Lauder. Ranks of elliptic curves over function fields. LMS J. Comput. Math. , 11:172–212, 2008.[12] Koh-ichi Nagao. Q ( T ) -rank of elliptic curves and certain limit coming from the local points. ManuscriptaMath. , 92(1):13–32, 1997. With an appendix by Nobuhiko Ishida, Tsuneo Ishikawa and the author.[13] Michael Rosen and Joseph H. Silverman. On the rank of an elliptic surface. Invent. Math. , 133(1):43–67,1998.[14] Michael O. Rubinstein. Elliptic curves of high rank and the Riemann zeta function on the one line. Exp.Math. , 22(4):465–480, 2013.[15] Joseph H. Silverman. The average rank of an algebraic family of elliptic curves.