Ranks of Elliptic Curves Cheat Sheet, Cheat Sheet of Probability and Statistics

Download Ranks of Elliptic Curves Cheat Sheet and more Cheat Sheet Probability and Statistics in PDF only on Docsity! RANKS “CHEAT SHEET” ALICE SILVERBERG This is a “cheat sheet”, which means that it consists of information packaged in a concise and efficient way so that it can easily be used as a quick reference. The topic is ranks of elliptic curves, mostly over Q. This is a slightly revised version of the handout I wrote as a supplement to my survey talk “Distributions of Ranks of Elliptic Curves” at MSRI’s Connections for Women: Arithmetic Statistics workshop in January of 2011. Updates might continue on my website [36]. I thank the organizers and participants of the MSRI workshop, and I thank the WIN2 organizers for the opportunity to publish this in the WIN2 Proceedings volume. 1. Mordell-Weil group, rank, and Tate-Shafarevich group An elliptic curve E over a field K is a smooth projective curve that has an affine equation of the form y2 + a1xy + a3y = x3 + a2x 2 + a4x+ a6, ai ∈ K. Discriminant: If E is y2 = x3 + Ax+B then ∆(E) := −16(4A3 + 27B2) 6= 0. Mordell-Weil Theorem. If K is finitely generated over the prime field, then the Mordell-Weil group E(K) is a finitely generated abelian group: E(K) ∼= Zrank(E(K)) ⊕ E(K)tors with rank(E(K)) ∈ Z≥0 and E(K)tors a finite abelian group. Tate-Shafarevich group (for E over a number field K): X(E/K) := ker [ H1(K,E)→ ∏ v H1(Kv, E) ] where H1(F,E) := H1(Gal(F̄ /F ), E(F̄ )), and the map is induced from the inclusions Gal(K̄v/Kv) ↪→ Gal(K̄/K). Tate-Shafarevich Conjecture. X(E/K) is finite. This work was supported by the National Science Foundation under grant CNS-0831004. I thank Manjul Bhargava, Noam Elkies, Joseph Silverman, Mark Watkins, and Yuri Zarhin for helpful comments. 1 2 ALICE SILVERBERG RANKS “CHEAT SHEET” 2. L-function, analytic rank, and BSD (Birch and Swinnerton-Dyer) Conjecture Fix E/Q. Below, p will denote primes. Replace E by an isomorphic curve with integer coefficients and |∆(E)| minimal and let ap := p+ 1−#E(Fp). Then L(E, s) := ∏ p-∆(E) (1− app−s + p1−2s)−1 ∏ p|∆(E) (1− app−s)−1. The product converges for s ∈ C with Re(s) > 3/2. Theorem 2.1 (Wiles et al. [43, 40, 5]). If E/Q, then L(E, s) has an analytic continu- ation to C and a functional equation relating L(E, s) and L(E, 2−s). More precisely, let NE denote the conductor of E and let Λ(E, s) := N s/2 E (2π)−sΓ(s)L(E, s). Then (1) Λ(E, s) = wEΛ(E, 2− s) with root number wE ∈ {±1}. Define rankan(E) := ords=1L(E, s). BSD I Conjecture. rank(E(Q)) = rankan(E). Theorem 2.2 (Kolyvagin, Gross-Zagier, Wiles et al. [27, 28, 20, 43, 40, 5]). If rankan(E) ≤ 1, then rank(E(Q)) = rankan(E) and X(E/Q) is finite. Theorem 2.3 (Bhargava-Shankar [4]). A positive proportion of elliptic curves E over Q satisfy rank(E(Q)) = rankan(E) = 0, and thus satisfy BSD I. Define Ω := ∫ E(R) dx |2y + a1x+ a3| ∈ R. For P = (x, y) ∈ E(Q), write x = u v with u, v ∈ Z in lowest terms, and define: Naive height: h(P ) := log max(|u|, |v|), ĥ(O) = 0. Néron-Tate height: ĥ(P ) := 1 2 lim n→∞ h(2nP ) 4n , ĥ(O) = 0. Define the Néron-Tate pairing, a bilinear form on E(Q), by 〈P,Q〉 := ĥ(P +Q)− ĥ(P )− ĥ(Q). With {P1, . . . , Pr} a Z-basis for E(Q)/E(Q)tors, define the regulator R := det(〈Pi, Pj〉)1≤i≤r,1≤j≤r. ALICE SILVERBERG RANKS “CHEAT SHEET” 5 6. Parity Parity Conjecture. rank(E) ≡ rankan(E) (mod 2). BSD I =⇒ Parity Conjecture. Theorem 6.1 (Monsky [32]). If E is an elliptic curve over Q and X(E/Q) is finite, then the Parity Conjecture holds for E. See [11] for results over other number fields. Equidistribution of Root Numbers Conjecture. The root numbers wE from (1) are 1 half the time and −1 half the time. Equidistribution of Root Numbers Conjecture + Parity Conjecture =⇒ the rank is even half the time and odd half the time. 7. Quadratic Twists Fix E/Q. If E : y2 = x3 + Ax + B and d ∈ Z 6=0, then the quadratic twist of E by d is Ed : y2 = x3 + Ad2x+Bd3. Let N∗(X) := #{squarefree d ∈ Z : |d| ≤ X, rank(Ed(Q)) is ∗}. Then N≥0(X) ∼ 12 π2 X. Trivial Bound. For each E/Q with all its 2-torsion defined over Q, there exists CE > 0 such that for all squarefree d ∈ Z with |d| > 2, rank(Ed(Q)) ≤ CE log|d| loglog|d| . Goldfeld Conjecture ([18]). The average rank of elliptic curves over Q in families of quadratic twists is 1 2 . Assuming the Parity and Goldfeld Conjectures, then: N0(X) ∼ N1(X) ∼ 6 π2 X, N≥2(X) = o(X). Theorem 7.1 (Heath-Brown [22]). Assuming BSD I and the Riemann Hypothesis for L-functions of elliptic curves, then the average rank of elliptic curves over Q in families of quadratic twists is ≤ 1.5. Theorem 7.2 (Heath-Brown [21]). The average rank of the quadratic twists Ed of E : y2 = x3 − x with d odd is ≤ 1.2645 . . .. See [44, 45, 46, 6] for related results. 6 ALICE SILVERBERG RANKS “CHEAT SHEET” Conjecture 7.3 (Conrey et al. [7]). N≥2,even(X) ∼ cEX 3/4(logX)bE with 4 possibili- ties for bE, depending on [Q(E[2]) : Q], and with 0.5 ≤ bE < 1.4. Theorem 7.4 (see [S5] for attributions). For some E/Q: N0(X) X, N1(X) X, N≥2(X) X 1 3 , N≥3(X) X 1 6 , N≥4(X)→∞. Assuming the Parity Conjecture: N≥1(X) ≥ 6 π2X for all sufficiently large X and N≥2(X) X 1 2 for all E/Q, while for some E/Q: N≥3(X) X 1 3 , N≥4(X) X 1 6 , and N≥5(X)→∞. 8. Selmer Groups and Selmer Ranks For E over a number field K, define the m-Selmer group: Sm(E/K) := ⋂ v res−1 v (κv(E(Kv)/mE(Kv))) ⊆ H1(K,E[m]) where the short exact sequence 0→ E[m]→ E(K̄) m−→ E(K̄)→ 0 induces 0→ E(K)/mE(K) κ−−−→ H1(K,E[m]) λ−−−→ H1(K,E(K̄))[m] → 0y yresv y 0→ E(Kv)/mE(Kv) κv−−−→ H1(Kv, E[m]) λv−−−→ H1(Kv, E(K̄v))[m] → 0 with κ(P ) := [σ 7→ σ(Q)−Q] where 2Q = P . This induces a short exact sequence of finite abelian groups killed by m: 0→ E(K)/mE(K) κ−→ Sm(E/K) λ−→ X(E/K)[m]→ 0. Define a “modified” p-Selmer rank: sp(E/K) := dimFpSp(E/K)− dimFpE(K)[p] ∈ Z≥0. Then sp(E/K) = rank(E(K)) + dimFpX(E/K)[p] ≥ rank(E(K)). If X(E/K)[p∞] is finite, then dimFpX(E/K)[p] is even, so sp(E/K) ≡ rank(E(K)) (mod 2). Define the p∞-Selmer group Sp∞(E/K) and p∞-Selmer rank sp∞(E/K): Sp∞(E/K) := lim−→Spn(E/K) ∼= (Qp/Zp)sp∞ (E/K) ⊕ (finite abelian p-group). There is a short exact sequence 0→ E(K)⊗Qp/Zp → Sp∞(E/K)→X(E/K)[p∞]→ 0. Since E(K)⊗Qp/Zp ∼= (Qp/Zp)rank(E(K)), if X(E/K)[p∞] is finite then sp∞(E/K) = rank(E(K)). ALICE SILVERBERG RANKS “CHEAT SHEET” 7 p-Selmer Parity Theorem (Monsky [32], Nekovář [33], Kim [25], Dokchitser-Dok- chitser [10]). For E/Q, sp∞(E/Q) ≡ rankan(E) (mod 2). Bhargava Conjecture. For each n > 1, and varying E/Q ordered by height, the average size of Sn(E/Q) is ∑ d|n d. For a proof when n = 2 see [3], for n = 3 see [4]; n = 4 and 5 are work in preparation by Bhargava & Shankar. Bhargava Conjecture for an infinite sequence of n + Parity Conjecture + Equidis- tribution of root numbers =⇒ Rank Distribution Conjecture. Theorem 8.1 (Mazur-Rubin [31] & Klagsbrun [26]). For E over a number field K with a real embedding, if E[2](K) = 0 and s ∈ Z≥0 then there are infinitely many quadratic twists Ed of E with s2(Ed/K) = s. For each prime p, let α(p) s := ηp s∏ j=1 p pj − 1 where ηp := ∞∏ j=0 1 1 + 1 pj = 1 2 ∞∏ j=0 ( 1− 1 p2j+1 ) . Then ∞∑ s=0 α(p) s = 1. As p→∞, α (p) 0 → 1 2 , α (p) 1 → 1 2 , and α(p) s → 0 for all s ≥ 2. α (p) s 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 s : p = 2 : p = 17 For example, when p = 2: α (2) 0 = η2 ≈ 0.21, α (2) 1 = 2η2 ≈ 0.42, α (2) 2 = 2η2 3 ≈ 0.28, α (2) 3 = 4η2 21 ≈ .08, α (2) 4 = 8η2 315 ≈ .01. Poonen-Rains Conjecture ([34]). Suppose s ∈ Z≥0, p is a prime, and K is a number field. Then the probability that an elliptic curve E over K has sp(E/K) = s is α (p) s . 10 ALICE SILVERBERG RANKS “CHEAT SHEET” [16] N. Elkies, j = 0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields, Number Theory Listserv posting, December 30, 2009, http://listserv.nodak.edu/cgi-bin/wa. exe?A2=ind0912&L=NMBRTHRY&F=&S=&P=14012 [17] N. D. Elkies, N. F. Rogers, Elliptic curves x3 + y3 = k of high rank, in Algorithmic number theory (ANTS-VI), ed. D. Buell, Lecture Notes in Comput. Sci. 3076, Springer, Berlin, 2004, 184–193. [18] D. Goldfeld, Conjectures on elliptic curves over quadratic fields, in Number theory, Carbon- dale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), ed. M. B. Nathanson, Lect. Notes in Math. 751, Springer, Berlin, 1979, 108–118. [19] G. Grigorov, A. Jorza, S. Patrikis, W. A. Stein, C. Tarnitǎ, Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves, Math. 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Math. 17 (2008), 105-125. [43] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Annals of Math. 141 (1995), 443–551. [44] G. Yu, Average size of 2-Selmer groups of elliptic curves. I, Trans. Amer. Math. Soc. 358 (2006), 1563-1584. [45] G. Yu, Average size of 2-Selmer groups of elliptic curves. II, Acta Arith. 117 (2005), 1-33. [46] G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (2005), 139-154. Department of Mathematics, University of California, Irvine, CA 92697 E-mail address: asilverb@math.uci.edu