On the vanishing of twists of L-functions of elliptic curves Chantal ...

On the vanishing of twists of L-functions ofelliptic curvesChantal DavidJack FearnleyHershy KisilevskyConcordia University

Twisted L-seriesLet E/Q be an elliptic curve with conductorN E and L-functionL E (s) =∞∑n=1a nn s.Functional Equation:(√ ) s NEΛ E (s) = Γ(s)L E (s) = w E Λ E (2 − s)2πThen, w E = −1 ⇒ L E (1) = 0.Let χ be a character of conductor f and letL E (s, χ) =Functional Equation:∞∑n=1a n χ(n)n s .Λ E (s, χ) =( √ ) s f NEΓ(s)L E (s, χ)2π= w E χ(N E ) τ(χ) 2Λ E (2 − s, χ).fwhere τ(χ) is the Gauss sum.1

Quadratic charactersLet χ d be a quadratic character of discriminantd. Then,χ d = χ d and L E (s, χ d ) = L Ed (s).The sign of the functional equation isw E χ d (−N E ) = ±1and L Ed (1) which vanishes for about half thediscriminants d.Conjecture (Birch and Swinnerton-Dyer)rank(E) = ord s=1 L E (s)We writeN ∗ (X) = # {|d| ≤ X : rank(E d ) is *}Conjecture (Goldfeld)N 0 (X) ∼ N 1 (X) ∼ 6 π 2 XN ≥2 (X) = o(X)In other words, the average rank is as small asit can be.2

Lower bounds and asymptotic for N ∗ (X)N ≥2 (X) ≫ X 1/2 (Gouvêa-Mazur)under the parity conjectureN ≥2 (X) ≫ X 1/7 log −2 (X) (Stewart-Top)N ≥2 (X) ≫ X 1/3 (Rubin-Silverberg)when E has rational 2-torsionN ≥3 (X) ≫ X 1/6 (Rubin-Silverberg)for some special EConjecture (Conrey, Keating, Rubinstein andSnaith) There are constants b E ≠ 0 and e Esuch thatwhen X → ∞.N ≥2,even (X) ∼ b E X 3/4 log e E X3

Higher order twistsLet k be an integer greater than 2, and considercharacters of order k.Questions:Infinitely many χ with L E (1, χ) = 0 ?Positive proportion of χ with L E (1, χ) = 0 ?Precise asymptotic ?Empirical vanishing for cubic twistsCurve Maximal Number of Number of(Torsion) conductor characters vanishingE11A (5) 2,023,513 320,795 1152E14A (6) 2,108,767 260,001 4347E15A (8) 399,979 51,890 807E32A (4) 300,217 47,577 117E36A (6) 283,051 36,718 346E37A (1) 279,211 41,991 559E37B (3) 364,723 54,830 1899E389A (1) 99,991 15,851 4084

Cubic twistsLet K/Q be a cyclic cubic field with charactergroup Ĝ = {1, χ, χ}. ThenL(E/K, s) = L E (s) L E (s, χ) L E (s, χ)and the vanishing of L E (s, χ) is related to therank of E over K.Theorem (Fearnley-Kisilevsky) If E has rational3-torsion, then# {K with f K ≤ X : rank E(K) > rank E(Q)}≫ X 1/2 .We denoteN(X) = {χ cubic with f χ ≤ X} ∼ c 3 XN E (X) = {χ ∈ N(X) with L E (s, χ) = 0} .Question:of N E (X)?What is the asymptotic behavior5

Random matricesPhilosophy for a family F of L-series with symmetrytype G(N):- Katz and Sarnak: statistics on the lowlyingzeroes of L f (s) in F should fit thoseof the eigenvalues of random matrices inG(N);- Keating and Snaith: statistics on the valuesof L f (s) in F should fit those of thecharacteristic polynomials of random matricesin G(N).Experimental and theoritical evidence:Montgomery, Odlyzko, Katz-Sarnak, Rudnick-Sarnak, Conrey-Ghost, Conrey-Farmer ...6

Families of L-functions and symmetry types• Unitary G(N) = U(N) :{ζ(1/2 + it), t ≥ 0}• Symplectic G(N) = Sp(2N) :{L(1/2, χ d ), χ d quadratic character}• Orthogonal G(N) = SO(2N) :{L(1/2, f), f ∈ S k (Γ 0 (N)) with k fixed}For the family of cubic twists, we have unitarysymmetries as- the signs of the functional equation areuniformly distributed on the unit circle (Heath-Brown and Patterson);- behavior of the moments of |L E (1, χ)|.7

MomentsLet k be a positive integer.2kth momentsM E (2k, X) = 1N(X)∑f χ ≤XWe consider the|L E (1, χ)| 2kwhere ∑ f χ ≤X runs over all cubic characters ofconductor f χ ≤ X.Conjecture (Keating and Snaith)M E (2k, X) ∼ a E (k) g k (2 log X) k2as X → ∞whereg k =k−1 ∏j=0j!(j + k)!and a E (k) is an arithmetical factor related tothe curve E.8

Empirical Moments1 2 3 4E11A E 2.878 22.02 274.3 4617.C 2.962 22.34 227.7 2288.E/C 0.972 0.985 1.205 2.017E14A E 2.196 11.76 104.6 1302.C 2.243 11.66 83.18 599.8E/C 0.979 1.008 1.257 2.171E15A E 2.609 15.86 149.4 1874.C 2.677 15.87 117.9 816.9E/C 0.974 1.000 1.266 2.294E32A E 1.928 7.863 49.23 407.2C 1.946 7.468 35.42 154.8E/C 0.991 1.052 1.389 2.630E36A E 1.792 6.491 35.34 253.6C 1.814 6.101 24.29 86.90E/C 0.988 1.063 1.454 2.919E37A E 3.196 29.50 441.3 8592.C 3.280 29.40 341.3 3547.E/C 0.974 1.003 1.292 2.421E37B E 1.656 6.060 36.15 311.3C 1.646 5.395 22.69 93.66E/C 1.006 1.123 1.593 3.32310

Random matricesKeating and Snaith: statistics on the values ofL f (s) in F should fit those of the characteristicpolynomials of random matrices in G(N).For each A ∈ G(N), letλ 1 = e iθ 1, . . . , λ N = e iθ Nbe the eigenvalues of A ordered such that0 ≤ θ 1 ≤ . . . ≤ θ N < 2πand letP A (λ) = det (A − λI)be the characteristic polynomial of A.Notice thatP A (1) = 0 ⇐⇒ λ 1 = e iθ 1 = 1 ⇐⇒ θ 1 = 011

The unitary groupJ. P. Keating, N. C. Snaith, Random matrixtheory and ζ(1/2 + it).Moments for |P A (1)| in U(N):M U (s, N) ==∫U(N) |P A(1)| s dHaarN∏j=1Γ(j)Γ(j + s)Γ 2 (j + s/2)is analytic for Re(s) > −1 with meromorphiccontinuation to the whole complex plane.Distribution function:P U (x, N) = 12πifor some c > −1.∫(c) M U(s, N)x −s−1 ds.For x smallP U (x, N) ∼ 1Γ(N)N∏j=1Γ(j) 2Γ 2 (j − 1/2) = R(N).12

When N → ∞R(N) → N 1/4 G 2 (1/2)where G is the Barnes G-function.Moments of L E (1, χ):M E (s, X) = 1N(X)∑f χ ≤X|L E (1, χ)| s .From the correspondance of Keating and Snaithwe conjecture that|L E (1, χ)| ↔ |P A (1)|,M E (s, X) = a E (s)M U (s, N)for the arithmetical factor a E (s) andN ∼ 2 log X(equating the densities of zeroes at a fixedheight).Probability distribution for small x:∫P E (X, x) = 12πiM E(s, X)x −s−1 ds(c)∼ a E (−1)R(N)13

Discretisation of L E (1, χ)Following Mazur-Tate-Teitelbaum,L algE (1, χ) = 2 f χ L E (1, χ)Ω τ(χ)= ∑χ(a)Λ(a, f χ )a mod fwhere Λ(a, f χ ) ∈ Z and Ω is a certain non-zerorational multiple of the real period Ω E . Clearly,L algE(1, χ) ∈ Z[ρ].Lemma|L algE (1, χ)| = ⎧⎪⎨⎪ ⎩n χ if w E = 1;√3 nχ if w E = −1;for some positive integer n χ .Then,|L E (1, χ)| = c En χ√fχ.14

Distribution of the integers n χCurve E11A E14A E15A E37BFactor 10 6 8 12Torsion 5 6 8 3n χ = 0 1152 4347 807 1899n χ = 1 1662 344 287 150n χ = 2 1117 440 229 136n χ = 3 2676 1379 414 1419n χ = 4 1328 336 660 171n χ = 5 1069 288 219 147n χ = 6 1711 2707 470 785n χ = 7 1827 390 327 188n χ = 8 1125 442 504 105n χ = 9 2578 2365 378 2853n χ = 10 631 293 174 88n χ = 11 1336 299 227 118n χ = 12 2183 2188 872 993n χ = 13 1607 365 274 149n χ = 14 1015 489 229 98n χ = 15 1625 1044 288 810Proposition Suppose that E is isogenous to acurve with rational 3-torsion. Then3 ν(f χ)−1 | n χ .15

LetB(f χ ) =⎧⎪⎨⎪⎩c E 3 ν(f χ)−1√fχif E(Q)[3] ≠ ∅;c E√fχotherwise.Then, Prob (|L E (1, χ)| = 0) is= Prob (|L E (1, χ)| < B(f χ ))=∫ B(fχ )0a E (−1/2) R(N) dx= a E (−1/2) R(N) B(f χ ).Summing the probabilitiesN E (X)∼⎧⎪⎨⎪⎩b E X 1/2 log 1/4 X E(Q)[3] = ∅;b E X 1/2 log 9/4 X E(Q)[3] ≠ ∅;16

Empirical results for E11A and E14A17