Dip. Matem. & Fisica. Notations. Università Roma Tre. Fields of characteristics 0. 1 Z is the ring of integers. 2 Q is the field of rational numbers

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1 On Never rimitive points for Elliptic curves Notations The 8 th International Conference on Science and Mathematical Education in Developing Countries Fields of characteristics 0 Z is the ring of integers University of Yangon Q is the field of rational numbers Myanmar 4 th -6 th December 05, R is the field of real numbers 4 C is the fields of complex numbers 5 For every prime p, F p = 0,,..., p } is the prime field; Z Q R C Z F p, n n(modp) surjective map Francesco appalardi Dipartimento di Matematica e Fisica & Roman Number Theory Association The Weierstraß Equation A Weierstraß equation E over a K (field) is an equation The definition of E(K) where A, B, C K E : y = x + Ax + Bx + C Let E/K elliptic curve and consider to be an extra point. Set E(K) = (x, y) K : y + = x + ax + b} } K } might be though as the vertical direction (line through points, Q E(K)) line through and Q if Q r,q : tangent line to E at if = Q projective or affine if #(r,q E(K)) #(r,q E(K)) = if tangent line, contact point is counted with multiplicity r,q : ax + b = 0 (vertical) r,q A Weierstraß equation is called elliptic curve if it is non singular! (i.e. 4A C A B 8ABC + 4B + 7C 0) r, E(K) =,, } We consider (most of times) simplified Weierstraß equation y = x + ax + b that are elliptic curves when 4a + 7b 0 4

2 History (from WIKIEDIA) xy y y x x x roperties of the operation + E Carl Gustav Jacob Jacobi (0//804 8/0/85) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. 0 0 Q - R R += +Q The addition law on E(K) has the following properties: (a) + E Q E(K) (b) + E = + E = (c) + E ( ) = (d) + E (Q + E R) = ( + E Q) + E R (e) + E Q = Q + E, Q E(K) E(K) E(K), Q, R E(K), Q E(K) Some of His Achievements: Theta and elliptic function Hamilton Jacobi Theory Inventor of determinants Jacobi Identity [A,[B, C]] + [B,[C, A]] + [C,[A, B]] = r,q E(K) =, Q, R} rr, E(K) =, R, R } +E Q := R r, E(K) =,, } (E(K),+ E) commutative group All group properties are easy except associative law (d) Geometric proof of associativity uses appo s := 5 6 Formulas for Addition on E Elliptic curves over C and over R E : y = x + Ax + B = (x, y ), = (x, y ) E(K) \ }, Addition Law E(C) = R/Z R/Z It is a compact Rieman surface of genus If x x x = x If = y 0 +E = λ = y = 0 +E = = y y x x λ = x + A y ν = yx yx x x, ν = x Ax B y + E = (λ x x, λ + λ(x + x ) ν) E(R) = R/Z R/Z ±} It is a circle or two circles 7 8

3 Elliptic curves over Q Elliptic curves over F p (Mordell ) If E/Q is an elliptic curve, then r N and G and finite abelian group G such that E(Q) = Z r G. E(F p) = Z/nZ Z/nkZ n, k N >0 In other words, E(Q) is finitely generated. (i.e. E(F p) is either cyclic (n = ) or the product of cyclic groups) (Mazur Torsion ) If Z/nZ denotes the cyclic group of order n, then the possible torsion subgroups (Weil) n p It is not known if r (the rank of E) is bounded. G = Tor(E(Q)) = Z/nZ with n 0 Z/Z Z/Z Z/nZ with n 4. (Hasse) Let E be an elliptic curve over the finite field F p. the order of E(F p) satisfies p + #E(F p) p. 9 0 From Elliptic curves over Q to Elliptic curves over F p Lang Trotter Conjecture for primitive points (Serre s Cyclicity Conjecture under the Riemann Hypothesis (976)) If E/Q then a, b Z s.t.: Let E/Q be an elliptic curve and assume GRH γ E, R 0 s.t. E : y = x + ax + b x #p x : Ē(Fp) is cyclic} γe, log x as x For all primes p 4a + 7b, we can consider the reduces curve Ē/Fp: Ē : y = x + āx + b. where ā = a mod p and b = b mod p. Given a certain property defined on finite groups, we consider Conjecture (Lang Trotter primitive points Conjecture (977)) Let E/Q, E(Q) with infinite order. α E, R 0 s.t. x #p x : Ē(Fp) = mod p } αe, log x as x π E(x,) = #p x : Ē(Fp) satisfies }. We are interested in studying the behaviour of π E(x,) and x for various properties. For most of the E s: ( ) If C = = , then γ l l(l ) (l+) E, = q C with q Q 0 ( ) If B = l l = , then α l l (l ) (l+) E, = q B with q Q 0 It is possible that α E, = 0 or that γ E, = 0 γ E, = 0 Z/Z Z/Z E(Q) if = kq, Q E(Q) and d = gcd(k,#tor(e(q)) >, then α E, = 0

4 Comparison between empirical data in Serre s Conjecture and Lang Trotter Conjecture Tests on Curves of rank, no torsion, Galois surjective l The notion of never primitive point π (x) = #p x : mod p = Ē(F p )} πcycl(x) = #p x : Ē(F p ) is cyclic} π label ( 5 ) B π ( 5 ) π( 5 ) π( 5 ) 7.a a a a a a a a π label cycl( 5 ) C π cycl( 5 ) π( 5 ) π( 5 ) 7.a a a a a a a a Let E/Q be an elliptic curve such that Z/Z Z/Z E(Q). A point E(Q) is called a never primitive if has infinite order for all l # Tor(E(Q)), is not the l th power of a rational point Q E(Q) mod p = Ē(Fp) for all p large enough Hence, given p, a primitive point modulo p satisfies mod p = Ē(Fp). A never primitive point never satisfies the above if Z/Z Z/Z E(Q), no point is ever primitive since Ē(Fp) is never cyclic we avoid such obvious cases we are interested in examples of curves with never primitive points 4 Twists with a Never rimitive point Other parametric families of curves with a never primitive point Given an elliptic curve E/Q with Weierstraß equation E : y = x + Ax + Bx + C ( - Jones, appalardi) Let s Q \ ±} and let E s : y = x 7(s ). and D Q, the twisted curve E D of E by D is s(s +, s(s 9)) E(Q) \ Tor(E(Q)) E D : y = x + ADx + BD x + CD. Tors(E s(q)) is trivial s is a never primitive point Let E/Q be an elliptic curve such that E(Q) contains a point of order. There D Z s.t. the twisted curve E D is such that E D(Q) contains a never primitive point. ( - Jones, appalardi) Let s Q \ 0, ±, ± }, and let Every elliptic curve with a point of order can be written in the form: E s : y = x s (s 8)x s (s 4 s + 4). E : y = x + ax + bx with a 4b 0 ( Set D = s(as + ) bs )., s Q except possibly when D is a perfect square, s(s +, 9s ) E(Q) \ Tor(E(Q)) Tors(E s(q)) is trivial D ( ( bs ), (as + + bs ) ( b s ) ) E D(Q) is never primitive. s is a never primitive point 5 6

5 Galois Action on the root sets Idea of the proof of The construction and its proof is based on the study of the Galois Action on the root-sets of : Lemma () Given E/Q, E(Q) and n N. and E[n] := Q C : nq = } := Q C : nq = } n Let E/Q be an elliptic curve, E(Q) \ Tor(E(Q)) and l be a prime such that is not an l-th power of a point in E(Q) Q(E[l]) = Q(ζ l, α /l ), Q( l ) R = Q,..., Ql} α Q Q(Q i) = Q((α i β) /l ), i =,..., l, β Q. Remark Note that E[n] is an abelian group Q( l ) = Q(ζl, α/l, β /l ) and is never primitive. The proofs of both s use the previous Lemma with l = E[n] = Z/nZ Z/nZ if R E[n] and S n, then R + S n n is a Z/nZ affine space. Gal(Q(E[n])/Q) Aut(E[n]) = GL(Z/nZ) Gal(Q( n )/Q) Aff( n ) = GL(Z/nZ) Z/nZ To verify the s one needs to compute the above Galois Groups for each elements of the family under consideration Lemma () Let s Z \ 0, ±, ±, ±} and consider E s, the elliptic curve in. Let α = s(s 9) and set ( ) T (s + 4sα + α ), 4 (α + sα + s α) E s(c). } E s[] :=,( s, ±4 s) ±T, ±T σ, ±T σ } where σ Gal(Q/Q( )) is such that σ( d) = e πi/ d d Q. Hence Q(E s[]) = Q(e πi/, s(s 9)). 7 8 Idea of the proof of Lemma () Let s Z \ 0, ±, ±, ±} and consider E s, the elliptic curve in. Set β = s (s + ), γ = s (s ) = αβ s(s + ), δ = (s ) (s + ) = α β s (s + ) and γ(x γ, y γ), β(x β, y β), δ(x δ, y δ) dove x β = s(s 8) + 4(s )β + 4β, y β = 4(s( s)( s) s(7 s)β (4 s)β ) x γ = s(s + 8) + 4(s + )γ + 4γ, y γ = 4(s( + s)( + s) + s(7 + s)γ + (4 + s)γ ) x δ = + (s + )δ + s s δ, y δ = s 9 + (s )δ + s + s δ. Q( γ) = Q(γ), Q( β) = Q(β), Q( δ) = Q(β) and } = β, σ β, σ β, γ, σ γ, σ γ, δ, σ δ, σ δ. Hence Q(E s[], ) = Q(eπi/, s(s 9), s (s )). The result follows from the previous lemmas 9