Galois representations on torsion points of elliptic curves NATO ASI 2014 Arithmetic of Hyperelliptic Curves and Cryptography

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1 Galois reresentations on torsion oints of ellitic curves NATO ASI 04 Arithmetic of Hyerellitic Curves and Crytograhy Francesco Paalardi Ohrid, August 5 - Setember 5, 04 Lecture - Introduction Let /Q be an ellitic curve It is well known since the time of Jacobi that, for any field etension K/Q, the set (K) of rojective K rational oints has a natural grou structure Furthermore, if K/Q is finite, (K), with resect to this grou structure, is finitely generated For any integer n, we consider the kernel of the multilication-by-n ma, n] = P (Q) : np = } which is called the n torsion subgrou It is art of the classical theory the fact that if n is an odd integer and P (, y, ) (Q) is non zero, then P n] if and only if is a root of the n division olynomial ψ n (X) ZX] which are defined by recursive formulas and are searable Furthermore This imlies easily that if n is odd, then deg ψ n = n A similar argument allows to conclude that () holds also for n even We set n] = Z/nZ Z/nZ () ] = n N n] which is the torsion subgrou of (Q) In virtue of (), we have that So, there is a rofinite grou structure The absolute Galois grou Aut(n]) = GL (Z/nZ) Aut( ]) = GL (Ẑ) Ẑ = lim Z/nZ G Q := Gal(Q/Q) = σ : Q Q, field automorhism} is also a rofinite grou and if K is any Galois etension of Q, then Gal(K/Q) = G Q /σ G Q : σ K = id K } So G Q admits as quotient any ossible Galois Grou of Galois etensions of Q and it is the rojective limit of its finite quotients For every integer n, we consider the n torsion field Q(n]) obtained by adjoining to Q all coordinates of all non zero oints in n] Finally we use G(n) to denote the Galois grou Gal(Q(n])/Q) If P = (, y, ) n] and σ G(n), then σp := (σ, σy, ) n] This roerty and the fact that the oeration in (Q) is defined by Q rational functions, rovides us with an inclusion which can be etended to ρ n : G(n) Aut(n]) = GL (Z/nZ) ρ : G Q Aut( ]) = l rime Aut(l ]) = l rime GL (Z l ) l ] = m N l m ] and Z l denoted the ring of l adic integers The above reresentation is an object of study during these three lectures The main result of the Theory is

2 Theorem (Serre s Uniformity Theorem) If does not have comle multilication (ie the only homomorhism (Q) (Q) which are defined by rational mas are the multilication-by-n mas), then the inde of ρ n (G(n)) inside Aut(n]) is bounded by a constant that deends only on This statement has several striking consequences among which: Corollary If does not have comle multilication, then for all l large enough G(l) = Aut(l]) One of the central roblem of this theory is to establish elicit bounds for l for which the conclusion of the above Corollary holds It is believed that it holds for all l > 37 The grou homomorhism ρ l : G Q Aut(l ]) obtained by comosing ρ with the rojection on the l-th comonent, is actually a continuous homomorhism of toological grous and it is called l adic reresentation The reresentation ρ l is unramified at all rimes l in the sense that for such rimes ρ l I = Id Zl, for a fied rime number and a fied rime of Q over, one defines the inertia subgrou I G Q as the set of those elements of G Q such that σ() mod, Z Serre s Uniformity Theorem is equivalent to the conjunction of the following two statements: For all rimes l, ρ l (G Q ) is an oen subgrou with resect to the l adic toology, For all but finitely many rimes l, ρ l (G Q ) = Aut(l ]) An imortant tool in the study of the above reresentations is the Frobenius element In general, in a Galois etension K of Q, for an unramified rime, one defines the Frobenius element as any element in the conjugation class of the Galois Grou Gal(K/Q) which is determined by the lift of the Frobenius automorhism of the finite field O K /P obtained as a quotient of the ring of integers O by any rime ideal P over Sometimes one calls Artin symbol, the conjugation class itself and denotes it by K/Q ] In the case of the division fields Q(n]), the Artin symbol ] can be thought as a conjugation class of matrices in GL (Z/nZ) The characteristic olynomial det( T ) turns out not to deend on n in the sense that Q(n])/Q ( ]) Q(n])/Q det mod n, ( ]) Q(n])/Q tr a mod n a = #(F ) During the first lecture we will introduce the above notions and elain some of their roerties Lecture - Serre s Oen Maing Theorem and its alications In most of Lecture we will assume that has no comle multilication During this lecture we will introduce more tools and notions necessary for later alications Chebotarev Density Theorem If K/Q is a finite Galois etension and C Gal(K/Q) is a union if conjugation classes of G = Gal(K/Q), ] then the Chebotarev Density Theorem redicts that the density of the rimes such that the Artin symbol C equals #C #G The Chebotarev Density Theorem has also a quantitative versions Let ] K/Q π C/G () := # : C Then (see Serre 0] and Murty, Murty & Saradha 7]), assuming that the Dedekind zeta function of K satisfies the Generalized Riemann Hyothesis, π C/G () = #C dt ( #G log + O ) #C log(m#g) M is the roduct of rimes numbers that ramify in K/Q An analogue version, indeendent on the Generalized Riemann Hyothesis can be found in 0] We will aly it in the secial case when K = Q(n]) we think at the element of G as by non singular matrices For eamle } K/Q

3 In the case when C = id}, the condition ] Q(n])/Q = id} is equivalent to the roerty that n] Ē(F ) (F ) is the grou of F -rational oints on the reduced curve In the case when C = G tr=r = σ G : tr σ = t}, and l is a sufficiently large rime so that Gal(Q(l])/Q) = GL (F l ), then l (l ) if r = 0 # GL (F l ) tr=r = l(l l ) otherwise These eamles will be elaborated during Lecture 3 Classification of ossible subgrous of GL (F l ) that can aear as image of Galois Part of the work of Serre consists in classifying the ossible images of G(l) More recisely, Serre roved in 9] that ρ l (G Q ) contains a subgrou of one of the following tyes: slit half Cartan subgrou : A cyclic subgrou of of l which can be reresented as ( ) } a 0 : a F 0 l, half Borel subgrou : A solvable grou that can be reresented as ( ) } a 0 : a F 0 b l, b F l, 3 non slit half Cartan subgrou : A cyclic subgrou of of l Furthermore if ρ l (G Q ) GL (F l ), then one of the following haens: ρ l (G Q ) is either contained in a Cartan subgrou or a Borel subgrou (uer triangular matrices) of GL (F l ), ρ l (G Q ) is contained in the normalizer of a Cartan subgrou and it is not contained in the Cartan subgrou of GL (F l ) We will conclude with some elicit eamles 3 The Definition of Serre s Curve It is in general not easy to comute the image ρ(g Q ) GL (Ẑ) Actually, it was showed by Serre that ρ(g Q) is always contained in an inde subgrou of GL (Ẑ) Such subgrou is called the Serre s Subgrou H and it is defined as H = π m (H m ) π m : GL (Ẑ) GL (Z/mZ) is the natural rojection, m is the Serre number of defined as the least common multile, disc(q( ))], and if ε denotes the signature ma (ie ε : GL (Z/mZ) GL (Z/Z) = S 3 ±}), then ( )} H m = σ GL (Z/mZ) : ε(a) = det A An ellitic curve /Q is called a Serre curve if ρ(g Q ) = H These curves are quite common and will be considered in the third lecture 3 Lecture 3 - The Lang Trotter Conjectures The third lecture is devoted to reviews of some alications of l adic reresentations to number Theory and in articular to the Lang Trotter Conjectures 3

4 3 Lang Trotter for rimitive oints Artin made a celebrated conjecture concerning the density of rimes for which a given integer is a rimitive root In the first art of this lecture, we will discuss an analogous conjecture for ellitic curves Let be an ellitic curve defined over Q with P (Q) a Q-rational oint of infinite order P is called rimitive for a rime if the reduction P of P mod generates the entire grou (F ) of F -rational oints on the reduced curve We set π,p () = # : and P is rimitive for } In 976, Lang and Trotter in 5] conjecture an asymtotic formula for π,p () and consequently an eression for the density of rimes for which P is rimitive More recisely, they conjecture that δ,p = π,p () δ,p log #C P,n µ(n) # Gal(Q(n], n P )/Q) n= Q(n], n P ) is the etension of Q(n]) obtained with all the coordinates of the oints Q ( Q) such that nq = P and C P,n are suitable defined union of conjugacy classes in Gal(Q(n], n P )/Q) The heuristic argument is based on the Chebotarev Density Theorem The Lang Trotter conjecture for rimite oints is still not known in any case The Generalized Riemann Hyothesis allows to deduce some analogue conjectures for CM ellitic curves We will discuss some of the known results and in articular those due to Guta, Murty and Murty 3] 3 Serre s Cyclicity Conjecture J P Serre has formulated a conjecture with a similar flavor Let /Q be an ellitic curve and let π cyclic () = # : (F ) is cyclic} The conjecture ostulates the validity of the asymtotic formula: π cyclic () δ cyclic log δ cyclic = n= µ(n) Gal(Q(n])/Q) Serre himself alied the Chebotarev Density Theorem, in analogy with the Hooley s work for Artin s Conjecture, and roved this conjecture as a consequence of the Generalized Riemann Hyothesis Furthermore, if has no CM, is a rational multile of the quantity δ cyclic l ( ) (l l)(l ) We will discuss this result and several more due to Guta and Murty 4] and to A Cojocaru ] 33 Lang Trotter for fied trace of Frobenius In an earlier ublication 6], Lang and Trotter considered, for a fied ellitic curve /Q and an integer r, the function π r () = # : and #(F ) = + r} and they conjecture that if either r 0 or if has no CM, then π r () C,r log C,r is the so called Lang Trotter constant which is defined as follows: C,r = π lim K m # Gal Q(K m ])/Q) trace=r m # Gal Q(K m ])/Q) 4

5 K m a sequence of integers with the roerty that every integer divides K m when m is large enough For eamle k m = m! has this roerty In virtue of the Oen Maing Theorem, we know that there eists an integer N, called the torsion conductor such that C,r = N # Gal Q(N ])/Q) trace=r l# GL (F l ) tr=r π # Gal Q(N ])/Q) # GL (F l ) l N As an alication of the theory of l adic reresentations and of the Chebotarev density Theorem, assuming the Generalized Riemann Hyoythesis, Serre in 0] showed that π() r 7/8 (log ) / if r 0 3/4 if r = 0 These results were imroved for r 0 by Murty, Murty and Sharadha 7] that showed, assuming the Gereralized Riemann Hyothsis, that π r () 4/5 /(log ) 34 Average Lang Trotter For every integer a, b such that 4a 3 + 7b 0, we let a A, b B (a, b) : y = 3 + a + b We will conclude the lecture with a discussion of the following statement which aeared in ] Let r be an integer, A, B > For every c > 0 we have (( π(a,b) r 4AB () = C dt r + O t log t A + ) ) 3/ + 5/ B AB + log c C r = π l # GL (F l ) tr=r # GL (F l ) References ] Cojocaru, Alina Carmen, Cyclicity f CM llitic Curves modulo Trans of the AMS 355, 7, (003) ] David, Chantal; Paalardi, Francesco, Average Frobenius Distribution of llitic Curves, Internat Math Res Notices 4 (999) ] Guta, Rajiv; Murty M Ram, Primitive oints on ellitic curves, Comositio Mathematica 58, n (986), ] Guta, Rajiv; Murty M Ram, Cyclicity and generation of oints mod on ellitic curves, Inventiones mathematicae 0 (990) ] Lang, Serge; Trotter, Hale, Frobenius distributions in GL -etensions Lecture Notes in Mathematics, Vol 504 Sringer-Verlag, Berlin New York, 976 6] Lang, Serge; Trotter, Hale, Primitive oints on ellitic curves Bull Amer Math Soc 83 (977), no, ] Murty, M Ram; Murty, V Kumar; Saradha, N, Modular Forms and the Chebotarev Density Theorem, American Journal of Mathematics, 0, No (988), ] Serre, Jean-Pierre, Abelian l-adic reresentations and ellitic curves With the collaboration of Willem Kuyk and John Labute Second edition Advanced Book Classics Addison-Wesley Publishing Comany, Advanced Book Program, Redwood City, CA, 989 9] Serre, Jean-Pierre, Proriétés galoisiennes des oints d ordre fini des courbes ellitiques (French) Invent Math 5 (97), no 4, ] Serre, Jean-Pierre, Quelques alications du théorème de densité de Chebotarev (French) Inst Hautes Études Sci Publ Math No 54 (98),