Symmetric Squares of Elliptic Curves: Rational Points and Selmer Groups

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1 Symmetric Squares of Eiptic Curves: Rationa Points and Semer Groups Nei Dummigan CONTENTS 1. Introduction 2. The Symmetric Square L-Function 3. The Boch-Kato Formua: Fudge Factors 4. Goba Torsion 5. The Construction of Eements in a Semer Group 6. Exampes Acknowedgments References We consider the Boch-Kato conjecture appied to the symmetric square L-function of an eiptic curve over Q, ats =2. In particuar, we use a construction of eements of order in a generaised Shafarevich-Tate group, which works when E has a rationa point of infinite order and a rationa point of order. The existence of the atter paces us in a situation where the recent theorem of Diamond, Fach, and Guo does not appy, but we find that the numerica evidence is quite convincing. 1. INTRODUCTION 2000 AMS Subject Cassification: Primary 11G40, 14G10; Keywords: Eiptic curve, symmetric square L-function, Boch-Kato conjecture Let E be an eiptic curve defined over Q and et L(E,s) be the associated L-function. (It is now known that E is moduar, so that L(E,s) has an anaytic continuation to the whoe compex pane.) The conjecture of Birch and Swinnerton-Dyer predicts that the order of vanishing of L(E,s) ats = 1 is the rank of the group E(Q) of rationa points, and aso gives an interpretation of the eading term in the Tayor expansion in terms of various quantities, incuding the order of the Shafarevich-Tate group. The Boch-Kato conjecture [Boch and Kato 90] is the generaisation to arbitrary motives of the eading term part of the Birch and Swinnerton-Dyer conjecture. Let L(Sym 2 (E),s) be the symmetric square L-function attached to an eiptic curve E/Q. Fach [Fach 93] ooked at the Boch-Kato conjecture for L(Sym 2 (E),s)ats =2, and transated it into a formua invoving ony rationa numbers, such as the degree of a moduar parametrisation, and the order of a generaised Shafarevich-Tate group. In [Fach 92] he appied Koyvagin s technique for bounding Semer groups, to Sym 2 (E) ats =2. Theorem 1 of [Fach 93] appies the resut of [Fach 92] to prove the -part of the Boch-Kato formua for a primes >3such that E has good reduction at, does not divide the degree of the moduar parametrisation, and the representation Ga(Q/Q) GL 2 (F )arisingfrom c AKPeters,Ltd /2001 $ 0.50 per page Experimenta Mathematics 11:4, page 457

2 458 Experimenta Mathematics, Vo. 11 (2002), No. 4 the action of Ga(Q/Q) onthe-torsion points of E is surjective. (Under these conditions, the -part of the generaised Shafarevich-Tate group is trivia.) Modified -Semer groups associated to Sym 2 (E) at s = 2 are intimatey connected with the deformation theory of the above Gaois representation. This connection ies at the heart of Wies s approach to the Shimura-Taniyama-Wei conjecture, that every eiptic curve E/Q is moduar. Athough the work of Tayor and Wies [Wies 95], [Tayor and Wies 95] does not actuay prove the reevant cases of the Boch-Kato conjecture (the Semer groups are defined differenty), it is ceary cosey reated. Diamond, Fach and Guo [Diamond et a. 01a, Diamond et a. 01b] have now proved a genera resut on the Boch-Kato conjecture (at s = 1) for the adjoint L- function of a newform of weight k 2. In the case that the newform has trivia character, this is equivaent to the symmetric square L-function (at s = k). Appying their resut to L(Sym 2 (E), 2) proves the -part of the Boch-Kato conjecture for primes 5whereE has good reduction and the representation of Ga(Q/Q) one[] is irreducibe. (It aso proves it for =3ifE has good reduction at 3 and the representation of Ga(Q/Q( 3)) on E[3] is absoutey irreducibe.) In [Dummigan 01a], I ooked at L(Sym 2 (f),s), where f is a Hecke eigenform of eve one and weight k, concentrating on the weights k =12, 16, 18, 20, 22, 26, for which f has rationa Fourier coefficients. For k = 18, 22, 26 (when k/2 is odd), the Boch-Kato conjecture turns out to be especiay interesting at the point s =(k 1)+(k/2). It is possibe to construct eements of order in the reevant Semer groups, where is an Eisenstein prime, and these primes do appear, at s =(k 1) + (k/2), when the critica vaues of the L-function are cacuated. [Dummigan 01b] deas with something simiar for a Hibert moduar form. The point s =(k 1) + (k/2) coincides with the point s = k deat with in [Diamond et a. 01a] ony when k =2. (Ofcourse,ifk =2,therearenononzeromoduar forms of eve one, but the eve one restriction is no onger necessary when we have an eiptic curve to work with.) The construction of [Dummigan 01a] can be made to work in the case that E(Q) has both a point of order and a point of infinite order. A suitabe point of infinite order gives rise, via the descent map, to a nonzero eement of H 1 (Q,E[]). Thanks to the existence of the rationa point of order, E[] is isomorphic to a Gaois submodue of Sym 2 (E[]), and we get a nonzero eement c H 1 (Q, Sym 2 (E[])). The main probem we face is to show that c (or rather its image in another group) satisfies a the oca conditions required for it to beong to the appropriate Boch-Kato Semer group. To faciitate this, we assume that E has good reduction at, andimpose some technica conditions which are satisfied by most of theexampesweookat(seetheprecisestatementof Theorem 5.1). Cremona and Mazur [2000] ook, among a strong Wei eiptic curves over Q of conductor N 5500, at those with nontrivia Shafarevich-Tate group (according to the Birch and Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate group has predicted eements of prime order m. Inmostcases,theyfind another eiptic curve, often of the same conductor, whose m-torsion is Gaois-isomorphic to that of the first one, and which has rank two. The rationa points on the second eiptic curve produce casses in the common H 1 (Q,E[m]). They show [Cremona and Mazur 02] that these ie in theshafarevich-tategroupofthefirst curve, so rationa points on one curve expain eements of the Shafarevich- Tate group of the other curve. Ceary, the construction of the present paper is anaogous to this. Ironicay, the rationa point of order which aows the construction to proceed causes the representation of Ga(Q/Q) to be reducibe, so the resuts of [Fach 92] and [Diamond et a. 01a] do not appy to the -part of the Boch-Kato conjecture for L(Sym 2 (E), 2). Therefore, it is appropriate to consider numerica evidence. We must take > 3, since the 2- and 3-parts of the fudge factors occurring in this instance of the Boch- Kato formua are not we understood. Since there cannot be a rationa -torsion point for >7[Mazur78], the ony reevant are = 5 and = 7. We concentrate on the case = 5, which occurs with much greater frequency in ists of eiptic curves ordered by conductor. With the exception of the order of the (generaised) Shafarevich-Tate group, a the quantities appearing in the -part of the Boch-Kato formua appear in, or may be cacuated from, the eiptic curve data in Cremona s tabes, for a conductors N These tabes may be found at persona/jec/ftp/data/index.htm. We find that the data fit we with both the conjecture and our construction. 2. THE SYMMETRIC SQUARE L-FUNCTION Let E/Q be an eiptic curve of conductor N. Let be a prime number, et T = im E[ n ]bethe-adic Tate modue of E, and V = T Q. Let A = V /T =

3 Dummigan: Symmetric Squares of Eiptic Curves: Rationa Points and Semer Groups 459 n=1e[ n ]. The absoute Gaois group Ga(Q/Q) acts continuousy on a of these modues, in a natura way. As Gaois modues, V H 1 (E,Q )(1). Let V =Sym 2 (V ),T =Sym2 (T )anda = V /T = n=1 A [ n ], where A [ n ]:=Sym 2 (E[ n ]). Note that if a b E[ n+1 ] E[ n+1 ], then (a b) =a b = a b is identified with a b E[ n ] E[ n ]. As Gaois modues, V Sym 2 (H 1 (E,Q ))(2). Let A = A. The L-function of the motive Sym 2 h 1 (E) isdefined, for (s) > 2, by a Dirichet series given by an Euer product L(Sym 2 E,s)= P r (r s ) 1. r The product is over a primes r, and the poynomia P r (X) :=det(1 Frob 1 r X V Ir ), where I r is an inertia subgroup at r, Frob r is an arithmetic Frobenius eement of Ga(Q/Q) and is any prime different from r. These Euer factors may be determined expicity, see [Coates and Schmidt 87] and [Watkins 02]. Suffice it to say here that if r is a prime of good reduction of E, then P r (X) =(1 αr 2X)(1 β2 r X)(1 rx), α r and β r being the eigenvaues of Frob 1 r on T. Foowing [Boch and Kato 90] (Section 3), for p = (incuding p = ) et Hf 1 (Q p,v )=ker(h1 (D p,v ) H1 (I p,v )). The subscript f stands for finite part. D p is a decomposition subgroup at a prime above p, I p is the inertia subgroup, and the cohomoogy is for continuous cocyces and coboundaries. For p = et Hf 1 (Q,V )=ker(h 1 (D,V ) H 1 (D,V B cris )) (see Section 1 of [Boch and Kato 90] for definitions of Fontaine s ring B cris ). Let Hf 1(Q,V ) be the subspace of eements of H 1 (Q,V ) whose oca restrictions ie in Hf 1(Q p,v )foraprimesp. There is a natura exact sequence 0 T V π A 0. Let Hf 1(Q p,a ) = π Hf 1(Q p,v ). Define the -Semer group Hf 1(Q,A ) to be the subgroup of eements of H 1 (Q,A ) whose oca restrictions ie in H1 f (Q p,a )for a primes p. Note that the condition at p = is superfuous uness =2. Inthefuture,p wi aways be a finite prime. Define the Shafarevich-Tate group Hf 1 X = (Q,A ) π Hf 1(Q,V ). Note that, since s = 2 is a noncentra critica point for L(Sym 2 E,s), it is conjectured that Hf 1(Q,V )istrivia, so the -Semer group shoud be identified with the -part of the Shafarevich-Tate group. 3. THE BLOCH-KATO FORMULA: FUDGE FACTORS Let f(z) = n=1 a nq n (q = e 2πiz )bethenormaised (a 1 = 1) newform of weight 2 and eve N associated with the eiptic curve E. (For p N, thenumberof points of E(F p )is1+p a p.) Let φ : X 0 (N) E be a moduar parametrisation. Let c be the associated Manin constant, i.e., φ ω = c 2πif(z)dz, whereω is a Néron differentia on E, chosen so that c is positive. The symmetric square L-function L(Sym 2 E,s) is cosey reated to the Rankin convoution n=1 a2 nn s, and L(Sym 2 E,2) may be evauated using the Rankin- Seberg method [Rankin 39], [Shimura 76]. Carefu comparison of this with the conjectura formua of Boch and Kato [Boch and Kato 90] eads to formua (10) of [Fach 93]: deg φ Nc 2 r S ± r r ± 1 = #X #H 0 (Q,A )#H 0 (Q,A ( 1)) c r. r (3 1) Here, S ± are certain sets of primes of bad additive reduction. In the exampes we ook at ater, E is aways semistabe, so S ± are empty. For precise definitions, see [Fach 93]. Likewise, the c r are certain fudge factors. The foowing corrected Lemma 1 of [Fach 93] provides us with what we need to know about them. If j is the j-invariant of the eiptic curve E and r is a finite prime, et d r = ord r (j). Lemma 3.1. c r =1for a but finitey many r. Up to powers of 2 and 3, andpowersofr if r is a prime of bad reduction, we have 1 if d r 0 or r = ; c r = #E(Q r )[d r ] if d r > 0. The proof is essentiay identica to the proof of Lemma1in[Fach93]. Toappytheemma,weneedto be abe to cacuate #E(Q r )[d r ]inthecased r > 0. In the proofoflemma1of[fach93],the-part of E(Q r )[d r ] is isomorphic to Z (1)/d r Z /d r as an I r -modue, but not necessariy as a Ga(Q r /Q r )-modue, so the formua given there is not aways correct. We need to use the foowing proposition, due to Tate. A pubished proof may be found in [Siverman 94] (see Lemma 5.1, Theorem 5.3, and Coroary 5.4). Proposition 3.2. Let E be an eiptic curve defined over Q r,withord r (j) =: d r < 0. There is a unique q Q r such that j(e) = 1 q q +...

4 460 Experimenta Mathematics, Vo. 11 (2002), No. 4 (i) If E has spit mutipicative reduction, then there is an isomorphism of groups, respecting the actions of Ga(Q r /Q r ): E(Q r ) Q r /qz. (ii) If E has either nonspit mutipicative reduction or additive reduction, then there exists a quadratic extension L of Q r such that E(Q r ) {u L /q Z : N L/Qr (u) q Z /q 2Z }. In the case of nonspit mutipicative reduction, the extension is unramified. With itte troube one may deduce the foowing emma, which tes us the -part of c r in certain cases. Lemma 3.3. As above, suppose that E/Q r with d r > 0. Let = r be an odd prime with a exacty dividing d r. (i) If E has spit mutipicative reduction, then #E(Q r )[ a ] = a+min{b,c}, where b = gcd( a,r 1) and c = max{e a : (q/r d r ) is a e -power (mod r)}. (ii) If E has nonspit mutipicative reduction, then #E(Q r )[ a ]=gcd( a,r+1). Note that j is the product of 1/q and a 1-unit in Q r. Since 1-units in Q r are e -powers, q/r dr mayberepaced by jr dr in the above emma. It may not be repaced by or /r dr, where is the minima discriminant of E/Q, since this may differ from q n=1 (1 qn ) 24 by mutipication by the 12 th power of some r-adic number which is not an th power. This is iustrated by the exampe 506D1 in Section 6.3. Cremona s tabe of Hecke eigenvaues incudes the eigenvaues of the Atkin-Lehner invoutions W p for p N. If p is a prime of mutipicative reduction, that reduction is spit or nonspit according as the eigenvaue of W p is 1 or +1 respectivey. Intheexampesweookatater, is a prime of good reduction, so certainy = r when d r > 0. Aso, since N, thefactorn in (3 1) has trivia -part. By Coroary 4.2 of [Mazur 78], if N is square-free and E is a strong Wei curve within its isogeny cass (as in a the exampes of conductor 8000 that we examine ater), then the Manin constant c is at worst a power of 2, and has trivia -part for odd prime. Infact,ifN is aso odd, it is now known that c = 1 [Abbes and Umo 96]. 4. GLOBAL TORSION Next we ook at the factors appearing in the denominator of (3 1). Lemma 4.1. If is an odd prime and E[] is an irreducibe representation of Ga(Q/Q) over F, then #H 0 (Q,A ) and #H 0 (Q,A ( 1)) both have trivia -part. Proof: Via the Wei pairing, E[] is dua to E[]( 1), as a Gaois modue. Hence E[] E[]( 1) Hom F (E[],E[]) as modues for F [Ga(Q/Q)]. Symmetric tensors correspond to inear maps of zero trace. H 0 (Q,E[] E[]( 1)) corresponds to Hom F [Ga(Q/Q)](E[],E[]), which by Schur s Lemma consists of scaar mutipes of the identity. (Note that, since is odd, E[], as an odd, irreducibe representation of Ga(Q/Q), is absoutey irreducibe.) The ony such map having zero trace is the zero map. Hence the -part of #H 0 (Q,A ( 1)) is trivia. Since E[] isnot isomorphic to E[](1) as a Gaois modue, #H 0 (Q,A ) aso has trivia -part. The foowing comes from Proposition 21 of [Serre 72], and provides us with a practica way of appying the previous emma. Proposition 4.2. Suppose that N is square-free (i.e., E is semistabe). If >3 is a prime, then E[] is irreducibe uness a p 1+p (mod ) for a primes p N. If E is semistabe and a p 1+p (mod ) for a primes p N, then the composition factors of E[] are F and F (1) (by Proposition 21 of [Serre 72]). Using E[] E[]( 1) Hom F (E[],E[]), it is easy to prove the foowing emma. Lemma 4.3. Let be an odd prime, E/Q an eiptic curve. (i) If E[] F F (1), then #H 0 (Q,A ( 1)) and #H 0 (Q,A ). (ii) If E[] has a submodue, but not a quotient isomorphic to F (i.e., if E has a rationa point of order, but is not -isogenous to an eiptic curve with a rationa point of order ), then #H 0 (Q,A ),but the -part of #H 0 (Q,A ( 1)) is trivia. (iii) If E[] has a submodue, but not a quotient isomorphic to F (1) (i.e., if E has no rationa point of order, but is -isogenous to an eiptic curve

5 Dummigan: Symmetric Squares of Eiptic Curves: Rationa Points and Semer Groups 461 with a rationa point of order ), then the -part of #H 0 (Q,A ( 1)) is trivia. If >3, then so is the -part of #H 0 (Q,A ). Lemma 4.4. Let be an odd prime, E/Q an eiptic curve with a prime r of spit mutipicative reduction such that (r 1) and 2 ord r (j). Then the divisibiities in the above emma are exact. (For the case of #H 0 (Q,A ) in (i), we must aso assume that >3.) Proof: We just prove part (ii) to iustrate the idea. There is an obvious Ga(Q/Q)-equivariant map from E[] toe[](1) which maps the quotient of E[] isomorphic to F (1) to the submodue of E[](1) isomorphic to F (1), and this spans H 0 (Q,A []). We need to show it is not divisibe by. Suppose for a contradiction that θ : E[ 2 ] E[ 2 ](1) is a Ga(Q/Q)-equivariant map dividing it by. Choose a Z/ 2 Z-basis {v 1,v 2 } for E[ 2 ] such that under the Tate parametrisation of E(Q r ), v 1 corresponds to a primitive 2 -root of unity, and v 2 corresponds to a chosen q 1/2, q being the Tate parameter of E. Since (r 1), the Ga(Q r /Q r )-modues F and F (1) are nonisomorphic, and v 1 must generate the quotient of E[] isomorphictof (1), and maps under θ to a point of order in E(Q r )(1). Hence θ(v 1 ), untwisted, is a point of order 2 in E(Q r ), but the hypotheses precude the existence of such a point. Remark 4.5. If E has a rationa point of order and r is a prime of mutipicative reduction such that (r +1), then necessariy it is spit mutipicative reduction, by something ike (ii) of Lemma THE CONSTRUCTION OF ELEMENTS IN A SELMER GROUP Theorem 5.1. Let > 3 be a prime, E/Q an eiptic curve with a prime r of spit mutipicative reduction such that (r 1) and 2 ord r (j). Suppose that E has a rationa point Q of order. Suppose aso that E has good reduction at, and that for any prime p of bad reduction, E[ ](Q p ) has order. Then the -torsion subgroup of the Semer group Hf 1(Q,A ) has dimension greater than or equa to the rank of E(Q). Proof: There is a natura injection ψ from E(Q)/E(Q) into H 1 (Q,E[]). Since E(Q) contains the point Q of order, E[] has a submodue isomorphic to F, with quotient F (1). Hence A [] := Sym 2 (E[]) has a submodue isomorphic to E[], with quotient F (2). Since H 0 (Q, F (2)) is trivia, we get an injection θ from H 1 (Q,E[]) to H 1 (Q,A []). Given the assumptions we have made, as in the proof of Lemma 4.4, H 0 (Q,A )= H 0 (Q,A []). It foows that the image in H 1 (Q,A []) of H 0 (Q,A )/H0 (Q,A ) (i.e., the kerne of the natura map from H 1 (Q,A []) to H 1 (Q,A )) is one-dimensiona. Hence the image of E(Q)/E(Q) in the -torsion of H 1 (Q,A ) has dimension at east as big as the rank of E(Q). For P E(Q), et c = ψ(p ),c = θ(c) andetd be the image in H 1 (Q,A )ofc. Assume P is chosen in such a way that d = 0. We need to show that, for every finite prime p, res p (d ) Hf 1(Q p,a ). Since > 3 is a prime of good reduction, one may prove the oca condition at p = using Fontaine-Lafaie modues, as in the proof of Proposition 9.2 of [Dummigan 01a]. Next consider a prime p = of good reduction. As is we-known (Proposition 2.1 in Chapter 8 of [Siverman 86]), the cass c H 1 (Q,E[]) is unramified at p. Hence the cass d H 1 (Q,A ) is unramified at p. Since A is unramified at p (a prime of good reduction), Hf 1(Q p,a ) is equa to (not just contained in) the kerne of the map from H 1 (Q p,a )toh1 (I p,a ), where I p is an inertia subgroup at p (see ine 3 of p. 125 of [Fach 90]). Hence d Hf 1(Q p,a ). Finay, suppose that p = is a prime of bad reduction. It is easy to check (using Tate curves) that Hf 1(Q p,v )={0}, so we need to show that res p(d )=0. Let E 1 (Q p ) be the kerne of reduction (mod p). Then E 1 (Q p )/E 1 (Q p )istrivia,ande(q p )/E 1 (Q p )isfinite, so the cass of P in E(Q p )/E(Q p ) may be represented by some -power torsion point R E(Q p ). (What we have reay done here is just to confirm that the image of E(Q p )inh 1 (Q p,a )ishf 1(Q p,a )={0}.) By assumption, R must be a mutipe of Q, soitsuffices to consider the case R = Q. Let π n : E[ n ] E[ n ] Sym 2 E[ n ]=A [ n ]bethe projection map, π n (a b) = 1 (a b + b a). Choose 2 S E[ 2 ] such that S = Q. Since Q E(Q), we aso have S σ = Q for any σ Ga(Q/Q) (orga(q p /Q p )). Then res p (c ) H 1 (Q p,a []) is represented by the cocyce σ π 1 ((S σ S) Q). Viewing it as an eement of H 1 (Q p,a [ 2 ]) via the natura incusion, it is represented by the cocyce σ π 2 ((S σ S) S) andaso by σ π 2 ((S σ S) S σ ), since both S = Q and S σ = Q. Expanding out the eft-hand factor, adding these expressions, and expoiting the symmetry of π 2,we see that 2res p (c ) H 1 (Q p,a [ 2 ]) is represented by the cocyce σ π 2 (S σ S σ S S), which is the image in

6 462 Experimenta Mathematics, Vo. 11 (2002), No. 4 H 1 (Q p,a []) of [π 1 (Q Q)] H 0 (Q p,a )/H0 (Q p,a ). This impies that res p (d ) H 1 (Q p,a )iszero,asrequired. 6. EXAMPLES 6.1 Rank Zero with Rationa Point of Order 5 We use Cremona s tabes to examine the first 14 semistabe strong Wei curves (ordered by conductor N) with 5 N, rank zero and a rationa point of order =5. We use his abeing for the curves. There woud be too many to go a the way to N = deg is the 5-part of the degree of the moduar parametrisation. Using Lemmas 3.1 and 3.3, we compute C, defined to be the 5-part of the product c r of oca fudge factors, and using Lemmas 4.3 and 4.4, we compute (or bound) D, defined to be the 5-part of #H 0 (Q,A )#H 0 (Q,A ( 1)). The minima discriminant is and #5-X? is the order of the 5-part of X predicted by the Boch-Kato formua (3 1). Note that since any bad reduction is mutipicative, d r,which was defined to be ord r (j), is aso ord r ( ). Name C D deg #5-X? 11A B C B C B C B A B D B A E By (3 1), the exponents of 5 in C and #5-X? add up to the same as those in D and deg. There is no particuar reason to expect eements of order 5 in X, and in each case the predicted order of the 5-part of X is 1. The presence of rationa 5-torsion forces #E(Q r )[5] to be divisibe by 5. This produces powers of 5 dividing those c r such that 5 d r. These are beautifuy baanced by powers of 5 dividing deg(φ). See especiay the exampe 246B No Rationa Point of Order 5, Moduar Parametrisation Degree Not Divisibe by 5 This seems to appy to most curves, and is not a very interesting case. A random seection of ten is 14A1, 26A1, 38A1, 57B1, 66A1, 69A1, 82A1, 102C1, 122A1, and 138B1. For each of these there is no bad r such that 5 d r, hence C = 1. Aso, in each exampe there is no congruence a p 1+p (mod 5) for a p 5N (this may be checked using Cremona s tabe of Hecke eigenvaues), so by Lemmas 4.2 and 4.1, we find that D = 1. Hence, the 5-part of #X is predicted to be trivia in a these exampes Dividing Moduar Parametrisation Degree Here are the first nine for which 5 deg(φ), foowed by the first five for which 5 2 deg(φ). As before, we ook ony at semistabe, strong Wei curves with 5 N. Name #E(Q) tors. C D deg #5-X? 46A C A B A B D B C B A C D B In the exampe 506D1, which has spit mutipicative reduction at 11, we find that 11 5 j = / is not a 5 th power (mod 11), so that #E(Q 11 )[5] is ony 5, not 5 2 (reca Lemmas 3.1 and 3.3). Aso for 506D1, the reduction at 2 is nonspit, so #E(Q 2 )[5] is ony 1, not 5. In severa of the other exampes, there are primes r of nonspit mutipicative reduction such that 5 d r. In each of the above exampes, since 5 deg(φ), according to (3 1) either C or #X has to be divisibe by 5, and it seems that it is sometimes one, sometimes the other (and sometimes both). In severa cases, 5 divides some d r without there being any rationa point of order 5, though not aways with the resut that 5 divides C, since the reduction at r may be nonspit. Note aso that Proposition 2 of [Fach 93] states that if > 3 is a prime of good reduction such that the natura map from Ga(Q/Q) toaut(e[]) is surjective, and if d r, (for some bad r such that d r > 0) then deg(φ). He argues that since d r, the representation of Ga(Q/Q) one[] is unramified at r. ByworkofRibet,

7 Dummigan: Symmetric Squares of Eiptic Curves: Rationa Points and Semer Groups 463 the moduar form f is then congruent (mod ) tosome other moduar form, of eve dividing N/r. Hence is a congruence prime and divides deg(φ). The condition about the image of Ga(Q/Q) issatisfied for semistabe curves not satisfying a p 1+p (mod ) forap N, by Proposition 21 of [Serre 72]. In particuar, it is satisfied in a the above exampes except 66C1. Of course, the main theorem of [Diamond et a. 01a] appies to a these exampes (except 66C1). 6.4 Positive Rank with Rationa Point of Order 5 We examine a semistabe strong Wei curves of conductor N 8000, with 5 N, rank at east one and a rationa point of order 5. Amazingy, the very ast curve in Cremona s main tabe is an exampe; in fact, it is the exampe with the argest vaue of C in Tabe 1. At the end of the tabe are severa exampes of arger conductor (and rank two or three), suppied by M. Watkins. (An asterisk signifies rank two, two asterisks signify rank three, otherwise the rank is one.) He found these using the parametrisation of eiptic curves with 5-torsion. He has checked in each case that the curve is not 5-isogenous to another one with rationa 5-torsion. Though a the exampes of conductor N 8000 are definitey strong Wei curves, it is not certain that these exampes of higher conductor are. See Section 3 of [Watkins 02] for a discussion of this probem, and Section 1 for his approach to cacuating the moduar degree. If some of these curves are not strong Wei, the numbers in the ast two coumns of the tabe shoud be mutipied by the 5-part of c 2.For the curves of rank three, that is the anaytic rank, which we assume is equa to the rank of E(Q). In stark contrast to the exampes in Section 6 (rank zero with a rationa point of order 5), here the predicted order of X is aways divisibe by 5. This is in keeping with Theorem 5.1, which aways produces a candidate for an eement of order 5 in X, though in 5 out of the 26 rank-one exampes the technica conditions of the theorem are not satisfied. For 2651C1 there is no prime r of mutipicative reduction such that 5 (r 1). For each of 302A1, 1717C1, 2786D1 and 2869B1, there is a prime p (151, 101, 199 and 151 respectivey) of bad reduction such that E(Q p ) has a point of order 25. Looking at the exampes of higher rank R =2or3, it appears that the conditions of Theorem 5.1 are not merey a technica convenience. For most of these curves, the theorem produces 5 R eements of 5-torsion in the Semer group, and the predicted order of the 5-part of X is at east 5 R. But the curve 5302I1 and the curve of conductor fai the condition #E(Q 11 )[5] = 5, and Name C D deg #5-X? 123A A G E E B E F E C C D B D E R A J K C F E D I E F K TABLE 1. Positive rank with rationa point of order 5. the predicted order of the 5-part of X is ony 5. It is easy to see in these two cases that the proof of Theorem 5.1 does at east suppy 5 eements of 5-torsion in the Semer group. Watkins has aso provided me with the foowing exampes of curves of rank two with a rationa point of order 7.

8 464 Experimenta Mathematics, Vo. 11 (2002), No. 4 Conductor C D deg #7-X? ACKNOWLEDGMENTS I am gratefu to J. Cremona, for making his data pubic and for doube-checking one of the moduar parametrisation degrees for me, and to M. Fach for correcting his Lemma 1 in [Fach 93], and for informing me of [Diamond et a. 01a]. I am indebted to M. Watkins for hepfu comments and for the spectacuar exampes of arge conductor mentioned in the ast section. I thank aso the referees for numerous usefu comments and substantia corrections. I am especiay gratefu to one of them for finding a mistake which, in an earier version of the paper, produced an apparent paradox which I had been unabe to resove. REFERENCES [AbbesandUmo96] A.Abbes,S.Umo. Àproposdea conjecture de Manin pour es courbes eiptiques moduaires. Compositio Math. 103 (1996), [Boch and Kato 90] S. Boch and K. Kato. L-Functions and Tamagawa Numbers of Motives. In The Grothendieck Festschrift Voume I, pp , Progress in Mathematics 86. Boston, MA: Birkhäuser Boston, [Cremona and Mazur 00] J. E. Cremona and B. Mazur. Visuaizing Eements in the Shafarevich-Tate Group. Experiment. Math. 9 (2000), [Cremona and Mazur 02] J. E. Cremona and B. Mazur. Visibe Evidence for the Birch and Swinnerton- Dyer Conjecture for Moduar Abeian Varieties of Rank Zero. Appendix to A. Agashe and W. Stein. Preprint. Avaiabe from Word Wide Web: ( [Coates and Schmidt 87] J. Coates and C. G. Schmidt. Iwasawa Theory for the Symmetric Square of an Eiptic Curve. J. Reine Angew. Math. 375/376 (1987), [Diamond et a. 01a] F. Diamond, M. Fach, and L. Guo. On the Boch-Kato Conjecture for Adjoint Motives of Moduar Forms. Math.Res.Lett.8 (2001), [Diamond et a. 01b] F. Diamond, M. Fach, and L. Guo. Adjoint Motives of Moduar Forms and the Tamagawa Number Conjecture. Preprint. Avaiabe from Word Wide Web: ( iguo/gpapers.htm), [Dummigan 01a] N. Dummigan, Symmetric Square L- Functions and Shafarevich-Tate Groups. Experiment. Math. 10 (2001), [Dummigan 01b] N. Dummigan. Vaues of a Hibert Moduar Symmetric Square L-Function. In Preparation. [Fach 93] M. Fach. On the Degree of Moduar Parametrisations. In Séminaire de Théorie des Nombres, Paris , edited by S. David, pp , Progress in Mathematics 116. Boston, MA: Birkhäuser Boston, [Fach 92] M. Fach. A Finiteness Theorem for the Symmetric Square of an Eiptic Curve. Invent. Math. 109 (1992), [Fach 90] M. Fach. A Generaisation of the Casses-Tate Pairing. J. reine angew. Math. 412 (1990), [Mazur 78] B. Mazur. Rationa Isogenies of Prime Degree. Invent. Math. 44 (1978), [Rankin 39] R. A. Rankin. Contributions to the Theory of Ramanujan s Function τ(n) and Simiar Arithmetica Functions. Proc. Cambridge Phios. Soc. 35 (1939), [Shimura 76] G. Shimura. The Specia Vaues of the Zeta Functions Associated with Cusp Forms. Comm. Pure App. Math. 29 (1976), [Serre 72] J.-P. Serre. Propriétés gaoisiennes des points d ordre fini des courbes eiptiques. Invent. Math. 15 (1972), [Siverman 86] J. H. Siverman. The Arithmetic of Eiptic Curves, GTM 106. New York: Springer-Verag, [Siverman 94] J. H. Siverman. Advanced Topics in the Arithmetic of Eiptic Curves, GTM 151. New York: Springer- Verag, [Tayor and Wies 95] R. Tayor and A. Wies. Ring- Theoretic Properties of Certain Hecke Agebras. Ann. Math. 141 (1995), [Watkins 02] M. Watkins. Computing the Moduar Degree of an Eiptic Curve. Preprint. Avaiabe from Word Wide Web: ( [Wies 95] A. Wies. Moduar Eiptic Curves and Fermat s Last Theorem. Ann. Math. 141 (1995), Nei Dummigan, University of Sheffied, Department of Pure Mathematics, Hicks Buiding, Hounsfied Road, Sheffied, S3 7RH, UK (n.p.dummigan@shef.ac.uk) Received September 27, 2001; accepted in revised form Apri 2, 2002.