Elliptic curves and lower bounds for class numbers
aa r X i v : . [ m a t h . N T ] A p r ELLIPTIC CURVES AND LOWER BOUNDS FOR CLASS NUMBERS
MICHAEL GRIFFIN AND KEN ONO
Abstract.
Ideal class pairings map the rational points of rank r ≥ E/ Q to theideal class groups CL( − D ) of certain imaginary quadratic fields. These pairings imply that h ( − D ) ≥
12 ( c ( E ) − ε )(log D ) r for sufficiently large discriminants − D in certain families, where c ( E ) is a natural constant. Thesebounds are effective, and they offer improvements to known lower bounds for many discriminants. Introduction and statement of results
Estimating class numbers h ( − D ) of imaginary quadratic fields Q ( √− D ) , which also countequivalence classes of integral positive definite binary quadratic forms of fundamental discrimi-nant − D , is one of the oldest problems in number theory. Gauss conjectured that h ( − D ) → + ∞ as D → ∞ . Heilbronn [12] confirmed this in the 1930s, and Siegel [17] shortly thereafter obtaineda nearly definitive solution. For ε > , he proved that there are constants c ( ε ) , c ( ε ) > c ( ε ) D − ε ≤ h ( − D ) ≤ c ( ε ) D + ε . Siegel’s lower bound is inexplicit; there is no known formula for c ( ε ). Therefore, his work couldnot even determine the class number 1 fundamental discriminants. Baker, Heegner, and Stark [1,11, 21] later famously determined this list: − D ∈ {− , − , − , − , − , − , − , − , − } . Thanks to work of Goldfeld, Gross and Zagier, modified and ingeniously optimized by Watkins,such lists are now known [22] for h ( − D ) ≤ E/ Q with analytic rank r ≥
3. Groundbreaking work by Gross and Zagier [9, 10] on the Birchand Swinnerton-Dyer Conjecture confirmed the existence of such curves ten years later, resultingin the effective lower bound [16](1.1) h ( − D ) > D ) Y p | D prime p = D (cid:18) − [2 √ p ] p + 1 (cid:19) . Here we obtain effective lower bounds for certain discriminants using elliptic curves in a com-pletely different way. Although we do not improve on (1.1) for all − D , the point of this noteis to highlight and make use of an interesting interrelationship between class groups and ellipticcurves that often leads to improved class number lower bounds. We employ ideal class pairings ,maps of the form E ( Q ) × E − D ( Q ) → CL( − D ) , where E − D is the − D -quadratic twist of E . Such maps were previously considered by Buell,Call, and Soleng [2, 3, 20]. The idea is quite natural. Throughout, suppose that E/ Q is given by(1.2) E : y = x + a x + a , where a , a ∈ Z , with j -invariant j ( E ) and discriminant ∆( E ), and suppose that E ( Q ) has rank r = r ( E ) ≥
1. If − D < E − D / Q be its quadratic twist (1.3) E − D : − D · (cid:16) y (cid:17) = x + a x + a . Furthermore, suppose that Q D = ( u, v ) ∈ E − D ( Q ) is an integer point, where v = 0, with v evenif − D is odd. Theorem 2.1 gives an explicit construction of the pairing. Therefore, the numberof SL ( Z )-inequivalent forms obtained by pairing points in E ( Q ) with Q D gives a lower boundfor h ( − D ).We derive lower bounds in terms of Ω r := π r / Γ (cid:0) r + 1 (cid:1) , the volume of the R r -unit ball, theregulator R Q ( E ), the diameter d ( E ) (see (3.3)), and the torsion subgroup E tor ( Q ) . We define(1.4) c ( E ) := | E tor ( Q ) | p R Q ( E ) · Ω r , and, in terms of the usual logarithmic heights (see Section 3) of j ( E ) and ∆( E ) , we define(1.5) δ ( E ) := 18 h W ( j ( E )) + 112 h W (∆( E )) + 53 . Finally, to facilitate the comparison with log( D ), we define(1.6) T E ( D, Q D ) := log (cid:18) D (1 + | u | ) (cid:19) − δ ( E ) . Theorem 1.1.
Assuming the hypotheses above, if (1 + | u | ) exp(4 δ ( E ) + d ( E )) < D ≤ (1+ | u | ) u v ,then h ( − D ) ≥ c ( E )2 · (cid:16) T E ( D, Q D ) r − r p d ( E ) · T E ( D, Q D ) r − (cid:17) . Although the hypotheses for Theorem 1.1 are satisfied by many E/ Q for each − D , we seekchoices that improve (1.1). In view of (1.6), we require choices where the height of Q D is smalland where c ( E ) is not too small. For each E , we offer a natural family of discriminants, those ofthe form − D E ( t ) := − t + a t − a ) , with t ∈ Z . In these cases, we choose Q − D E ( t ) := ( − t, Theorem 1.2. If ε > , then there is an effectively computable constant N ( E, ε ) < such thatfor negative fundamental discriminant of the form − D E ( t ) , where t ∈ Z and − D E ( t ) < N ( E, ε ) , we have h ( − D E ( t )) ≥ (cid:18) c ( E ) √ r − ε (cid:19) · log( D E ( t )) r . Remark.
Theorems 1.1 and 1.2 are stated under the assumption that the discriminants arefundamental for reasons of aesthetics. There is a straightforward modification that offers lowerbounds for the class number of the corresponding imaginary quadratic field when − D (resp. − D E ( t ) ) is not fundamental. We note that Theorem 2.1, which is the source of the lower bounds,holds for all discriminants. Namely, the proof gives lower bounds for the Hurwitz-Kronecker class For reasons which will become apparent later, we choose this nonstandard normalization. Goldfeld conjectures [8] that asymptotically half of the E − D have rank 1, and so such points are plentiful.The integrality of Q D is easily satisfied by changing models of E and E − D by clearing denominators if necessary. LLIPTIC CURVES AND LOWER BOUNDS FOR CLASS NUMBERS 3 number H ( − D ) (for example, see p. 273 [4] ), the class number of discriminant − D quadraticforms, which counts each class C with multiplicity / Aut( C ) . Thankfully, Hurwitz class num-bers satisfy a particularly nice multiplicative formula relative to class numbers with fundamentaldiscriminant. If − D = − D f , where − D is a negative fundamental discriminant, then (1.7) H ( − D ) = h ( − D ) ω ( − D ) · X d | f µ ( d ) χ − D ( d ) σ ( f /d ) , where µ ( · ) is the M¨obius function, ω ( − D ) is half the number of units in Q ( √− D ) , χ − D ( · ) is the corresponding Kronecker character, and σ ( n ) is the sum of positive divisors of n . As aresult, a lower bound for H ( − D ) (resp. H ( − D E ( t )) ) leads to a lower bound for the class numberof the corresponding imaginary quadratic field. Remark.
A classical theorem of Hooley (see Ch. IV of [13] ) gives asymptotic formulas for thenumber of square-free values of irreducible cubic polynomials f ( t ) ∈ Z [ t ] . Namely, it is generallythe case (i.e. barring trivial obstructions arising from congruence conditions) that a positiveproportion of the values of f at integer arguments are square-free. Using this fact we can quantifythe frequency with which Theorem 1.2 improves on (1.1) for large − D E ( t ) (i.e. t → + ∞ ) when r ( E ) ≥ . A famous example of Elkies [5] has r ( E ) ≥ , and so we obtain the effective lowerbound h ( − D ) ≫ ε (log D ) − ε which holds for ≫ ε X many explicit fundamental discriminants − X < − D < . Example.
For E : y = x − x + 1 , we have | E tor ( Q ) | = 1 , r ( E ) = 3 , and R E ( Q ) ∼ . . . . .Therefore, for large fundamental discriminants of the form − D E ( t ) = − t − t − , we have h ( − D E ( t )) > · (log( D E ( t )) . We give infinite families of E/ Q using the discriminant ∆ a,b := − b − a ) curves(1.8) E a,b : y = x − a x + b . For integers t , we let D a,b ( t ) := 4( t − a t − b ) . For positive integers a, b , we let(1.9) c (2) a,b := Ω · b h ( P (2)max ) and c (3) a,b := Ω √ · b h ( P (3)max ) , where P (2)max ∈ { (0 , b ) , ( − a, b ) } ⊂ E a,b ( Q ) and P (3)max ∈ { (0 , b ) , ( − a, b ) , ( − b , ab ) } ⊂ E a,b ( Q ) arechosen to have the largest canonical height. Theorem 1.3. If a and b are positive integers, then the following are true:(1) If a ≫ b (resp. b ≫ a ), then r ( E a,b ( Q )) ≥ . Moreover, if ε > , then for sufficientlylarge fundamental discriminants − D a,b ( t ) < in absolute value we have h ( − D a,b ( t )) ≥ ( c (2) a,b − ε ) · log( D a,b ( t )) . (2) If a ≫ b (resp. b ≫ a ), then r ( E a,b ( Q )) ≥ . Moreover, if ε > , then for sufficientlylarge fundamental discriminants − D a,b ( t ) < in absolute value we have h ( − D a,b ( t )) ≥ ( c (3) a,b − ε ) · log( D a,b ( t )) . These calculations were performed using
SageMath . MICHAEL GRIFFIN AND KEN ONO
Three Remarks. (1) Theorem 1.3 is effective. One can make explicit b ≫ a a ≫ b . Moreover, we notethat − D n +1 ,n ( −
1) = − n covering all − D ≡ t ∈ Z + when a and b are small, orwhen − D a,b ( t ) is suitably composite).(3) Theorem 1.3 (2) often provides a log D power improvement to (1.1). Florian Luca has noted,for each 0 < c < /
2, that the effective lower bound h ( − D ) ≫ c log( D ) − c holds for ≫ X · exp (cid:0) log( X ) c (cid:1) many explicit − X < − D <
0. The idea is that integers ofthe form N = t − a t − b have unique representations with t ∈ [ X/ , X ] , a ∈ [ y/ , y ], and b ∈ [ z/ , z ] , where y = o ( X ) and z = o ( X ), and one then lets y = z = exp (cid:0) log( X ) c (cid:1) andcounts cubes b .This note is organized as follows. In Section 2 we prove Theorem 2.1, a result which providesthe ideal class pairings, and determines conditions guaranteeing SL ( Z )-inequivalence. Using thisresult, the proof of Theorem 1.2 is reduced to effectively counting rational points with boundedheight, which we address in Section 3. In Section 4 we state and prove Theorem 4.1, a resultwhich implies Theorem 1.2. Theorem 1.1 follows the proof of Theorem 4.1 mutatis mutandis .Finally, in Section 5 we prove Theorem 1.3. Acknowledgements
The second author thanks the NSF (DMS-1601306) and the Thomas Jefferson fund at theU. Virginia. The authors thank the referee, N. Elkies, D. Goldfeld, B. Gross, F. Luca, K.Soundararajan, D. Sutherland and J. Thorner for useful comments concerning this paper.2.
Elliptic curves Ideal class pairings
Works by Buell, Call, and Soleng [2, 3, 20] offered elliptic curve ideal class pairings, whichproduce discriminant − D integral positive definite binary quadratic forms from points on E ( Q )and E − D ( Q ). We offer a generalization and minor correction of Theorem 4.1 of [20]. Assume the notation from Section 1. Let P = ( AC , BC ) ∈ E ( Q ) , with A, B, C ∈ Z , and Q = ( uw , vw ) ∈ E − D ( Q ) , with u, v, w ∈ Z , not necessarily in lowest terms . Moreover, supposethat v = 0, with v even if − D is odd. If we let α := | Aw − uC | and G := gcd( α, C v ) , thenwe shall show that there are integers ℓ for which F P,Q ( X, Y ) defined below is a discriminant − D positive definite integral binary quadratic form.(2.1) F P,Q ( X, Y ) = αG · X + 2 w B + ℓ · αG C v · XY + (cid:0) w B + ℓ · αG (cid:1) + C v D C v · αG · Y For example, Fujita and Nara [6] show for b = 1 that a ≥ This corrects sign errors in the discriminants in Theorem 4.1 of [20], and also ensures the resulting quadraticforms are integral when C = 1. Moreover, this theorem allows for both even and odd discriminants. Thanks to (1.3), every Q has such a representation where gcd( u, w ) and gcd( v, w ) divide D . LLIPTIC CURVES AND LOWER BOUNDS FOR CLASS NUMBERS 5
Theorem 2.1.
Assuming the notation and hypotheses above, F P,Q ( X, Y ) is well defined (e.g.there is such an ℓ ) in CL( − D ) . Moreover, if ( P , Q ) and ( P , Q ) are two such pairs for which F P ,Q ( X, Y ) and F P ,Q ( X, Y ) are SL ( Z ) -equivalent, then α G = α G or α α G G > D/ . Example.
For E : y = x − x + 9 , we have points P := (0 , and P := ( − , . Using Q :=( − , ∈ E − ( Q ) and ℓ = 2 , we obtain the inequivalent discriminant − forms F P ,Q ( X, Y ) =3 X + 12 XY + 14 Y and F P ,Q ( X, Y ) = X + 8 XY + 22 Y . It turns out h ( −
24) = 2 .Proof.
A calculation shows that F P,Q ( X, Y ) has discriminant − D . We now show that there areintegers ℓ for which F P,Q ( X, Y ) is integral, and that all such choices preserve SL ( Z )-equivalence.To this end, let f ( x ) := x + a x + a , so that B = C f (cid:0) AC (cid:1) and − v D = 4 w f (cid:0) uw (cid:1) . Notethat α = (cid:12)(cid:12) w C (cid:0) AC − uw (cid:1)(cid:12)(cid:12) , which divides(2.2) w B + C v D = w C (cid:18) f (cid:18) AC (cid:19) − f (cid:16) uw (cid:17)(cid:19) . Since G = gcd( α, C v ), we have that G | w B . Let H := gcd(2 w B, C v ) . Then G | H , andso C v/H , which divides C v /G , is relatively prime to α/G . Choose k ∈ Z so that αkG ≡ − w BH − C vDH (mod C vH ) . If α/G is odd, k can be found by inverting α/G (mod C vH ) . If α/G is even, then − C vD ≡ w B (mod 2), and so k may be found by inverting α/ G (mod C vH ) . We take ℓ ≡ Hk (mod 2 C v )or (mod C v ) depending on whether k is defined (mod C vH ) or (mod C vH ) respectively. Theconditions on ℓ imply that the coefficient of XY in F P,Q ( X, Y ) has the same parity as − D . Thenumerator of the Y term, (2 w B + ℓα/G ) + C v D , is divisible by 4 C v . By (2.2), it is alsodivisible by 4 α . Therefore, it is divisible by 4 C v α/G , and so F P,Q ( X, Y ) is integral.We now determine the inequivalence of F P ,Q ( X, Y ) and F P ,Q ( X, Y ). For i = 1 and 2 we let A i , B i , C i , α i , and G i be the corresponding quantities for these two pairs of points. Note that F P ,Q ( X, Y ) = G α "(cid:18) α G X + 2 w B + ℓ · αG C v Y (cid:19) + D Y . Since ℓ was chosen so that ℓ · αG is defined modulo 2 C v, its choice does not affect SL ( Z )-equivalence. If ( a bc d ) ∈ SL ( Z ) and F P ,Q ( X, Y ) = F P ,Q ( aX + bY, cX + dY ) , then the leadingterms satisfy α G = G α "(cid:18) α aG + 2 w B + ℓ · αG C v c (cid:19) + D c . If c = 0, then a = 1, and the equation reduces to α G = α G . If c = 0, both terms inside thesquare brackets are positive, and together are at least D/
4, so α G ≥ G α D/ (cid:3) Heights on elliptic curves
To deduce Theorem 1.2 from Theorem 2.1, we use estimates for the number of rational pointson elliptic curves with bounded height. Here we recall the facts we require. Each rational point P ∈ E ( Q ) has the form P = ( AC , BC ), with A, B, C integers such that gcd(
A, C ) = gcd(
B, C ) = 1.
MICHAEL GRIFFIN AND KEN ONO
The naive height of P is H ( P ) = H ( x ) := max( | A | , | C | ) . The logarithmic height (or Weil height)is h W ( P ) = h W ( x ) := log H ( P ) , and the canonical height is given by(3.1) b h ( P ) = lim n →∞ h W ( nP ) n . Logarithmic and canonical heights are generally close. A theorem of Silverman [19] boundsthe differences between these heights in terms of the logarithmic heights of j ( E ) and ∆( E ). Theorem 3.1 (Theorem 1.1 of [19]) . If P ∈ E ( Q ) , then − h W ( j ( E )) − h W (∆( E )) − . ≤ b h ( P ) − h W ( P ) ≤ h W ( j ( E )) + h W (∆( E )) + 1 . . Asymptotics for the number of rational points on an elliptic curve with bounded height arewell known (for example, see [14, Prop 4.18]). If E ( Q ) has rank r ≥ r = π r / Γ (cid:0) r + 1 (cid:1) ,then in terms of the regulator R Q ( E ) and | E tor ( Q ) | , we have(3.2) { P ∈ E ( Q ) | b h ( P ) ≤ T } ∼ | E tor ( Q ) | p R Q ( E ) · Ω r T r = c ( E ) T r . Using an argument of Landau which estimates the number of lattice points in r -dimensionalspheres (for example, see [15]), one can show that the error term in the asymptotic is O( T r − r +1 ) . To prove Theorem 1.2, we require effective lower bounds for the number of points with boundedheight. To this end, if { P , . . . , P r } is a basis of E ( Q ) /E tor ( Q ), then its diameter is(3.3) d ( E ) = max δ i ∈{± , } b h r X i =1 δ i P i ! . It is the largest square-distance between any two vertices of the parallelopiped in R r constructedfrom vectors v , . . . v r which have v i · v j = h P i , P j i := (cid:16)b h ( P i + P j ) − b h ( P i ) − b h ( P j ) (cid:17) . Proposition 3.2.
Assume the notation and hypotheses above. If d = d ( E ) is the diameter ofany basis of E ( Q ) /E tor ( Q ) , then for T > d ( E ) / we have { P ∈ E ( Q ) | b h ( P ) ≤ T } ≥ c ( E ) (cid:16) T r − r √ d · T r − (cid:17) . Proof.
Let B = { P , . . . , P r } be any basis for E ( Q ). We must count points on the lattice Λ ∈ R r , generated by v , v , . . . , v r for which v i · v j = h P i , P j i . The number of points in the subgroup of E ( Q ) generated by B with canonical height bounded by T is the number of points in Λ ∩ B ( T ),where B ( R ) is the closed ball in R r centered at the origin of radius R .For each point λ ∈ Λ, let P λ be the half-open parallelepiped given by P λ = ( λ + r X i =1 x i v i | x i ∈ [0 , ) . If P λ intersects B ( T − d ) , then λ ∈ B ( T ) . Therefore, we have (cid:16) Λ ∩ B ( T ) (cid:17) ≥ Vol (cid:16) B ( T − d ) (cid:17) Vol( P λ ) = Ω r Vol( P λ ) · (cid:16) T − d (cid:17) r ≥ Ω r Vol( P λ ) · (cid:16) T r − r √ d · T r − (cid:17) . LLIPTIC CURVES AND LOWER BOUNDS FOR CLASS NUMBERS 7
In the last inequality we used the binomial expansion and the fact that T ≥ d/
4. Since wehave R Q ( E ) := | det( h P i , P j i ) ≤ i,j ≤ r | , it follows that Vol( P λ ) = p R E ( Q ) . To complete the proof,we note that torsion points have height zero, and so we may multiply the last estimate by | E tor ( Q ) | . (cid:3) These same arguments can be used to give lower bounds for the number of points of boundedheight generated from any linearly independent points in E ( Q ). Proposition 3.3.
Assume the notation and hypotheses above. Suppose G is a subgroup of E tor ( Q ) , and that B = { P , . . . , P m } is a set of linearly independent points in E ( Q ) listed inascending order by height. If T > d ( B ) / , then { P ∈ E ( Q ) | b h ( P ) ≤ T } ≥ | G | qb h ( P m ) m · Ω m (cid:18) T m − m q b h ( P m ) T m − (cid:19) . Proof.
The proof of Proposition 3.2 works with two modifications. Note that d ( B ) ≤ m b h ( P m ),and that the volume of the parallelopiped for B satisfies Vol( B ) ≤ Q mi =1 b h ( P i ) / ≤ b h ( P r ) m . (cid:3) Proof of Theorems 1.1 and 1.2
Theorems 1.1 and 1.2 are proven in the same way. For simplicity, we first consider Theorem 1.2,which pertains to fundamental discriminants − D E ( t ) = − t + a t − a ), and where we havechosen to pair the points in E ( Q ) with Q t := ( − t, ∈ E − D E ( t ) ( Q ). We obtain the preciseTheorem 4.1, which in turn implies Theorem 1.2.We show that points P ∈ E ( Q ) with canonical height b h ( P ) ≤ T E ( t ), where(4.1) T E ( t ) := 14 log (cid:18) D E ( t )( t + 1) (cid:19) − δ ( E ) , map to inequivalent forms F P,Q t ( X, Y ) ∈ CL( − D ). Theorem 4.1.
Assume the hypotheses above. If T E ( t ) ≥ d ( E ) / and − D E ( t ) is a negativefundamental discriminant for which ( t + 1) exp(4 δ ( E ) + d ( E )) ≤ D E ( t ) ≤ t ( t + 1) , then h ( − D E ( t )) ≥ c ( E )2 (cid:16) T E ( t ) r − r p d ( E ) T E ( t ) r − (cid:17) . Remark.
Since D E ( t ) is cubic, the conclusion holds for all but finitely many − D E ( t ) . Moreover,the proof works for any m independent points in E ( Q ) thanks to Proposition 3.3.Deduction of Theorem 1.2 from Theorem 4.1. Due to the ( t + 1) in (4.1), we have T E ( t ) ∼ log( D E ( t )) /
12, and the result follows. (cid:3)
Proof of Theorem 4.1.
We suppose that t ∈ Z + satisfies ( t + 1) exp(4 δ ( E ) + d ( E )) ≤ D E ( t ) ≤ t ( t + 1) . Proposition 3.2 implies that(4.2) { P ∈ E ( Q ) | b h ( P ) ≤ T E ( t ) } ≥ | E tor ( Q ) | p R Q ( E ) · Ω r (cid:16) T E ( t ) r − r p d ( E ) T E ( t ) r − (cid:17) . We show that these points map to inequivalent forms when paired with Q t = ( − t, ∈ E − D E ( t ) ( Q ). MICHAEL GRIFFIN AND KEN ONO
Suppose that P = ( A C , B C ) , P = ( A C , B C ) ∈ E ( Q ) satisfy b h ( P i ) ≤ T E ( t ), and let F := F P ,Q t ( X, Y ) and F := F P ,Q t ( X, Y ). Since gcd( A i , C i ) = 1 , their leading terms are α i G i = α i = | A i + tC i | . Thanks to Theorem 3.1, we have that h W ( P i ) ≤ (cid:16)b h ( P i ) + h W ( j ( E )) + h W (∆( E )) + 0 . (cid:17) ≤ T E ( t ) − log(2) = log (cid:12)(cid:12)(cid:12)(cid:12) D E ( t )4( t + 1) (cid:12)(cid:12)(cid:12)(cid:12) . We observe that α i ≤ ( t + 1) H ( P i ). By Theorem 3.1, we have H ( P i ) = exp( h W ( P i )) ≤ √ D E ( t )2( t +1) , which gives α i ≤ p D E ( t ) . Hence, we find that α α ≤ D E ( t ) , and so by Theorem 2.1, F and F are inequivalent unless α = | A + tC | = | A + tC | = α . However, by hypothesis D E ( t ) ≤ t ( t +1) , and so | A i | ≤ H ( P i ) ≤ t . Since C i >
0, this means α i = | A i + tC i | = A i + tC i . If α = α , then A ≡ A (mod t ) , and by the bounds on | A i | we have that A = A . This thenimplies that C = C , and so P = ± P , which explains the further factor of 1 / (cid:3) Proof of Theorem 1.1.
We follow the proof of Theorem 4.1, noting instead that Q D = ( u, v ) ∈ E − D ( Q ) , with u, v ∈ Z , where v = 0, and v is even if − D is odd. Arguing as before, we find that F and F are inequivalent unless α G = | A − uC | G = | A − uC | G = α G , and that | A i | ≤ H ( P i ) ≤ | u | v ≤ | u | . Since C i >
0, this implies that A − uC and A − uC have the same signs. If α G = α G ,then A G − A G = u ( C G − C G ) . The left hand side is divisible by, but not exceeding, | u | ; and so it must be 0. This implies A G = A G , and C G = C G , and so A C = A C . Hence, we have P = ± P , which explains the furtherfactor of 1 / (cid:3) A nice family of elliptic curves
Theorem 1.3 is a simple consequence of the following proposition.
Proposition 5.1. If a and b are positive integers, then the following are true:(1) If a ≫ b (resp. b ≫ a ), then (0 , b ) and ( − a, b ) are independent points in E a,b ( Q ) .(2) If a ≫ b (resp. b ≫ a ), then (0 , b ) , ( − a, b ) , and ( − b , ab ) are independent points in E a,b ( Q ) .Proof. This follows easily from Silverman’s specialization theorem for elliptic curves. Supposethat E t / Q ( t ) is an elliptic curve which is not isomorphic over Q ( t ) to an elliptic curve definedover Q . For w ∈ Q , we let σ w be the specialization map ( t → w ): σ w : E t ( Q t ) → E w ( Q ) . Generally, E w is an elliptic curve over Q . Silverman’s theorem (see Th. C of [18]) states, for allbut finitely many w ∈ Q , that σ w is an injective homomorphism between elliptic curves. Theclaims follows immediately by viewing a and b as indeterminates respectively. (cid:3) Proof of Theorem 1.3.
In view of Proposition 5.1 and Proposition 3.3, the proof follows the proofof Theorem 4.1 mutatis mutandis . (cid:3) LLIPTIC CURVES AND LOWER BOUNDS FOR CLASS NUMBERS 9
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