aa r X i v : . [ m a t h . N T ] A ug A Constructive Proof of Masser’s Theorem
Alexander J. Barrios
Abstract.
The Modified Szpiro Conjecture, equivalent to the abc
Conjecture, states that foreach ǫ >
0, there are finitely many rational elliptic curves satisfying N ǫE < max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) where c and c are the invariants associated to a minimal model of E and N E is the conductor of E .We say E is a good elliptic curve if N E < max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) . Masser showed that there are infinitelymany good Frey curves. Here we give a constructive proof of this assertion.
1. Introduction
By an
ABC triple, we mean a triple of positive integers ( a, b, c ) such that a, b, and c are relativelyprime positive integers with a + b = c . The ABC
Conjecture [
CR01 , 5.1] states that for any ǫ >
ABC triples that satisfy rad( abc ) ǫ < c where rad( n ) denotes theproduct of the distinct primes dividing n . We say that an ABC triple is good if rad( abc ) < c . Forinstance, the triple (1 , ,
9) is a good
ABC triple and more generally the triple (cid:0) , k − , k (cid:1) is agood ABC triple for each positive integer k [ CR01 ]. In 1988 , Oesterl´e [
Oes88 ] proved that the
ABC
Conjecture is equivalent to the modified Szpiro conjecture which states that for ǫ >
0, thereare only finitely many elliptic curves E such that N ǫE < max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) where N E denotes theconductor of the elliptic curve and c and c are the invariants associated to a minimal model of E . As with ABC triples, we define a good elliptic curve to be an elliptic curve E that satisfiesthe inequality N E < max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) . In the special case of Frey curves, that is, a rational ellipticcurve that has a Weierstrass model of the form y = x ( x − a ) ( x + b ) where a and b are relativelyprime integers, Masser [ Mas90 ] showed that there are infinitely many good Frey curves. In thisarticle, we provide a constructive proof of Masser’s Theorem. Moreover, the torsion subgroup of aFrey curve can only take on four possibilities due to Mazur’s Torsion Theorem [
Maz77 ], namely E ( Q ) tors ∼ = C × C N where C m denotes the cyclic group of order m and N = 1 , , , or 4. Withthis we state our main theorem: Theorem 1
For each of the four possible torsion subgroups T = C × C N where N = 1 , , , or4, there are infinitely many good elliptic curves such that E ( Q ) tors ∼ = T .This is equivalent to Theorem 6.3, where the main theorem is given in its constructive form.As a consequence we get examples akin to the infinitely many good ABC triples (cid:0) , k − , k (cid:1) foreach positive integer k . For each of the four possible T , we use rational maps of modular curvesto construct a recursive sequence of ABC triples P Tj = ( a j , b j , c j ) such that if P Tj is a good ABC
Mathematics Subject Classification.
Primary 11G05.
Key words and phrases.
Number Theory, Elliptic Curves, Arithmetic Geometry. triple satisfying certain congruences, then P Tj is a good ABC triple for each nonnegative integer j .Once this is proven, we prove our main Theorem by showing that the associated Frey curve F P Tj : y = x ( x − a j ) ( x + b j )is a good elliptic curve for each positive integer j with F P Tj ( Q ) tors ∼ = T .
2. Certain Polynomials
In this section we establish a series of technical results which will ease the proofs in the sectionsthat are to follow. Let T = C × C N where N = 1 , , ,
4. For each T let A T = A T ( a, b ) , B T = B T ( a, b ) , C T = C T ( a, b ) , D T = D T ( a, b ) , A rT = A rT ( a, b ) , B rT = B rT ( a, b ) , C rT = C rT ( a, b ) ,U T = U T ( a, b, r, s ) , V T = V T ( a, b, r, s ) , and W T = W T ( a, b, r, s ) be the polynomials in R = Z [ a, b, r, s ] defined in Table 6.For a fixed T , the polynomials A T , B T , C T , and D T are homogenous polynomials in a and b of the same degree m T . In particular, we have the equalities a m T A T (cid:0) , ba (cid:1) = A T ( a, b ) a m T B T (cid:0) , ba (cid:1) = B T ( a, b ) a m T C T (cid:0) , ba (cid:1) = C T ( a, b ) a m T D T (cid:0) , ba (cid:1) = D T ( a, b ) . The first result can be verified via a computer algebra system and we note that we are considering A T (1 , t ) , B T (1 , t ) , C T (1 , t ) , D T (1 , t ) as functions from R to R . Lemma . For T = C × C N with N = 1 , , , , let f T , g T : R → R be the function in thevariable t defined in Table 6. Let θ T be the greatest real root of f T ( t ) . The (approximate) value of θ T is found in Table 6. Then for each T , (1) A T ∈ R ; (2) A T + B T = C T ; (3) U T B T + V T C T = W T ; (4) f T (cid:0) ba (cid:1) = B T ( a,b ) A T ( a,b ) − ba ; (5) g T ( t ) = C T (1 , t ) − D T (1 , t ) ; (6) f T ( t ) , g T ( t ) , A T (1 , t ) , B T (1 , t ) , C T (1 , t ) , D T (1 , t ) > for t > θ T ; (7) For T = C × C N for N = 1 , , f T ( t ) , g T ( t ) , A T (1 , t ) , B T (1 , t ) , C T (1 , t ) , D T (1 , t ) > for t in (0 , .
3. Good
ABC
Triples
Definition . By an
ABC triple, we mean a triple P = ( a, b, c ) such that a, b, and c arerelatively prime positive integers with a + b = c . We say P = ( a, b, c ) is good if rad( abc ) < c . Lemma . For each T = C × C N , let P = ( a, b, a + b ) be an ABC triple with a even and ba > θ T where θ T is as defined in Lemma 2.1. Suppose further that a ≡ if N = 3 . Then ( A T , B T , C T ) is an ABC triple with A T ≡ , B T ≡ , and B T A T > θ T . Moreover, if N = 3 , then A T ≡ . Proof.
Since a and b are relatively prime, there exist integers r and s such that ra n + sb n =1 , for any positive integer n . Therefore, by Lemma 2.1, gcd( B T , C T ) divides 32 if N = 3 andgcd( B T , C T ) divides 48 if N = 3. Since a is even and a ≡ N = 3, we conclude thatgcd( B T , C T ) = 1. Next, observe that f T (cid:18) ba (cid:19) = B T (cid:0) , ba (cid:1) A T (cid:0) , ba (cid:1) − ba . CONSTRUCTIVE PROOF OF MASSER’S THEOREM 3
Since ba > θ T , we have by Lemma 2.1 that f T (cid:0) ba (cid:1) is positive and therefore B T A T > ba > θ T . By Lemma2.1 we also have that A T + B T = C T for each T and therefore ( A T , B T , C T ) is an ABC triple. Since a is even it is easily verified that A T ≡ N = 3, A T ≡ a ≡ T , B T ≡ b k mod 4 for some integer k . Since b is odd,it follows that B T ≡ (cid:3) Lemma . Let P = ( a, b, a + b ) be a good ABC triple and assume the statement of Lemma3.2. Then ( A T , B T , C T ) is a good ABC triple.
Proof.
Since a is assumed to be even, we have that rad(2 n ax ) = rad( ax ) for some integer x .Therefore rad( A T ) = rad( A rT ) , rad( B T ) = rad( B rT ) , rad( C T ) = rad( C rT ) . Since ( a, b, a + b ) is a good ABC triple, we have that rad( ab ( a + b )) < a + b . From this and thefact that rad (cid:0) xy k (cid:1) = rad( xy ) ≤ xy for positive integers k, x, y , we have that for each T , we attainrad( A T B T C T ) = rad( A rT B rT C rT ) < | D T | . Since ba > θ T , D T (cid:0) , ba (cid:1) is positive by Lemma 2.1. In particular, D T is positive since a m T D T (cid:0) , ba (cid:1) = D T where m T is the homogenous degree of D T . Now observe that C T − rad( A T B T C T ) > C T − D T = a m T (cid:18) C T (cid:18) , ba (cid:19) − D T (cid:18) , ba (cid:19)(cid:19) > A T , B T , C T ) is a good ABC triple sincerad( A T B T C T ) < C T . (cid:3) Proposition . Let ( a , b , c ) be a good ABC triple with a even. For each T define thetriple P Tj recursively by P Tj = ( a j , b j , c j ) = ( A T ( a j − , b j − ) , B T ( a j − , b j − ) , C T ( a j − , b j − )) for j ≥ . Assume further that b a > θ T and that b ≡ if T = C × C . Then for each j ≥ , P Tj is agood ABC triple with a j ≡ , b j ≡ , and b j a j > θ T . Additionally, if T = C × C ,then a j ≡ . Proof.
This follows automatically from Lemmas 3.2 and 3.3. (cid:3)
4. Frey Curves
As before, we suppose T = C × C N and define for t ∈ P , the mapping X t as the mappingwhich takes T to the elliptic curve X t ( T ) where the Weierstrass model of X t ( T ) is given in Table 1.Our parameterizations for T = C × C N where N = 3 , HLP00 , Table 3] whichexpands the implicit expressions for the parameters b and c in [ Kub76 , Table 3] to express theuniversal elliptic curves for the modular curves X (2 , N ) in terms of a single parameter t . Similarly,our model for T = C × C differs by a linear change of variables from the model given for W in[ Sil97 , §
4] which parameterizes elliptic curves E with C × C ֒ → E ( Q ( i )) tors . In particular, X t ( T )is a one-parameter family of elliptic curves with the property that if t ∈ K for some field K , then X t ( T ) is an elliptic curve over K and T ֒ → X t ( T )( K ) tors .For T = C × C , define(4.1) X t ( T ) : y = x + (cid:0) t − t + 6 t − t + 1 (cid:1) x − t ( t − (cid:0) t + 1 (cid:1) x. Lemma . If t ∈ Q such that X t ( T ) is an elliptic curve, then T ֒ → X t ( T )( Q ) tors . ALEXANDER J. BARRIOS
Table 1.
Universal Elliptic Curve X t ( T ) X t ( T ) : y + (1 − g ) xy − f y = x − f x f g T t − t +12 t − t +22( − t ) C × C − t +14 t − t +10( t +3) ( t − − t +10( t +3)( t − C × C t +16 t +6 t +1(8 t − t +16 t +6 t +12 t (4 t +1)(8 t − C × C Proof.
Recall that the modular curve X (2 , N ) (with cusps removed) for N = 2 , , E, P, Q ) where E is an elliptic curve having full 2-torsion, P and Q are torsion points of order 2 and 2 N , respectively, and h P, N · Q i = E [2].For T = C × C N where N = 3 ,
4, we note that our parameterizations are those of the universalelliptic curve for the modular curve X (2 , N ) [ HLP00 , Table 3]. Thus
T ֒ → X t ( T ) ( Q ) tors .For T = C × C , let t ′ = t t − so that X t ( T ) is equal to the Weierstrass model given for the universal elliptic curve over X (2 , HLP00 , Table 3] with parameter t ′ . Hence T ֒ → X t ( T )( Q ) tors .For T = C × C , let t = ba and consider the admissible change of variables x a x and y a y . This gives a Q -isomorphism between X t ( T ) and the elliptic curve y = x (cid:0) x − ab (cid:0) a + b (cid:1)(cid:1) (cid:16) x + ( a − b ) (cid:17) which has (cid:10)(cid:0) ab (cid:0) a + b (cid:1) , (cid:1) , (0 , (cid:11) ∼ = C × C . Thus T ֒ → X t ( T )( Q ) tors . (cid:3) Definition . For an
ABC triple P = ( a, b, c ), let F P = F P ( a, b ) be the Frey curve givenby the Weierstrass model F P : y = x ( x − a ) ( x + b ) . Lemma . Let ( a, b, c ) be an ABC triple which satisfies the assumptions of Lemma 3.2. Thenfor each T , the Frey curve F P with P = ( A T , B T , C T ) has torsion subgroup F P ( Q ) tors ∼ = T . Proof.
Let X t ( T ) be as defined in Table 1 for T = C × C N for N = 2 , , N = 1. In addition, let u T , r T , s T , w T , and t T be as defined in Table 5. We now proceedby cases.Case I. Suppose T = C × C N for N = 2 , ,
4. Then the admissible change of variables x u T x + r T and y u T y + u T s T x + w T gives a Q -isomorphism from F P onto X t T ( T ). Inparticular, T ֒ → F P ( Q ) tors by Lemma 4.1. By Mazur’s Torsion Theorem [ Maz77 ] we concludethat F P ( Q ) tors ∼ = C × C N for N = 3 , F P ( Q ) tors is isomorphic to either C × C or C × C if T = C × C . For the latter, we observe that our model for X t ( T ) parametrizes ellipticcurves E over Q ( i ) with C × C ֒ → E ( Q ( i )) tors [ Sil97 , § Kam92 ] we conclude that E ( Q ( i )) tors ∼ = C × C . Thus X t ( T )( Q ( i )) tors ∼ = C × C and therefore C × C ֒ → X t ( T )( Q ( i )) tors . Hence X t ( T )( Q ) tors ∼ = C × C .Case II. Suppose T = C × C and T = C × C . Then there is a 2-isogeny φ : X t ( T ) → X t ( T )obtained by applying V´elu’s formulas [ V´71 ] to the elliptic curve X t ( T ) and its torsion point 2 P where P = (0 ,
0) is the torsion point of order 4 of X t ( T ). CONSTRUCTIVE PROOF OF MASSER’S THEOREM 5
Next, observe that via the First Isomorphism Theorem:(4.2) |X t ( T )( Q ) tors | |X t ( T )( Q )[ φ ] | = |X t ( T )( Q ) tors | [ X t ( T )( Q ) tors : φ ( X t ( T )( Q ) tors )] . By Case I above we have that |X t ( T )( Q ) tors | = 8 which implies that the only prime dividing |X t ( T )( Q ) tors | is 2 since φ is a 2-isogeny.Next, we consider the admissible change of variables x u T x + r T and y u T y + u T s T x + w T which gives a Q -isomorphism from F P onto X t T ( T ). In particular, C × C ֒ → F P ( Q ) tors by Lemma4.1. By the proof of Lemma 4.1, X t ( T ) is Q -isomorphic to the elliptic curve given by the Weierstrassmodel y = x (cid:0) x − ab (cid:0) a + b (cid:1)(cid:1) (cid:16) x + ( a − b ) (cid:17) . This model satisfies the assumptions of [
Ono96 , Main Theorem 1] and therefore we have that X t ( T )( Q ) tors ∼ = C × C if 8 ab (cid:0) a + b (cid:1) is not a square. If it were a square we would have anontrivial integer solution to the Diophantine equation x − y = z since8 ab (cid:0) a + b (cid:1) + ( a − b ) = ( a + b ) . This contradicts Fermat’s Theorem and therefore Y t ( T )( Q ) tors ∼ = C × C . (cid:3) Theorem . Let T = C × C N for N = 1 , , , and consider the sequence of good ABC triples P Tj defined in Proposition 3.4. Then for each j ≥ , the Frey curve F P Tj determined by P Tj has torsion subgroup F P Tj ( Q ) tors ∼ = C × C N . Proof.
In Proposition 3.4, we saw that each P Tj satisfies the assumptions of Lemma 3.2.Consequently, the Theorem follows from Lemma 4.3. (cid:3) The case of N = 2 , BTW10 ] as part of the Mathematical Sciences Research Institute Undergraduate Program.
5. Examples of Good ABC Triples
Definition . For an
ABC triple P = ( a, b, c ), define the quality q ( P ) of P to be q ( P ) = log( c )log(rad( abc )) . In particular, P is a good ABC triple is equivalent to q ( P ) > Example . For T = C × C N where N = 1 , P = (cid:0) , , (cid:1) . Then P is a good ABC triple since q ( P ) ≈ . ABC triple results in two distinctinfinite sequences of good
ABC triples P Tj .For T = C × C , let P = (cid:0) , , (cid:1) . Then P is a good ABC triple since q ( P ) ≈ . > θ T . By Proposition 3.4, this good ABC triple results in an infinitesequence of good
ABC triples P Tj .For T = C × C , let P = (cid:0) , , (cid:1) . Then P is a good ABC triple since q ( P ) ≈ . > θ T . By Proposition 3.4, this good ABC triple results in an infinite sequence ofgood
ABC triples P Tj .Table 2 gives a and b of P Tj = ( a j , b j , c j ) as well as the quality q (cid:0) P Tj (cid:1) for j = 1 , ,
3. Wenote that the values of a j and b j are not given for j ≥ T = C × C N for N = 3 ,
4, we only compute q (cid:0) P Tj (cid:1) for j = 1 , ALEXANDER J. BARRIOS
Table 2.
Table for Example 5.2
T a b q (cid:0) P T (cid:1) q (cid:0) P T (cid:1) q (cid:0) P T (cid:1) C × C . . . C × C . . . C × C
61 5 · . . − C × C · · · · . . −
6. Infinitely Many Good Frey Curves
Recall that the
ABC
Conjecture is equivalent to the modified Szpiro conjecture which statesthat for every ǫ > E satisfying N ǫE < max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) where N E is the conductor of E and c and c are the invariants associated to a minimal model of E . The following definition gives the analog of good ABC triples and the quality of an
ABC triplein the context of elliptic curves.
Definition . Let E be a rational elliptic curve with minimal discriminant ∆ min E and asso-ciated invariants c and c . Define the modified Szpiro ratio σ m ( E ) and Szpiro ratio σ ( E ) of E to be the quantities σ m ( E ) = log max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) log N E and σ ( E ) = log (cid:12)(cid:12) ∆ min E (cid:12)(cid:12) log N E where N E is the conductor of E . We say that E is good if σ m ( E ) > P = ( a, b, c ) be an ABC triple with a even and b ≡ T = C × C N where N = 1 , , ,
4, let A T = A T ( a, b ) , B T = B T ( a, b ) , C T = C T ( a, b ) , and D T = D T ( a, b ) be as definedin Table 6. Assume further that a ≡ T = C × C . Then the elliptic curve F T = F T ( a, b )given by the Weierstrass model F T : y = x ( x − A T )( x + B T )satisfies F T ( Q ) tors ∼ = T by Lemma 4.3. Moreover, the congruences on A T and B T imply that theFrey curve F T is semistable with minimal discriminant ∆ T = (cid:0) − A T B T C T (cid:1) [ Sil09 , Exercise8.23]. Consequently, the conductor N T of F T satisfies N T = rad(∆ T ) < | D T | and the invariant c ,T = c ,T ( a, b ) associated with a global minimal model of F T is as given in Table 3. Table 3.
The Invariant c of F T c ,T Ta + 60 a b + 134 a b + 60 a b + b C × C a + 14 a b + b C × C a + 228 a b + 30 a b − a b + b C × C a − a b + 12 a b + 8 a b + 230 a b + 8 a b + 12 a b − a b + b C × C Lemma . Let P = ( a, b, c ) be a good ABC triple satisfying a ≡ , b ≡ , and ba > θ T where θ T is as given in Lemma 2.1. Assume further that a ≡ if T = C × C .Then the Frey curve F T = F T ( A T , B T ) is good and F T ( Q ) tors ∼ = T . Proof.
By Lemma 4.3, F T ( Q ) tors ∼ = T . Since F T is a Frey curve we have that the invariants c and c associated to a global minimal model of F T satisfy max (cid:8)(cid:12)(cid:12) c (cid:12)(cid:12) , c (cid:9) = c since c and ∆ min F T are always positive [ Sil09 , Lemma VIII.11.3]. The congruences on a and b imply that c = c ,T .It, therefore, suffices to show that c ,T − N T > N T is the conductor of F T . Since F T issemistable, N T = rad( A T B T C T ) < D T CONSTRUCTIVE PROOF OF MASSER’S THEOREM 7 by Lemma 3.3. Note that D T is positive since ba > θ T . Thus(6.1) c ,T − N T D T (1 , t ) > c ,T (1 , t ) − D T (1 , t ) D T (1 , t ) for t = ba Lastly, for each T , the polynomial c ,T (1 , t ) − D T (1 , t ) is positive on the open interval ( θ T , ∞ )from which we conclude that F T is a good elliptic curve. (cid:3) Theorem . For each T , let P T = ( a , b , c ) be a good ABC triple satisfying a ≡ , b ≡ , and b a > θ T where θ T is as given in Lemma 2.1. Assume further that a ≡ if T = C × C . For j ≥ , define P Tj recursively by P Tj = ( a j , b j , c j ) = ( A T ( a j − , b j − ) , B T ( a j − , b j − ) , C T ( a j − , b j − )) . Then for each j , the Frey curve F T ( a j , b j ) is good and F T ( a j , b j ) ( Q ) tors ∼ = T . Proof.
By Proposition 3.4, P Tj = ( a j , b j , c j ) satisfies a j ≡ b j ≡ , and b j a j > θ T for each j . For T = C × C , if a ≡ a j ≡ j . Hence P Tj isa good ABC triple for each j by Proposition 3.4. Therefore the result follows by Lemma 6.2. (cid:3) In Example 5.2 we began with a good
ABC triple P = ( a , b , c ). For each T , we constructedan infinite sequence of good ABC triples P Tj = ( a j , b j , c j ). By Theorem 6.3, each Frey curve F T ( a j , b j ) ( Q ) tors is a good elliptic curve with torsion subgroup isomorphic to T . Table 4 lists themodified Szpiro ratios of the Frey curves corresponding to P Tj . Due to computational limitations,we could only compute these ratios up to j = 3. Table 4.
Example of Good Frey Curves
T C × C C × C C × C C × C σ m ( F T ( a , b )) 6 . . . . σ m ( F T ( a , b )) 6 . . . . σ m ( F T ( a , b )) 6 . . A L E XAN D E R J . B A RR I O S
7. Table of Polynomials
Table 5.
Admissible Change of Variables for Lemma 4.3
T u T r T s T w T t T C × C a ba C × C a − b ) − ab ( a − b ) ( a − b ) − ab ( a − b ) (cid:0) a + b (cid:1) ba C × C a − b − a ( a + b ) ( − a + b ) 5 a − b a − a b + 4 a b a + ba + b C × C a ( a + b )( b − ab − a ) ab ( a + b ) ( a + b ) ( b − ab − a ) a +4 a b − b a ( a + b )( b − ab − a ) ab ( a + b ) ( a + b ) ( b − ab − a ) a b − a ) Table 6.
Polynomials and Rational Functions
T C × C C × C C × C C × C A T ab (cid:0) a + b (cid:1) (2 ab ) a b (2 ab ) B T ( a − b ) (cid:0) a − b (cid:1) ( a + b ) ( b − a ) (cid:0) a − a b + b (cid:1)(cid:0) a + b (cid:1) C T ( a + b ) (cid:0) a + b (cid:1) (3 a + b )( b − a ) (cid:0) a − b (cid:1) D T b − a b − a (cid:0) b − a (cid:1)(cid:0) b − a (cid:1) (cid:0) a − a b + b (cid:1)(cid:0) b − a (cid:1) A rT ab (cid:0) a + b (cid:1) ab ab ab B rT ( a − b ) a − b ( a + b )( b − a ) (cid:0) a − a b + b (cid:1)(cid:0) a + b (cid:1) C rT a + b a + b (3 a + b )( b − a ) a − b f T (1 − t ) t (1+ t ) − t ( − t ) (2 t ) − t (1+ t ) ( t − t − t ( − t + t )( t ) (2 t ) − tg T t + 6 t + 4 t + 2 2 t + 2 4 t + 8 t −
12 2 t + 6 t − t + 2 θ T . . U T a r + 20 a br + 29 ab r +16 b r + 16 a s + 29 a bs +20 ab s + 5 b s a r + 2 b r +2 a s + b s − a r + 144 a br − ab r +24 b r − a s +6 a bs − b s a r − a b r + 20 a b r − b r − a s +20 a b s − a b s + 4 b sV T − a r + 20 a br − ab r +16 b r + 16 a s − a bs + 20 ab s − b s − a r + 2 b r +2 a s − b s a r + 144 a br +117 ab r + 24 b r − a s − a bs + b s − a r + 15 a b r + 44 a b r +26 b r + 26 a s +44 a b s + 15 a b s − b sW T (cid:0) ra + sb (cid:1) (cid:0) ra + sb (cid:1) (cid:0) ra + sb (cid:1) (cid:0) ra + sb (cid:1) CONSTRUCTIVE PROOF OF MASSER’S THEOREM 9
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