Size and cancellations in Sato Tate sequences Florian Luca November 13, 2013 Florian Luca Size and cancellations in Sato Tate sequences
Example of interest for us: Elliptic curves Let E be an elliptic curve over the field of rational numbers given by the minimal global Weierstraß equation : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x + A 6 (1) and let ∆ be its discriminant. For each prime p we put a p = p + 1 − # E ( F p ) , where E ( F p ) is the reduction of E modulo p . If p | ∆ , then E ( F p ) has a singularity and we put 0 for the case of a cusp , a p = 1 for the case of a split node , − 1 for the case of a non–split node . Florian Luca Size and cancellations in Sato Tate sequences
We have | a p | ≤ 2 √ p . The L -function associated to E is given by 1 1 � � L ( s , E ) = 1 − a p p − s + p 1 − 2 s . 1 − a p p − s p ∤ ∆ p | ∆ The infinite product above is convergent for Re ( s ) > 3 / 2 and therefore we can expand it into a series � a n n − s . L ( s , E ) = n ≥ 1 Florian Luca Size and cancellations in Sato Tate sequences
Other example of interest for us: The Ramanujan τ -function Let τ ( n ) be the Ramanujan function given by τ ( n ) q n = q � � ( 1 − q i ) 24 ( | q | < 1 ) . n ≥ 1 i ≥ 1 Ramanujan observed but could not prove the following three properties of τ ( n ) : (i) τ ( mn ) = τ ( m ) τ ( n ) whenever gcd ( m , n ) = 1. (ii) τ ( p r + 1 ) = τ ( p ) τ ( p r ) − p 11 τ ( p r − 1 ) for p prime and r ≥ 1. (iii) | τ ( p ) | ≤ 2 p 11 / 2 for all primes p . These conjectures were proved by Mordell and Deligne. Florian Luca Size and cancellations in Sato Tate sequences
Fibonacci numbers Let { F m } m ≥ 0 be the Fibonacci sequence given by F 0 = 0 , F 1 = 1 and F m + 2 = F m + 1 + F m for all m ≥ 0 . Let { a n } n ≥ 1 be the sequence of coefficients of the L -function of an elliptic curve E . We put A E = { n : | a n | = F m } , and for a positive x we put # A E ( x ) = # ( A E ∩ [ 1 , x ]) . Florian Luca Size and cancellations in Sato Tate sequences
Before we start, we remark that there could be many n such that a n is a Fibonacci number simply because it may happen that a p = 0 for some prime p , in which case n = p ℓ with any positive integer ℓ coprime to p has the property that a n = 0 = F 0 . To discard this instance, let M E = { n : a n � = 0 } . Putting M E ( x ) = M E ∩ [ 1 , x ] , we have # M E ( x ) ≫ x in case of non CM curves (Serre, 1981 ). Theorem (L., Yalc ¸iner) Let E be a non-CM curve with non-trivial 2 -torsion. The estimate � � � # M E ( x ) � x # N E ( x ) = O = O ( log x ) 0 . 0007 ( log # M E ( x )) 0 . 0007 holds for all x ≥ 2 . The implied constant depends on E. Florian Luca Size and cancellations in Sato Tate sequences
Later, we proved a more general result. Theorem (L., Oyono, Yalc ¸iner) Let E be an elliptic curve defined over and u = { u m } m ≥ 0 be a nondegenerate binary recurrent sequence. There is a positive number c = c ( E , u ) depending on E and u such that the estimate � # M E ( x ) � # N E ( x ) = O ( log x ) c holds for all x ≥ 2 . The implied constant depends on E. Florian Luca Size and cancellations in Sato Tate sequences
Squares in a certain sequence Again, { a n } n ≥ 1 is the sequence of coefficients of the L -function of an elliptic curve E . We studied the set N E = { n : n 2 − a n 2 + 1 = � } . The reason we studied this is because if we replace n 2 by p and consider the “extreme case” a p = ± 2 √ p , then p − a p + 1 = p ± 2 √ p + 1 = ( √ p ± 1 ) 2 looks like a “perfect square”. Florian Luca Size and cancellations in Sato Tate sequences
Theorem (L., Yalc ¸iner) Let E be a non CM curve for which the Sato–Tate conjecture holds. The estimate � x � # N E ( x ) = O ( log x ) 0 . 00001 holds for all x ≥ 2 . The implied constant depends on E. Note that if p | ∆ and a p = ± 1, and ℓ ≥ 1, then a p ℓ = ( a p ) ℓ = ( ± 1 ) ℓ , which implies that n = p ℓ ∈ N E . Moreover if all prime factors p of n divide ∆ and have a p = ± 1, then n ∈ N E . However, the set of such positive integers n is very thin since the number of such integers n ≤ x is O (( log x ) c ) for some constant c ≤ ω (∆) . Florian Luca Size and cancellations in Sato Tate sequences
Elliptic Carmichael numbers Again, { a n } n ≥ 1 is the sequence of coefficients of the L -function of an elliptic curve E . Slightly relaxing a definition of Silverman, we say that a positive integer n is an E -Carmichael number if it is not a prime power; for any prime divisor p | n we have p ∤ ∆ ; for any point P ∈ E ( F p ) we have ( n + 1 − a n ) P = O p , (2) where both the equation and the group law are considered over F p . Florian Luca Size and cancellations in Sato Tate sequences
For a real x ≥ 1, let N E ( x ) be the number of E -Carmichael numbers n ≤ x . Theorem (L., Shparlinski) Let E be a non CM curve. For a sufficiently large x N E ( x ) ≪ x ( log log log x ) 1 / 2 ( log log log log x ) 1 / 4 . ( log log x ) 1 / 4 Florian Luca Size and cancellations in Sato Tate sequences
Sato–Tate sequences Let A ST be the class of infinite sequences { a n } n ≥ 1 of real numbers, which satisfy the following properties: Multiplicativity: a mn = a m a n , whenever gcd ( m , n ) = 1 . Sato-Tate distribution: for any prime p , a p ∈ [ − 2 , 2 ] , and for the angles ϑ p ∈ [ 0 , π ) defined by a p = 2 cos ϑ p , and α ∈ [ 0 , π ) , we have � α # { p ≤ x : p prime , ϑ p ∈ [ 0 , α ] } = 2 sin 2 ϑ d ϑ. lim π ( x ) π x →∞ 0 Growth on prime powers: There exist a constant ̺ > 0 such that for any integer a ≥ 2 and prime p we have | a p a | ≤ p ( a − 1 ) / 2 − ̺ . Florian Luca Size and cancellations in Sato Tate sequences
The above properties are known to hold both for the Ramanujan function τ ( n ) / n 11 / 2 as well as for a n / n 1 / 2 , where { a n } n ≥ 1 is the sequence of coefficients arising of an L -function of an elliptic curves with certain conditions, like a non-integral j -invariant. Theorem (L., Shparlinski) For any sequence { a n } n ≥ 1 ∈ A ST , the inequality | a n | ≤ ( log n ) − 1 / 2 + o ( 1 ) holds for almost all positive integers n. Theorem (L., Shparlinski) For any sequence { a n } n ≥ 1 ∈ A ST , we have � � a n = o | a n | ( x → ∞ ) . n ≤ x n ≤ x Florian Luca Size and cancellations in Sato Tate sequences
The proof of the result involving Fibonacci numbers The proof goes in various steps. Florian Luca Size and cancellations in Sato Tate sequences
Removing n with a large square full part Recall that s is a square full number if p 2 | s whenever p | s . Put y = log x . For each n we write � t ( n ) = p and s ( n ) = n / t ( n ) . p � n p ∤ 6 ∆ Then s ( n ) = ab , where a is square free and a | 6 ∆ and b is squarefull up to factors of 2 and 3. We put N 1 ( x ) = { n ≤ x : s ( n ) > y } . (3) Then x x # N 1 ( x ) ≪ y 1 / 2 = ( log x ) 1 / 2 , (4) where we used that the counting function of the number of square full numbers s ≤ t is O ( t 1 / 2 ) . Florian Luca Size and cancellations in Sato Tate sequences
Removing smooth n Let P ( n ) be the largest prime factor of n . Put � log x log log log x � z = exp . log log x We let N 2 ( x ) = { n ≤ x : P ( n ) ≤ z } . (5) From known results from the distribution of smooth numbers, in this range for z and x , it is known that # N 2 ( x ) = x exp ( − ( 1 + o ( 1 )) u log u ) as x → ∞ , where u = log x / log z = log log x / log log log x . Hence, u log u = ( 1 + o ( 1 )) log log x , as x → ∞ , showing that � x � # N 2 ( x ) = x exp ( − ( 1 + o ( 1 )) log log x ) = O . (6) ( log x ) 1 / 2 Florian Luca Size and cancellations in Sato Tate sequences
Removing n with too few prime factors Let α ∈ ( 0 , 1 ) to be found later and consider the set N 3 ( x ) = { n ≤ x : ω ( n ) < ( 1 − α ) log log x } . (7) The results from the book Divisors of Hall, Tenebaum, show that x # N 3 ( x ) ≪ ( log x ) β , (8) where � � e β = 1 − ( 1 − α ) log . 1 − α ) Florian Luca Size and cancellations in Sato Tate sequences
The final argument Assume that n ∈ N 4 ( x ) = N E ( x ) \ ( N 1 ( x ) ∪ N 2 ( x ) ∪ N 3 ( x )) . Since n �∈ N 1 ( x ) , we may write n = up 1 · · · p ℓ , u ≤ y , p 1 < · · · < p ℓ , gcd ( u , p 1 · · · p ℓ ) = 1 . Furthermore, p i ∤ 6 ∆ for any i = 1 , . . . , ℓ . Assume that x is large enough so that z > y . Then P ( n ) = p ℓ . Write F m = a n = a u a p 1 · · · a p ℓ . Let ε > 0 be arbitrary. Note that since log u log y ω ( u ) ≪ log log u ≪ log log y = o ( log log x ) as x → ∞ , it follows that ω ( u ) < ε log log x holds whenever x is sufficiently large. Florian Luca Size and cancellations in Sato Tate sequences
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