Small Algebraic Values of Twists of Elliptic L -Functions
aa r X i v : . [ m a t h . N T ] J a n SMALL ALGEBRAIC VALUES OF TWISTS OF ELLIPTIC L -FUNCTIONS HERSHY KISILEVSKYWITH AN APPENDIX BY JUNGBAE NAM
Abstract.
We consider heuristic predictions for small algebraic values of twistsof the L -function of an elliptic curve E/ Q by Dirichlet characters. We providecomputational evidence for these predictions and consequences of them for in-stances of an analogue of the Brauer-Siegel theorem. Introduction
Let E be an elliptic curve defined over the rational field Q and let L ( E/ Q , s ) denoteits L -function. Then the Birch & Swinnerton-Dyer Conjecture relates the value of L ( E/ Q , s ) at s = 1 to algebraic invariants of E ( Q ) . In particular, it predicts thatthe order of vanishing of L ( E/ Q , s ) at s = 1 is equal to the Z -rank of the Mordell-Weil group E ( Q ) . Similar conjectures are made for E over finite extensions of K/ Q , and there are many results and conjectures in this area along with heuristicmodels predicting the statistical frequency of vanishing of L ( E/K, s ) at s = 1 , and of the existence of a point of infinite order in E ( K ) (see e.g. [MR]),. In thecase of abelian extensions K/ Q , these questions can be investigated via the twists L ( E, s, χ ) of L ( E/ Q , s ) by Dirichlet characters χ associated to the extension K/ Q .Some models for vanishing or non-vanishing of L ( E, , χ ) are reviewed in § § L ( E, , χ ),and some consequences for analogues of the Brauer-Siegel theorem in this context.In this article we will generally be concerned with “orders of growth” rather thanasymptotics, and will use the notation f ( x ) ∼ g ( x ) to indicate that f = O ( g )and g = O ( f ) as x → ∞ (or that 0 < c < | f ( x ) /g ( x ) | < C as x → ∞ withoutspecifying c or C ). In an appendix by Jungbae Nam we provide the results ofnumerical computations in support of the consequent conjectures. We would liketo acknowledge very useful input from Evan Dummit and Andrew Granville to theproof of Lemma 10.1. Preliminaries and Notation
Let f be a positive integer, either odd or divisible by 4 . Then the multiplicativegroup ( Z / f Z ) ∗ is naturally isomorphic with the Galois group G = Gal( Q ( ζ f ) / Q ) ofthe cyclotomic field of f th roots of unity. The group of characters b G of G can thenbe identified with the group of Dirichlet characters \ ( Z / f Z ) ∗ . Since each Dirichletcharacter modulo f is induced from a unique primitive character of conductor f χ dividing f , we may identify G = \ ( Z / f Z ) ∗ with the set of primitive characters χ ofconductor f χ dividing f . The trivial character with f χ = 1 will be denoted 1 , andfor χ ∈ \ ( Z / f Z ) ∗ let ord( χ ) denote the order of χ. In the following we restrict to using only primitive characters as this allows forsimpler functional equations and factorisations of L -functions (without extra Eulerfactors).For k ≥ A k := { χ | χ k = 1 } and B k := { χ ∈ A k | ord( χ ) = k } . Also let A k ( f ) = { χ ∈ A k | f χ = f } and B k ( f ) = { χ ∈ B k | f χ = f } . For
X > , let A k ( X ) = { χ ∈ A k | f χ ≤ X } and B k ( X ) = { χ ∈ B k | f χ ≤ X } so that A k ( X ) = S f ≤ X A k ( f ) and B k ( X ) = S f ≤ X B k ( f ) . Let a k ( f ) = A k ( f )) and b k ( f ) = B k ( f )) so A k ( X )) = X f ≤ X a k ( f ) and B k ( X )) = X f ≤ X b k ( f )For N ≥ A k,N ( X ) := [ f ≤ X gcd( f ,N )=1 A k ( f ) and B k,N ( X ) := [ f ≤ X gcd( f ,N )=1 B k ( f ) . We consider the series F k ( s, A k ) = X f a k ( f ) f s and F k ( s, B k ) = X f b k ( f ) f s . MALL VALUES 3 Multiplicativity
Fix k ≥ f = Q p b p be the prime factorization of f ∈ Z with f odd or 4 | f . The Chinese Remainder Theorem states that( Z / f Z ) × ≃ Y p | f ( Z /p b p Z ) × so that \ ( Z / f Z ) × ≃ Y p | f \ ( Z /p b p Z ) × , and hence every χ ∈ \ ( Z / f Z ) × factors uniquely as χ = Q p | f χ p with χ p ∈ \ ( Z /p b p Z ) × . Also the conductor f χ of χ is equal to f if and only if the conductor f χ p of χ p is p b p for each prime p dividing f . Finally, χ ∈ A k ( f ) ⇐⇒ χ k = 1 ⇐⇒ χ kp = 1 for every p | f ⇐⇒ χ p ∈ A k ( p b p ) for every p | f and therefore a k ( f ) = Q p | f a k ( p b p ) . Calculation of a k ( p b ) . Note that ( Z /p b Z ) × ≃ Z / ( p − Z × Z /p b − Z for p > ≃ Z / Z × Z / b − Z for p = 2 . Let k = p a k ′ with ( p, k ′ ) = 1 , and let d p = gcd( p − , k ′ ) = gcd( p − , k ) . Case a).
Suppose that gcd( p, k ) = 1 . Then for 1 = χ ∈ A k ( p b ) and p > , wemust have b = 1 , and the order ord( χ ) divides d p . Hence in this case a k ( p ) = d p − a k ( p b ) = 0 for b ≥ . For p = 2 , gcd(2 , k ) = 1 implies that a k (2 b ) = 0 for all b ≥ . H. KISILEVSKY
Case b).
Suppose that p | k so k = p a p k ′ with a p = ν p ( k ) ≥ . Let p > . For χ ∈ A k ( p b ) we have χ = χ × χ where the order ord( χ ) divides d p and is prime to p and f χ = p, and where χ = 1 if b = 1 , and the orderord( χ ) = p b − and f χ = p b if b > . Therefore a k ( p b ) = d p − b = 1 and a k ( p b ) = ( p b − − p b − ) d p − ≤ b ≤ a + 1 and a k ( p b ) = 0 for b ≥ a + 2 . Now let p = 2, then any Dirichlet character χ = 1 with f χ = 2 b must have b ≥ . Then a k ( p ) = a k (4) = 1 , a k ( p b ) = 2 · b − − · b − = 2 b − for 3 ≤ b ≤ a + 2 and a k ( p b ) = 0 for b ≥ a + 3 . F k ( s, A k )Let N ≥ F k ( s, A k ) = X f a k ( f ) f s and F k,N ( s, A k ) = X gcd( f ,N )=1 a k ( f ) f s . By the multiplicativity of the coefficients a k ( f ) we have the Euler product facoriza-tion F k ( s, A k ) = Y gcd( p,k )=1 (cid:18) d p − p s (cid:19) × (cid:18) s + 22 s + · · · + 2 a ( a +2) s (cid:19) × Y = p | k (cid:18) d p − p s + ( p − d p − p s + · · · + p a p − ( p − d p − p ( a p +1) s (cid:19) where the 2 − Euler factor is 1 if k is odd.For functions f ( s ) , g ( s ) , write f ( s ) ≈ g ( s ) if f ( s ) = g ( s ) h ( s ) for h ( s ) analytic near s = 1 , and h (1) = 0 . Then since F k,N ( s, A k ) differs from F k ( s, A k ) by a finitenumber of Euler factors, we see that F k,N ( s, A k ) ≈ F k ( s, A k )and we have F k ( s, A k ) ≈ Y gcd( p,k )=1 (cid:18) d p − p s (cid:19) ≈ Y p ∤ k (cid:18) − p s (cid:19) − ( d p − . MALL VALUES 5
Then since d p = P d | d p φ ( d ) we can write (cid:18) − p s (cid:19) − ( d p − = Y = d | d p (cid:18) − p s (cid:19) − φ ( d ) and therefore F k ( s, A k ) ≈ Y p ∤ k (cid:18) − p s (cid:19) − ( d p − ≈ Y p ∤ k Y = d | d p (cid:18) − p s (cid:19) − φ ( d ) . The factor (1 − /p s ) − φ ( d ) appears if and only if p ∤ k, = d | k , and p ≡ d )so that F k ( s, A k ) ≈ Y d | kd =1 Y p ≡ d ) (cid:18) − p s (cid:19) − φ ( d ) ≈ Y d | kd =1 ζ d ( s )where ζ d ( s ) is the Dedekind ζ -function of the d th cyclotomic field Q ( e πi/d ) . Since F k,N ( s, A k ) ≈ F k ( s, A k ) it follows from the Ikehara–Tauberian theorem that A k,N ( X )) = X f ≤ X gcd( f ,N )=1 a k ( f ) ∼ c k X log τ ( k ) − ( X )where c k is a non-zero constant and τ ( k ) is the number of divisors of k. Also byinclusion-exclusion we see that B k,N ( X )) = A k,N ( X )) + X = d | k µ ( d ) A k/d,N ( X )) . Since for 1 = d | k the asymptotic power of log X in A k/d,N ( X )) is stricly smallerthan that in A k,N ( X )), we see that B k,N ( X )) = X f ≤ X gcd( f ,N )=1 b k ( f ) ∼ c k X log τ ( k ) − ( X ) . H. KISILEVSKY Algebraic Values
Let E be an elliptic curve defined over Q of conductor N E = N, and let L ( E/ Q , s )be its L -function. For a Dirichlet character χ with gcd( f χ , N E ) = 1 , let L ( E, s, χ )denote the L -function L ( E/ Q , s ) twisted by χ. It is known that there are realnumbers Ω ± E such that L ( E, , χ ) = Ω χ τ ( χ ) X a mod f χ χ ( a ) c ( a, f χ ; E )where Ω χ equals Ω ± according as χ ( −
1) = ± c ( a, f χ ; E ) are integers thatdo not depend on χ (but only on a, f χ and E ).Following [MTT] we define the algebraic part of L ( E, , χ ) by(6.1) L alg E ( χ ) := 2 τ ( χ ) L ( E, , χ )Ω χ = X a mod f χ χ ( a ) c ( a, f χ ; E ) . Then L alg E ( χ ) is an algebraic integer in the cyclotomic field Q ( χ ) generated over Q by the values of χ and satisfies σ ( L alg E ( χ )) = L alg E ( χ σ ) for all σ ∈ Gal( Q ( χ ) / Q ) . Noting that χ ( −
1) = χ ( −
1) we see that Ω χ = Ω χ , so from the functional equation,we have L alg E ( χ ) = 2 τ ( χ ) L ( E, , χ )Ω χ = 2 τ ( χ ) w E χ ( N E ) τ ( χ ) f χ Ω χ L ( E, , χ )= w E χ ( N E ) τ ( χ ) τ ( χ ) f χ · τ ( χ ) L ( E, , χ )Ω χ = w E χ ( N E ) χ ( − f χ f χ · L alg E ( χ )= w E χ ( − N E ) L alg E ( χ )If z ∈ C ∗ is any non-zero complex number satisfying z = w E χ ( − N E ) z, then itfollows that L alg E ( χ ) /z = α ∈ R is real. Let ζ = w E χ ( − N E ) . Then ζ = ζ n is aprimitive n th root of unity for some n ≥ k, where k is the order of χ. Suppose now that χ is a complex Dirichlet character of order k ≥ . MALL VALUES 7 If ζ = ± , choose z = 11 + ζ so that z = ζ
11 + ζ = ζ z, and L alg E ( χ ) = 11 + ζ α χ with α χ ∈ O + χ where O + χ is the ring of integers in Q ( χ ) + , the maximal real subfieldof Q ( χ ) . If ζ = − , let c be the least positive integer such that the order of χ ( c ) is equal to k , the order of χ. Choose z = 1 χ ( c ) − χ ( c ) so that z = − z = ζ z and L alg E ( χ ) = 1 χ ( c ) − χ ( c ) α χ with α χ ∈ O + χ . If ζ = 1 , then L alg E ( χ ) = α χ with α χ ∈ O + χ . We have proved the following:
Proposition 6.1.
Let E/ Q be an elliptic curve defined over Q , and let χ be aprimitive Dirichlet character of order k ≥ and conductor f χ , with gcd( f χ , N E ) = 1 . Let ζ = w E χ ( − N E ) and let ζ k be a primitive k th root of unity. Then L alg E ( χ ) =
11 + ζ α χ if ζ = ± χ ( c ) − χ ( c ) α χ if ζ = − α χ if ζ = 1 , where α χ ∈ O + χ are real cyclotomic integers satisfying σ ( α χ ) = α χ σ for all σ ∈ Gal( Q ( χ ) / Q ) . H. KISILEVSKY Probabilities for L ( E, , χ ) = 0We are interested in the frequency of vanishing of L ( E, , χ ) = 0 as χ varies, andwe consider probability models to try to describe this. Since L ( E, , χ ) = 0 ⇐⇒ L alg E ( χ ) = 0 , it is sufficient to find models for the vanishing of L alg E ( χ ) . Na¨ıve model.
The GRH (or the Lindel¨of hypothesis) implies that for ǫ > ,L ( E, , χ ) = O ( f ǫχ )so from (6.1) we have L alg E ( χ σ ) = O ( f + ǫχ )for all σ ∈ Gal( Q ( χ ) / Q ) . Hence γ ( α χ ) = O ( f + ǫχ ) for all γ ∈ Gal( Q ( χ ) + / Q ) . Lemma 7.1.
Let K/ Q be an extension of degree [ K : Q ] = d, and let O K be thering of integers of K. Let { σ , . . . , σ d } be the distinct embeddings of K and supposethat { ω , . . . , ω d } is an integral basis for O K . If α = P i a i ω i ∈ O K satisfies | σ i ( α ) | = O ( f + ǫ ) for ≤ i ≤ d, then | a i | = O ( f + ǫ ) for ≤ i ≤ d, where all implied constants depend only on K. Proof.
Let { η , . . . , η d } be the dual basis to { ω , . . . , ω d } , and let M = max {| σ i ( η j ) |} . Then | a j | = | Tr K/ Q ( αη j ) | ≤ X i | σ i ( α ) | M = O ( f + ǫ ) . (cid:3) Suppose now that χ is a character of order k > K = Q ( χ ) + and d =[ K : Q ] = φ ( k ) / , and fix a basis { ω , . . . , ω d } , of the ring of integers O K = O + χ . Applying Lemma 7.1 to the real cyclotomic integers α χ = P i a i ω i we see that a i = O ( f + ǫ ) for 1 ≤ i ≤ φ ( k ) / . Assuming that each value of a i is equally likely,and that L ( E, , χ ) = 0 if and only if a i = 0 for each 1 ≤ i ≤ φ ( k ) / , we get Prob ( L ( E, , χ ) = 0) is of the order of C/ f φ ( k ) / χ In the case that k = ord( χ ) = 2 the non-zero values L alg E ( χ ) are essentially theorders of the Shafarevich–Tate groups of E f (the twist of E by f ), which are squares. MALL VALUES 9
Therefore for quadratic characters χ such that χ ( − N E ) = +1 ( so that L ( E, s, χ )does not have a central zero forced by the functional equation), we have L ( E, , χ )is (essentially) a square integer less than f / χ and Prob ( L ( E, , χ ) = 0) is of the order of C/ f / χ . Random Matrix Theory model.
In [DFK1, DFK2] we outline a proba-bilistic model based on the model of Katz-Sarnak ( [KS]) which gives the somewhatmore precise estimate
Prob ( | L ( E, , χ ) | ≤ c/ f χ ) is of the order of C log ( f χ ) / f / χ . Since | L ( E, , χ ) | = | L ( E, , χ ) | , the φ ( k ) / | L ( E, , χ ) | are acted on by G + = Gal( Q ( χ ) + / Q ) and we may view them as being | L ( E, , χ γ ) | for γ ∈ G + . Assuming that for given χ, the values | L ( E, , χ γ ) | behave as identically distributedindependent random variables for γ ∈ Gal( Q ( χ ) / Q ) we obtain Prob ( L ( E, , χ ) = 0) is of the order of C log a ( f χ ) / f φ ( k ) / χ for some constants C, C , C , and a .8. Frequency of Vanishings for Characters of order k To compute an estimate for the frequency of vanishing of L ( E, , χ ) for primitiveDirichlet characters χ of order k, with conductor f χ relatively prime to N = N E , we compute the sums X χ ∈B k,N ( X ) Prob ( L ( E, , χ ) = 0)for X >> , and determine the convergence of X χ ∈B k gcd( f χ ,N )=1 Prob ( L ( E, , χ ) = 0) . For the first sum we have X χ ∈B k,N ( X ) Prob ( L ( E, , χ ) = 0) = X f
Note that since B k ( f ) consists of all primitive characters mod f of order k , then ∪ k B k ( f ) is the set of all primitive characters mod f and therefore is a subset of theset of all characters mod f . Hence X k b k ( f ) = |{∪ k B k ( f ) }| ≤ | \ ( Z / f Z ) × | = φ ( f ) ≤ | f | . MALL VALUES 11
If the series X k ≥ k X χ ∈B k gcd( f χ ,N )=1 Prob ( L ( E, , χ ) = 0)converges for some k , it converges absolutely and so its value would equal thevalue of the re-arranged series X gcd( f ,N )=1 X k ≥ k b k ( f ) ≤ X f φ ( f ) f φ ( k ) / ≤ X f f φ ( k ) / − which converges absolutely for φ ( k ) / > , ı.e. for φ ( k ) ≥ . Since there areonly finitely many k with φ ( k ) = 6 or 8 and by (8.1) for each of these k thereare only a finite number of χ ∈ B k,N for which L ( E, , χ ) = 0 , the Borel-Cantellilemma predicts the conjecture of Mazur and Rubin that { χ ∈ ∪ φ ( k ) > B k,N ( X ) | L ( E, , χ ) = 0 } is finite . Probabilities for “small” non-zero vaues
In this section we shall be concerned with a probabilistic model for “small non-zerovalues” of L alg E ( χ ) . From Proposition 6.1 we see that L alg E ( χ ) =
11 + ζ α χ if ζ = ± χ ( c ) − χ ( c ) α χ if ζ = − α χ if ζ = 1 , where α χ ∈ O + χ are real cyclotomic integers satisfying σ ( α χ ) = α χ σ for all σ ∈ Gal( Q ( χ ) / Q ) . We are interested in the norms A χ = Nm Q ( χ ) + / Q ( α χ ) ∈ Z with respect to Gal( Q ( χ ) + / Q ) . We note that A χ = 0 ⇐⇒ γ ( α χ ) = 0 for all γ ∈ Gal( Q ( χ ) + / Q ) ⇐⇒ α χ γ = 0 for all γ ∈ Gal( Q ( χ ) + / Q ) . It follows from the GRH (or the Lindel¨of hypothesis) that γ ( α χ ) = O ( f + ǫχ ) (andalso γ ( β χ ) = O ( f + ǫχ )) for all γ ∈ Gal( Q ( χ ) + / Q ) . For a totally real field F/ Q , with [ F : Q ] = n , let φ : F −→ R n be the usual mapwhere for α ∈ F, φ ( α ) = ( σ ( α ) , σ ( α ) , . . . , σ n ( α )) ∈ R n where 1 = σ , σ , . . . , σ n are the distinct embeddings of F into R . Then the image of the ring of integers φ ( O F ) ⊂ R n is a sublattice of R n . We will be interested in the case that F = Q ( χ ) + ,and O F = O + χ . Lemma 10.1.
Let n ≥ be a positive integer and let < L ≤ M n be real numbers.Define subsets T ⊆ R ⊂ R n by R = R ( M ) = { x = ( x , x , . . . , x n ) ∈ R n | ≤ x i ≤ M, ≤ i ≤ n } and T = T ( L, M ) = { x = ( x , x , . . . , x n ) ∈ R | x x · · · x n ≤ L } . Then µ ( T ) µ ( R ) = LM n P n − (cid:18) LM n (cid:19) where P m ( x ) = P mj =0 (( − x ) j /j ! is the m th Taylor polynomial of e − x at x = 0 . Thenthe order of growth of the ratio of their measures as M → ∞ is µ ( T ) µ ( R ) ∼ Ln n − log n − ( M )( n − M n . Proof. (due to A. Granville)Clearly µ ( R ) = M n , so we compute µ ( T ). Re-scaling and letting x ′ i = x i /M wesee that µ ( T ) = M n Z ≤ x ′ ,x ′ ,...,x ′ n ≤ x ′ x ′ ··· x ′ n ≤ LMn dx ′ dx ′ · · · dx ′ n = M n I ( n ) . and so µ ( T ) /µ ( R ) = I ( n ) . Set C = L/M n and x ′ j = e − y j , so that dx ′ j = e − y j dy j , and then I ( n ) becomes I ( n ) = Z y ,y ,...,y n ≥ y + y + ··· + y n ≥ log(1 /C ) e − ( y + y + ··· + y n ) dy dy · · · dy n = Z x ≥ log(1 /C ) e x · Z y + y + ··· + y n − ≤ x dy dy dy n − dx. Let J n ( x ) := Z y + y + ··· + y n ≤ x dy dy · · · dy n MALL VALUES 13
Then J ( x ) = x and J n ( x ) = Z xy =0 J n − ( x − y ) dy = Z xt =0 J n − ( t ) dt = x n /n ! by induction . Hence we have I ( n ) = Z x ≥ log(1 /C ) e − x x n − ( n − dx. Integrating by parts we see Z x ≥ A e − x x n n ! dx = (cid:20) − e − x x n n ! (cid:21) ∞ A + Z x ≥ A e − x x n − ( n − dx, so by induction we have I ( n ) := Z ≤ x ,x ....,x n ≤ x x ··· x n ≤ C dx dx · · · dx n = C n − X m =0 ( − log( C )) m m ! . Recalling that C = L/M n yields the statement. (cid:3) In order to compute an estimate for the frequency with which A χ = Nm Q ( χ ) + / Q ( α χ )assumes “small” non-zero values for primitive Dirichlet characters χ of order k, wemust compute the sums X χ ∈B k,N ( X ) Prob (0 < | A χ | ≤ L )for X >> , and determine the convergence of X χ ∈B k gcd( f χ ,N )=1 Prob (0 < | A χ | ≤ L ) . For characters χ of order k and conductor f χ , the GRH (or the Lindel¨of hypothesis)implies that the image φ ( α χ ) (resp. ϕ ( β χ )) lies in R ′ ( M ) where R ′ ( M ) = { x = ( x , x , . . . , x n ) ∈ R n | | x i | ≤ M, ≤ i ≤ n } with M ∼ f + ǫχ and n = φ ( k ) / Q ( χ ) + : Q ], so that taking into account thepossible signs of the x i we have µ ( R ′ ( M )) = 2 n µ ( R ( M )) . Similarly, µ ( T ′ ( L, M )) = n µ ( T ′ ( L, M )) where T ′ ( L, M ) = { x = ( x , x , . . . , x n ) ∈ R ′ ( M ) | | Y i x i | ≤ L } . So supposing that the number of lattice points in a region is proportional to thevolume of the region, for fixed k and n = φ ( k ) / , and as M → ∞ we have Prob (0 < | A χ | ≤ L ) = µ ( T ′ ( L, M )) µ ( R ′ ( M )) ∼ L log n − ( f χ ) f φ ( k ) / χ . Then for the first sum we have X χ ∈B k,N ( X ) Prob (0 < | A χ | ≤ L ) = X f
For k = 3 , we have B = τ (3) + φ (3) / − , so the predicted growth rate for“small” non-zero algebraic values of cubic twists L alg ( E, χ,
1) is of the order CX / . For k = 5 , we have B = τ (5) + φ (5) / − , so the predicted growth ratefor “small” non-zero algebraic values of quintic twists L alg ( E, χ,
1) is of the order C log ( X ) . These predictions seem to agree with the experimental computations in the casesbelow. 11.
Brauer-Siegel Consequences In §
10 we saw that taking L to be a “small” fixed constant the model predictsthat { χ ∈ B k | gcd( f χ , N ) = 1 , < | A χ | ≤ L } is finite for φ ( k ) ≥ . Considering larger values of k (say for φ ( k ) ≥ L = L ( f ) = f α , then since L ≤ M n = X φ ( k ) / , we have 0 ≤ α ≤ φ ( k ) /
4. Thenfrom (10.1) above we find that the model predicts that { χ ∈ B k | gcd( f χ , N ) = 1 , < | A χ | ≤ f α } is finite for α < φ ( k ) / − φ ( k ) / − ≤ α ≤ φ ( k ) / . Theprediction implies that { χ ∈ B k,N ( X ) | < | Nm Q ( χ ) + / Q ( α χ ) | ≤ f α } grows as log B +1 X if α = φ ( k ) / − X α − ( φ ( k ) / − log B X if φ ( k ) / > α >φ ( k ) / − X → ∞ . For any such character χ (attached to the cyclic extension K = K χ / Q ) we mayview E/K is an abelian variety over Q whose “ χ -component” is also an abelianvariety A ( χ ) over Q with L -function equal to L ( A ( χ ) , s ) = Y ≤ i APPENDIX: COMPUTATIONAL RESULTS Jungbae Nam In this section, we present computational results to support the statistical predic-tions above for the statistics of some small integer values A χ associated with thecentral critical L -values for L ( E, s, χ ) for E : 11a1 , B k,N ( X ) for k = 3with X = 3 × and k = 5 with X = 10 . Here we let N = N E for a given E .Then, we compute the L -values at s = 1 by the following well-known formula: L ( E, , χ ) = X n ≥ ( a n χ ( n ) + w E c χ a n χ ( n )) n exp (cid:18) − πn f χ √ N E (cid:19) , where a n is the coefficients of L ( E, s ) and c χ = χ ( N E ) τ ( χ ) / f χ . Knowing that χ ( − 1) = 1 for k odd, we use non-zero complex values Ω χ = Ω + computed bySageMath [S + 09] for 11a1 and 14a1. Using those values we compute L alg E ( χ ) andthen A χ ∈ Z using Proposition 6.1.In our numerical computations, the numerical values of L ( E, , χ ) for 11a1 and 14a1with f χ ≤ × were already computed by Jack Fearnley for the articles [DFK1]and [DFK2], and we used his L -values. Those central L -values for k = 3 with2 × ≤ f χ ≤ × and k = 5 with f χ ≤ were obtained by using around40 threads in the cluster of the Centre de Recherches Math´ematiques (CRM) for acouple of months. The codes to evaluate L ( E, , χ ) were created by using Cythonand SageMath [S + A χ . Table 1 shows B k,N ( X )) for E, k and X used inour experimental computations. k X × Table 1. B k,N ( X )) for each E and k = 3 with X = 3 × and k = 5 with X = 10 .Moreover, for the support of our predictions on the frequencies of integer values,we use the values of A χ divided the greatest common multiple (gcd), depending on k and E , of all non-zero A χ for χ ∈ B k,N ( X ). For E : 11a1 and 14a1, those gcdsare not trivial. By abuse of notations we denote those values of A χ by A χ .Given k , E with N = N E and an integer l , denote n k,E ( x ; l ) = { χ ∈ B k,N ( x ) | A χ = l } . Then, in particular, n k,E ( x ; 0) = { χ ∈ B k,N ( x ) | L ( E, , χ ) = 0 } . Let’s considerthe frequencies of vanishings we predicted. Define a function on the real numbers x > f k,E ( x ) = x / log / ( x ) if k = 3 , E : 11a1 x / log / ( x ) if k = 3 , E : 14a1log / ( x ) if k = 5 , E : 11a1log / ( x ) if k = 5 , E : 14a1 . Note that the larger powers of log( x ) come from the existences of k -torsion pointsof E ; 11a1 has 5-torsion and 14a1 has 3-torsion points and we numerically findmore vanishings for these elliptic curves than those without k -torsion points. Referto [DFK1] and [DFK2] for these choices of powers of log( x ). Then, our predictionsassert n k,E ( x ; 0) ∼ c k,E f k,E ( x ) as x → ∞ with a constant c k,E depending on k and E . In order to see this, we compute theratios n k,E ( x ; 0) /f k,E ( x ). Figure 1 and 5 depict those ratios for 11a1 and 14a1, k = 3 and x ≤ × . Furthermore, Figure 9 and 13 do the same for k = 5 and x ≤ .Now, let’s consider the frequencies of some small non-zero integer values we pre-dicted for twists for k = 3 , 5. Define a function on the real numbers x > g k ( x ) = x / if k = 3log ( x ) if k = 5 . We choose | l | = 1 , , . . . , k = 3 and 1 , , , , , , , , 25 for k = 5. Notethat for k = 5, those values of | l | are the first 9 possible absolute values of non-zero A χ . Then, we compute the ratios n k,E ( x ; l ) /g k ( x ). For k = 3 with x ≤ × ,Figure 2, 3 and 4 depict those ratios for 11a1. Figure 6, 7 and 8 do the same for14a1. For k = 5 with x ≤ , Figure 10, 11 and 12 depict those ratios for 11a1.Figure 14, 15 and 16 do the same for 14a1. MALL VALUES 19 Similarly, for an integer L > 0, denote s k,E ( x ; L ) = { χ ∈ B k,N ( x ) | < | A χ | ≤ L } . Then, the above figures for n k,E ( x ; L ) seem to support our predictions assert s k,E ( x ; L ) ∼ d k,E g k ( x ) as x → ∞ with a constant d k,E depending on k and E . Note that these s k,E ( x ; L ) are thesum of n k,E ( x ; l ) over 0 < | l | ≤ L . We compute the ratios s k,E ( x ; L ) /g k ( x ) for L = 1 , , . . . , k = 3 and 1 , , . . . , 25 for k = 5. Figure 17 and 18 depict thoseratios for 11a1 and 14a1, k = 3 and x ≤ × . Furthermore, Figure 19 and 20 dothe same but for k = 5 and x ≤ . Note that as expected, the lines in Figure 19and 20 lie higher as L increases. Figure 1. n k,E ( x ; l ) /f k,E ( x ) for k = 3 and x ≤ × where l = 0 and f k,E ( x ) = x / log / ( x ). Figure 2. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± g k ( x ) = x / . The upper line is for l = 1 and thelower one is for l = − MALL VALUES 21 (a) l = ± 2: upper for l = − l = 2. (b) l = ± 3: upper for l = 3 and lower for l = − l = ± 4: upper for l = 4 and lower for l = − 4. (d) l = ± 5: upper for l = − l = 5. Figure 3. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± , ± , ± , ± g k ( x ) = x / . (a) l = ± 6: upper for l = − l = 6. (b) l = ± 7: upper for l = 7 and lower for l = − l = ± 8: upper for l = − l = 8. (d) l = ± 9: upper for l = 9 and lower for l = − Figure 4. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± , ± , ± , ± g k ( x ) = x / . MALL VALUES 23 Figure 5. n k,E ( x ; l ) /f k,E ( x ) for k = 3 and x ≤ × where l = 0 and f k,E ( x ) = x / log / ( x ). Figure 6. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± g k ( x ) = x / . The upper line is for l = 1 and thelower one is for l = − (a) l = ± 2: upper for l = 2 and lower for l = − 2. (b) l = ± 3: upper for l = 3 and lower for l = − l = ± 4: upper for l = 4 and lower for l = − 4. (d) l = ± 5: upper for l = − l = 5. Figure 7. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± , ± , ± , ± g k ( x ) = x / . MALL VALUES 25 (a) l = ± 6: upper for l = 6 and lower for l = − 6. (b) l = ± 7: upper for l = 7 and lower for l = − l = ± 8: upper for l = 8 and lower for l = − 8. (d) l = ± 9: upper for l = 9 and lower for l = − Figure 8. n k,E ( x ; l ) /g k ( x ) for k = 3 and x ≤ × where l = ± , ± , ± , ± g k ( x ) = x / . Figure 9. n k,E ( x ; 1) /f k,E ( x ) for k = 5 and x ≤ where l = 0 and f k,E ( x ) = log / ( x ). Figure 10. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± g k ( x ) = log ( x ). The upper line is for l = 1 andthe lower one is for l = − MALL VALUES 27 (a) l = ± 4: upper for l = − l = 4. (b) l = ± 5: upper for l = 5 and lower for l = − l = ± 9: upper for l = 9 and lower for l = − 9. (d) l = ± 11: upper for l = 11 and lower for l = − Figure 11. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± , ± , ± , ± 11 and g k ( x ) = log ( x ). (a) l = ± 16: upper for l = − 16 and lower for l = 16. (b) l = ± 19: upper for l = − 19 and lower for l = 19.(c) l = ± 20: upper for l = − 20 and lower for l = 20. (d) l = ± 25: upper for l = 25 and lower for l = − Figure 12. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± , ± , ± , ± 25 and g k ( x ) = log ( x ). MALL VALUES 29 Figure 13. n k,E ( x ; l ) /f k,E ( x ) for k = 5 and x ≤ where l = 0 and f k,E ( x ) = log / ( x ). Figure 14. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± g k ( x ) = log ( x ). The upper plot is for l = − l = 1. (a) l = ± 4: upper for l = 4 and lower for l = − 4. (b) l = ± 5: upper for l = − l = 5.(c) l = ± 9: upper for l = − l = 9. (d) l = ± 11: upper for l = − 11 and lower for l = 11. Figure 15. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± , ± , ± , ± 11 and g k ( x ) = log ( x ). MALL VALUES 31 (a) l = ± 16: upper for l = − 16 and lower for l = 16. (b) l = ± 19: upper for l = 19 and lower for l = − l = ± 20: upper for l = 20 and lower for l = − 20. (d) l = ± 25: upper for l = 25 and lower for l = − Figure 16. n k,E ( x ; l ) /g k ( x ) for k = 5 and x ≤ where l = ± , ± , ± , ± 25 and g k ( x ) = log ( x ). Figure 17. s k,E ( x ; L ) /g k ( x ) for k = 3 and x ≤ × where L = 1 , , . . . , g k ( x ) = x / . The lines for L lie higher as L increases. Figure 18. s k,E ( x ; L ) /g k ( x ) for k = 3 and x ≤ × where L = 1 , , . . . , g k ( x ) = x / . The lines for L lie higher as L increases. MALL VALUES 33 Figure 19. s k,E ( x ; L ) /g k ( x ) for k = 5 and x ≤ where L = 1 , , , , , , , , 25 and g k ( x ) = log ( x ). The linesfor L lie higher as L increases. Figure 20. s k,E ( x ; L ) /g k ( x ) for k = 5 and x ≤ where L = 1 , , , , , , , , 25 and g k ( x ) = log ( x ). The linesfor L lie higher as L increases. References [DFK1] C. David, J. Fearnley and H. Kisilevsky, On the vanishing of twisted L -functions ofelliptic curves , Experiment. Math. (2004), no. 2, 185–198.[DFK2] C. David, J. Fearnley and H. Kisilevsky, Vanishing of L -functions of elliptic curves overnumber fields , Ranks of Elliptic Curves and Random Matrix Theory, London Mathemat-ical Society Lecture Note Series, , Cambridge University Press (2007), pp. 247-259.[KS] Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and mon-odromy , American Mathematical Society Colloquium Publications, vol. 45, AmericanMathematical Society, Providence, RI, 1999.[MR] B. Mazur and K. Rubin, Arithmetic Conjectures Suggested by the Statistical Behaviorof Modular Symbols , arXiv1910.12798, 2019[MTT] B. Mazur, J. Tate, and J. Teitelbaum, On p -adic analogues of the conjectures of Birchand Swinnerton-Dyer , Invent. Math. (1986), no. 1, 1–48.[S + SageMath, the Sage Mathematics Software System (Version 8.6.rc0) , The Sage Devel-opers, 2019, .(H. Kisilevsky) Department of Mathematics and Statistics and CICMA, Concor-dia University, 1455 de Maisonneuve Blvd. West, Montr´eal, Quebec, H3G 1M8,CANADA E-mail address : [email protected] (Jungbae Nam) Department of Mathematics and Statistics and CICMA, Concor-dia University, 1455 de Maisonneuve Blvd. West, Montr´eal, Quebec, H3G 1M8,CANADA E-mail address ::