Biases in Moments of Dirichlet Coefficients of Elliptic Curve Families
aa r X i v : . [ m a t h . N T ] F e b Biases in Moments of Dirichlet Coefficientsof Elliptic Curve Families
Yan (Roger) Weng
Peddie SchoolHightstown, NJ, USAunder the direction of
Steven. J. Miller
Williams CollegeWilliamstown, MA, USAFebruary 10, 2021
IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES
YAN (ROGER) WENGA
BSTRACT . Elliptic curves arise in many important areas of modern number theory. One way to study themis take local data, the number of solutions modulo p , and create an L -function. The behavior of this globalobject is related to two of the seven Clay Millenial Problems: the Birch and Swinnerton-Dyer Conjecture andthe Generalized Riemann Hypothesis. We study one-parameter families over Q ( T ) , which are of the form y = x + A ( T ) x + B ( T ) , with non-constant j -invariant. We define the r th moment of an elliptic curveto be A r,E ( p ) := 1 p X t mod p a t ( p ) r , where a t ( p ) is p minus the number of solutions to y = x + A ( t ) x + B ( t ) mod p . Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil groupof rational solutions.Michel proved that pA ,E ( p ) = p + O ( p / ) . Based on several special families where computations can bedone in closed form, Miller in his thesis conjectured that the largest lower-order term in the second moment thatdoes not average to is on average negative. He further showed that such a negative bias has implications in thedistribution of zeros of the elliptic curve L -function near the central point. To date, evidence for this conjectureis limited to special families. In addition to studying some additional families where the calculations can bedone in closed form, we also systematically investigate families of various rank. These are the first general testsof the conjecture; while we cannot in general obtain closed form solutions, we discuss computations whichsupport or contradict the conjecture. We then generalize to higher moments, and see evidence that the biascontinues in the even moments.
1. Introduction 21.1. Pythagorean Triples 31.2. Elliptic Curves Preliminaries 41.3. Elliptic Curve L -function 61.4. Moments of the Dirichlet Coefficients of the Elliptic Curve L -functions 91.5. Summary of Results 112. Biases in second moments of elliptic curve families 132.1. Systematic investigation for the second moments sums 132.2. First and second moments of the family y = 4 x + ax + bx + c + dt y = 4 x + (4 m + 1) x + n · tx y = x − t x + t Date : February 10, 2021.
Appendix B. Forms of 4th and 6th moments sums 44B.1. Tools for higher moments calculations 44B.2. Form of 4th moments sums 45B.3. Form of 6th moments sums 46Appendix C. Mathematical code 48C.1. Finding formulas for second moment sums 48References 51Contents 1. I
NTRODUCTION
Elliptic curves play a major role in many problems modern number theory for a variety of reasons. First,they are part of a tradition going back thousands of years: counting how many rational points satisfy apolynomial. The best known example of this are the Pythagorean triples: finding integer sides a, b and c of aright triangle require a + b = c , which is equivalent to finding rational points on the circle x + y = 1 . Amore recent application was the discovery that elliptic curves could be used to create strong crypto-systems.In this paper we explore some properties related to elliptic curves, in particular to biases in the distributionof the Dirichlet coefficients of the associated L -functions. To put our work in perspective, we go back alittle over a hundred years to the International Congress of Mathematicians in Paris in 1900. David Hilbert[BY] presented 10 of 23 problems (many of which are still open) that he believed would be great guides formathematics in the coming years. To start this century, the Clay Mathematics Institute proposed 7 problems[MP]. One has since been solved (the Poincare Conjecture), but the other 6 are open; our work intersectstwo of these: the Generalized Riemann Hypothesis, and the Birch and Swinnerton-Dyer Conjecture.We briefly summarize our problem, expanding into greater detail below. An elliptic curve is of the form y = x + ax + b with a, b ∈ Z , and more generally we can consider one-parameter families y = x + a ( T ) x + b ( T ) with a ( T ) , b ( T ) ∈ Z [ T ] . If we specialize T to an integer t we get an elliptic curve E t ,and for each prime p we can count how many ( x, y ) solve y ≡ x + a ( t ) x + b ( t ) mod p . We let a t ( p ) be related to this count, and build a function L ( E t , s ) = X n a t ( n ) /n s (we define how to get a t ( n ) fromthe prime values later); this is a generalization of the famous Riemann Zeta Function ζ ( s ) = X n /n s . Itturns out that the set of rational solutions to an elliptic curves forms a commutative group, and the Birch andSwinnerton-Dyer Conjecture states the order of vanishing of L ( E t , s ) at s = 1 equals the geometric rank ofthe group. Further, the Generalized Riemann Hypothesis states that the non-trivial zeros of L ( E t , s ) havereal part equal to 1.It turns out that the moments of the Dirichlet coefficients a t ( p ) of the elliptic curve L -function are related tothese problems. We define the r th moment of the one-parameter family E : y = x + a ( T ) x + b ( T ) to be A r,E ( p ) := 1 p X t mod p a t ( p ) r (1.1)(we will see later that a t ( p ) = a t + ℓp ( p ) , and thus it suffices to sum over t mod p ). By work of Rosen andSilverman [RS] the first moment is related to the rank of the one-parameter family. Michel [Mic] provedthat if the elliptic curve doesn’t have complex multiplication then the second moment is of size p , or better pA E (2) = p plus lower order terms of size p / , p, p / and 1. In his thesis Miller [Mi1, Mi2] discoveredthat in every family of elliptic curves where pA ,E ( p ) could be computed in closed form, the first lower We are using the normalization that the critical strip is < Re( s ) < , so the central point is s = 1 / . Assuming the Tate Conjecture [Ta], they prove a conjecture of Nagao [Na] linking these two quantities.
IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 3 ( x , y )(0,-1)(0,0) F IGURE
1. A line through (0 , − with slope m , hitting the circle at ( x, y ) .order term that did not average to zero had a negative average. This has implications for the distribution ofzeros of elliptic curves, connecting our work to the Generalized Riemann Hypothesis. This bias has beenseen by many others, both in families of elliptic curves as well as other automorphic forms [Mi1, Mi3]We continue these investigations in two new directions. The first is prior studies concentrated on specificfamilies where the computations could be done in closed form. Unfortunately, what this means is thatperhaps the observed bias is merely a consequence of being such a special family that the computation canbe done. While we do compute the bias in closed form for several new families, we also launch the firstsystematic investigation of general one-parameter families. In some cases the numerics do suggest closedform answers for the second moment’s expansion, at least for primes in certain congruence classes. We alsoexplore, for the first time, higher moments, to see if the biases persist.For the rest of the introduction, after motivating our problem we quickly review some needed facts on ellipticcurves; for more detail see [Kn, ST, Ta]. We then describe in detail our results.1.1. Pythagorean Triples.
For thousands of years, there have been efforts to find integral and rationalpoints on curves. We start with curves that are easy to analyze and also important: circles. It turns out thatthe set of rational points on a unit circle is closely related to primitive Pythagorean triples.The Pythagorean theorem is one of the most important results in geometry, as it allows us to measure thedistance between two points. Given any right triangle with sides a, b and hypotenuse c , we have a + b = c ;the triple is primitive if gcd( a, b, c ) = 1 . It is easy to find some solutions, such as (3 , , or (5 , , .Interestingly, one can generate all triples if we can find just one. Theorem 1.1 (Pythagorean triples) . For all primitive Pythagorean triples ( a, b, c ) there exist integers p and q with p > q > such that a = 2 pq, b = p − q and c = p + q .Proof. Given a + b = c we divide by c . Letting x = a/c and y = b/c we see our problem is equivalentto looking for rational solutions on the circle x + y = 1 . Note it is easy to find some points on this curve: ( ± , or (0 , ± . We use the point (0 , − and draw a line through that point with slope m ; see Figure 1.The equation of our line is y = mx − . It intersects the circle at (0 , − and one other point, say ( x, y ) .We find that by simultaneously solving x + y = 1 and y = mx − . After some algebra we find x = 2 mm + 1 , y = m − m + 1 . (1.2) YAN (ROGER) WENG
Note that x and y are rational numbers if and only if m is a rational number; this is because we are solving x + ( mx − = 1 , which is a quadratic with rational coefficients and one root ( x = − ) rational, forcingthe other root to be rational. Thus the point on the unit circle is ( x, y ) = (cid:18) mm + 1 , m − m + 1 (cid:19) . (1.3)As this is on the circle, its distance from the origin must be 1, and thus we see (2 m ) + ( m − = ( m + 1) . (1.4)As our slope m is a rational number, we may write m = p/q with gcd( p, q ) = 1 , and thus find (2 pq ) + ( p − q ) = ( p + q ) . (1.5)We can also argue in the opposite direction; given a point ( x, y ) on the circle with rational coordinates, theline from (0 , − to it has rational slope. Thus we see there is a 1-to-1 correspondence between points withrational coordinates and lines with rational slope m . (cid:4) Building on our successful analysis of the circle, we could look at other quadratic forms (such as ellipsesor hyperbolas). While there is a lot of interesting mathematics here, such as Pell’s equation for certainhyperbola’s, we instead move on to the next simplest polynomial to study: cubics.1.2.
Elliptic Curves Preliminaries.
There are many degree three equations we can study. An elliptic curveis given by y + a xy + a y = x + a x + a x + a , (1.6)with a , a , a , a , a ∈ Z ; we often impose some conditions so the curve is not degenerate. For example, ifwe had y = x ( x − we could send y xy and “reduce” to studying y = x − , a parabola. Instead ofhaving our coefficients in Z we could have them in a field; other common choices are the rationals Q as wellas the integers modulo p . The integers and rationals are of characteristic zero, while the integers modulo p are of characteristic p ( p is always a prime unless stated otherwise).When the elliptic curve is defined over field K whose characteristic is neither 2 nor 3, we can convert theequation to a more useful form, called the Weierstrass form, which is of the form y = f ( x ) for some degree3 polynomial f .Consider the elliptic curve E : y + a xy + a y = x + a x + a x + a . Define b = a + 4 a b = 2 a + a a b = a + 4 a b = a a + 4 a a − a a a + a a − a c = b − b c = − b + 36 b b − b . (1.7)The first simplification is that after some algebra we find our curve is equivalent to y = 4 x + b x +2 b x + b when char( K ) = 2 (we need to be able to invert 2 in the algebra). The second simplification gives us y = x − c x − c when char( k ) = 3 . Thus, frequently we study curves of the form y = x + ax + b .The discriminant of the elliptic curve y = x + ax + b is given by ∆ = − a + 27 b ) . The discriminantis nonzero when x + ax + b has three distinct roots. In order to avoid the cases when the curves degenerate,we impose the condition a + 27 b = 0 . In an elliptic function that has complex multiplication, the IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 5 endomorphism ring is isomorphic to a subring in an imaginary quadratic field. The j -invariant of an ellipticcurve is given by j ( E ) = 2 a a + 27 b .We let E ( Q ) denote the set of rational points on the elliptic curve E . Interestingly, this is not just a set, butalso a group. We sketch the proof.Let P = ( x , y ) and Q = ( x , y ) be two distinct points in E ( Q ) (if they are the same point we have tomodify the construction slightly). Write the line connecting them by y = αx + β ; as our points have rationalcoordinates, α and β are rational. The line intersects the elliptic curve at another point, unless the line istangent to the curve; see Figure 2. R ( P , Q ) P + QP Q F IGURE
2. The addition law of the elliptic curve.To find the third point where the line hits the cubic, we substitute for y with the equation of the line, andfind x + ax + b = ( αx + β ) . As this cubic has rational coefficients and two rational roots ( x = x or x ), the third root is also rational. Let ( x , y ) be this point. We define P + Q to be ( x , − y ) , which isthe reflection of ( x , y ) about the x -axis. With this definition, it turns out that the set of rational pointsforms a group. It is easy to see that it is a commutative group due to the geometric nature of the construction(the line through P and Q is the same as the line through Q and P ). Interestingly, what is hard is showingassociativity: ( P + Q ) + R = P + ( Q + R ) ; this can be done through tedious algebra, or by appealing tohigh level results (the Rieman-Roch Theorem).Now that we know E ( Q ) is a group, it is natural to ask what is its structure. By the Mordell-Weil Theoremwe can write it as Z r g ⊕ T E ; here r g is the geometric rank of the group, and T E is a finite torsion group.Mazur [Ma] proved that there are only 15 possibilities for the torsion group: if C k is the cyclic group with k elements, the possibilities are C n for n ∈ { , . . . , , } and the direct sum of C with C , C , C or C .We end with a brief remark about why we care about points on rational curves. For years RSA was thegold standard for encryption, and was based on working with ( Z /pq Z ) ∗ with p, q distinct primes. The hope,which was realized through elliptic curves, was that using a more complicated group could lead to moresecure encryption. It is thus an important question to know what types of groups can arise. To date thelargest observed rank is 28, and some recent conjectures suggest (contrary to the initial thoughts) that themaximum rank might be bounded. YAN (ROGER) WENG
Elliptic Curve L -function. We now introduce the L -function of an elliptic curve, and discuss itsconnections with the geometric rank. The basic idea is that we use local information to build a globalobject, and then use the global object to get information about the local inputs. We illustrate this with twowell-known example: Binet’s formula for the Fibonacci numbers, and the Riemann zeta function.1.3.1. Generating Functions and Fibonacci Numbers.
For the Fibonacci numbers F n +1 = F n + F n − (withinitial conditions F = 0 , F = 1 ), we use the recurrence relation to build a generating function, and thenuse the closed form expression of the generating function to get Binet’s formula for individual Fibonaccinumbers. Theorem 1.2 (Binet’s Formula) . We have F n = 1 √ " √ ! n − − √ ! n . (1.8) Proof.
We define the generating function by g ( x ) = X n> F n x n . (1.9)Using the recurrence relation and some algebra, we find X n ≥ F n +1 x n +1 = X n ≥ F n x n +1 + X n ≥ F n − x n +1 X n ≥ F n x n = X n ≥ F n x n +1 + X n ≥ F n x n +2 X n ≥ F n x n = x · X n ≥ F n x n + x · X n ≥ F n x n g ( x ) − F x − F x = x ( g ( x ) − F x ) + x g ( x ) g ( x ) = x − x − x . (1.10)By partial fraction expansion, g ( x ) = x − x − x = 1 √ " ( √ ) x − ( √ ) x − ( − √ ) x − ( − √ ) x , (1.11)and then using the geometric series formula to expand each denominator yields the coefficient of x n is F n = 1 √ " √ ! n − − √ ! n . (1.12) (cid:4) The reason we are able to obtain a closed form expression for the individual Fibonacci numbers is that wehave a closed form expression for the global object, the generating function. Before giving the elliptic curve L -function we first discuss the most basic example, the Riemann zeta function: ζ ( s ) := ∞ X n =1 n s , (1.13)which converges for Re( s ) > . IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 7
Riemann Zeta Function.
Theorem 1.3 (Euler product formula for the Riemann zeta function) . For
Re( s ) > we have ζ ( s ) = ∞ X n =1 n s = Y p prime (cid:18) − p s (cid:19) − . (1.14) Proof.
We sketch the argument, not worrying about convergence (which can be justified for such s ); fordetails see for example [MT-B]. Y p prime (cid:18) − p s (cid:19) − = (cid:18) − s (cid:19) − (cid:18) − s (cid:19) − (cid:18) − s (cid:19) − · · · = (cid:18) s + 14 s · · · (cid:19) (cid:18) s + 19 s · · · (cid:19) (cid:18) s + 125 s · · · (cid:19) · · · . (1.15)By the Fundamental Theorem of Arithmetic, every positive integer can be written uniquely as a product ofprime powers (with the primes in increasing order). Thus Y p prime (cid:18) − p s (cid:19) − = 1 + 12 s + 13 s + 14 s + 15 s · · · = ∞ X n =1 n s . (1.16) (cid:4) The Riemann Zeta Function is useful as connects the integers and the primes. For example, as we take thelimit as s approaches 1 the sum-definition of the Riemann zeta function tends to the harmonic series, whichdiverges. Because this must equal the Euler product, we conclude that there are infinitely many primes (asotherwise the product would stay finite). With a little more work, we can also show that X p /p divergesand even obtain a growth rate on X p ≤ x /p from knowing X n ≤ x /n is log( x ) plus lower order terms. Thus theprimes are far more numerous than the perfect squares (whose reciprocal sum converges). This is just thestart; by appealing to complex analysis (specifically contour integration) we obtain formulas connecting thenumber of primes up to x with the distribution of zeros and poles of ζ ( s ) (though initially defined just for Re( s ) > , ζ ( s ) can be continued to be analytic everywhere save for a pole of residue 1 at s = 1 ).1.3.3. Linear and Quadratic Legendre Sums.
Before we can define the elliptic curve L -function, we firstneed to introduce some terminology; we will also state some relations that play a central role in our studies. Definition 1.4 (Legendre symbol) . Let p be an odd prime number, and let a be an integer. We say a is aquadratic residue modulo p if it is congruent to a non-zero perfect square modulo p and is a non-quadraticresidue modulo p otherwise. We define the Legendre symbol (cid:18) ap (cid:19) by (cid:18) ap (cid:19) = if a is a quadratic residue modulo p − if a is a non-quadratic residue modulo p if a ≡ modulo p . (1.17)In the sums below, we write X x ( p ) to mean a sum over x modulo p : thus it is p − X x =0 . YAN (ROGER) WENG
Lemma 1.5 (Linear Legendre Sums) . We have X x ( p ) (cid:18) ax + bp (cid:19) = 0 (1.18) if p ∤ a ; if p | a the sum is just p (cid:18) bp (cid:19) . The proof is given in Appendix A.
Lemma 1.6 (Quadratic Legendre Sums) . Assume a p ) . Then X x ( p ) (cid:18) ax + bx + cp (cid:19) = − (cid:18) ap (cid:19) if p ∤ b − ac ( p − (cid:18) ap (cid:19) if p | b − ac . (1.19)The proof is given in Appendix A.Unfortunately, while we have great success with linear and quadratic legendre sums, there is no closed formsolution for cubic and higher (for more on these and related sums, see [BEW]). Our inability to handle inparticular cubic sums makes the analysis of elliptic curves exceedingly difficult.1.3.4. Elliptic Curve L -functions. Consider an elliptic curve E : y = x + ax + b . Let us look for pairs ( x, y ) that solve this modulo p . If x + ax + b is a non-zero square modulo p then there are two distinct y that work for this x , if it is zero then only one y works, while if it is not a square modulo p no y work. Thusthe number of solutions is X x ( p ) (cid:20)(cid:18) x + ax + bp (cid:19) + 1 (cid:21) = p + X x ( p ) (cid:18) x + ax + bp (cid:19) . (1.20)We define a E ( p ) to be p minus the number of solutions mod p; Thus a E ( p ) := − X x ( p ) (cid:18) x + ax + bp (cid:19) . (1.21)Note that if we assume the x + ax + b are as likely to be a quadratic residue as a non-residue, we wouldexpect the number of solutions to be around p . Thus a E ( p ) measures the fluctuations in the number ofsolutions from the expected number. Hasse proved that | a E ( p ) | < √ p . Remark 1.7 (Philosophy of Square-Root Cancellation) . The cancellation in Hasse’s theorem is an exampleof the
Philosophy of Square-Root Cancellation : if you have a sum of N signed terms all of order B , then weexpect the size of the sum to be of order B √ N (up to sat a few powers of log N ). Perhaps the most famousexample of this is the Central Limit Theorem: if we toss a fair coin N times and record 1 for each head and-1 for each tail, we expect the sum to be zero with fluctuations of size √ N . For elliptic curves, we have asum of p signed terms, each of size 1; the resulting sum is of size √ p . We now use this local data over the primes to build an elliptic curve L -function through an Euler product.We sketch the construction; in particular, we do not go into details on the conductor of the elliptic curve, N E . For our purposes it is enough to know that it is an integer and χ N E ( p ) = 0 if p | N E and 1 otherwise.The L -function of the elliptic curve E is L ( E, s ) = Y p (cid:18) − a E ( p ) p s + χ N E ( p ) p s − (cid:19) − . (1.22) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 9
The next conjecture, one of the seven Clay Millenial Problems, gives one of the most important applicationsof the above.
Conjecture 1.8 (Birch and Swinnerton-Dyer) . The Taylor expansion of L ( E, s ) at s = 1 has the form L ( E, s ) = c ( s − k + higher order terms , with c = 0 and k the geometric rank of the group of rationalsolutions E ( Q ) . In other words, the Birch and Swinnerton-Dyer Conjecture tells us that the order of vanishing of L ( E, s ) at s = 1 (the analytic rank) equals the geometric rank of the elliptic curve.1.4. Moments of the Dirichlet Coefficients of the Elliptic Curve L -functions. We call the a E ( p ) theDirichlet Coefficients of the elliptic curve L -function; if we have a one-parameter family we write a t ( p ) for a E t ( p ) . We study how these vary in a family.Consider the one-parameter family of elliptic curves over Q ( T ) given by E : y = x + a ( T ) x + b ( T ) where a ( T ) , b ( T ) are polynomials in Z [ T ] ; we often call E an elliptic surface. The r -th moment of theDirichlet coefficients is A r,E ( p ) = 1 p X t mod p a t ( p ) r . (1.23)1.4.1. First Moment and Rank.
The first moment is A ,E ( p ) = 1 p X t mod p a t ( p ) . (1.24) Theorem 1.9 (Rosen-Silverman) . For an elliptic surface E (a one-parameter family over Q ( T ) ), lim x →∞ x X p ≤ x A ,E ( p ) log pp = rank( E ) , (1.25) where rank( E ) is the rank of the group of rational points of E over Q ( T ) . The Rosen-Silverman theorem tells us that a bias in the first moment is responsible for the geometric rankof elliptic curves. Given how important the rank of an elliptic curve is, it is thus natural to study highermoments to see if there are biases there as well, and if so what they might tell us about elliptic curves.1.4.2.
Second Moment and the Bias Conjecture.
The second moment is A ,E ( p ) = 1 p X t mod p a t ( p ) . (1.26)Michel [Mic] proved that if the elliptic curve does not have j ( T ) constant then A ,E ( p ) = p + O ( p / ) , (1.27)where the lower order terms have size p / , p, p / and . We are using big-Oh notation: f ( x ) = O ( g ( x )) if for all x sufficiently large there is some C such that | f ( x ) | ≤ C | g ( x ) | .In his thesis Miller [Mi1, Mi2] studied several families where A ,E ( p ) could be computed in closed form.In all of these, the first lower term that did not average to zero had a negative average. He further showedthat such a negative bias has implications in the distribution of zeros of the elliptic curve L -function near thecentral point. In particular, it helped explain a small amount of the observed excess rank, thus highlightinghow important these terms are. He conjectured that this bias exists in every family, and so far this has beenobserved in every family studied to date [HKLM, MMRW, Mi4]. Conjecture 1.10 (Negative Bias Conjecture (Miller)) . The largest lower order term in the second momentexpansion of a one-parameter family that does not average to 0 has a negative average.
The main idea in previous work has been to look at the triple sum p X t ( p ) X x ( p ) X y ( p ) (cid:18) x + a ( t ) x + bp (cid:19)(cid:18) y + a ( t ) y + b ( t ) p (cid:19) . (1.28)There is no hope in executing the x or y sum first; the idea is to switch orders and sum over t . By carefullychoosing a ( t ) and b ( t ) one is able to get a closed form solution for this t -sum, and if one is fortunate theresulting x and y sums can also be done in closed form. This is typically the case if, after changing variablesin the sums, we have a quadratic in t sum. Then by our results on quadratic Legendre sums, its value isa function of the discriminant; here the discriminant will be a function of x and y . In many cases we cancarefully choose the polynomials so we can easily determine when the discriminant is zero modulo p , andthus we can evaluate the resulting x and y sums, and obtain closed form expressions for the second moment.Of course, it is possible that the bias is present because the families investigated using the above techniquesare clearly not generic. It is thus important to study all families of elliptic curves, and not just specialfamilies where the sums can be executed. All previous work has been focused on trying to find as general aspossible families where we can execute these sums in closed form. This has been done successfully for manyone and two-parameter families, see for example [ACFKKMMWWYY, HKLM, MMRW, Mi3, Mi4, Wu];note biases in two-parameter families were first studied by Wu [Wu] last year in her S.-T. Yau High SchoolScience project. This work continues that program by moving in two new directions.First, while we do look at some additional families, we give the first systematic investigation of one-parameter families of given rank. Thus most of the time we are unable to obtain closed form expressions, andinstead we have to try to determine if a bias exists or not. Second, we look at whether or not the biases per-sist to higher moments. This introduces a tremendous computational challenge, as the bias conjecture statesthat the first lower order term that does not average to zero has a negative average; if there is a larger lowerorder term that does average to zero, it will drown out this bias and make it impossible to see numerically.For example, we will show later the fourth moment is of size p with lower order terms of size p / , p and so on. Assume we have a family where there is a term of size p / but it averages to zero, and thereis a term of size p that has a negative average. If we look at the observed fourth moment minus the mainterm ( p ) and divide by p / , the contribution from the p / term will average to zero. Similar to ourcoin tossing example from before and the Central Limit Theorem, it will oscillate around zero and if wesum P such primes, the average will be of size √ P /P = 1 / √ P ; the √ P comes from summing P signedterms and assuming square-root cancellation, while the P in the denominator comes from taking an average.Meanwhile, the contribution from the lower order term of size p with a negative bias is hidden; since weare dividing by p / each term here will be of size / √ p , and thus its average will be of order √ P /P or / √ P .This creates tremendous challenges in our studies. We will investigate higher moments, and sometimes wewill still be able to see the bias. This can happen for example if there is no p / term, and we can thusdivide by p . While these are not proofs, these results indicate interesting phenomena that are worth furtherstudy. In other families we can at least try to show that the numerics suggest that the first lower order termaverages to zero. We can do this by looking at many averages over consecutive blocks. For example, ittakes a few days to compute the results for the first 1000 primes in a family. We can break into 20 blocksof 50 consecutive primes, and by the Central Limit Theorem each one of these blocks should be of size 0with fluctuations of size √ if it truly averages to zero. We can investigate if this happens, as well as lookand see if roughly half the blocks have a small positive and half a small negative average. Additionally, wecan look at the grand average of the first 1000 primes; if the first lower order term does average to zero we IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 11 expect to see something on the order of / √ ≈ . (thus anything two or three times this, positive ornegative, is suggestive but not a proof of averaging to zero).1.5. Summary of Results.
We have systematically investigated general one-parameter families. For somefamilies, we are able to find the second moments expansions by separating primes into different congruenceclasses, which suggests that there is often a closed-form polynomial expression. We are then able to provethe results mathematically. The following table summarizes the numerical results for the second moments’expansions of the families we studied; see Figure 3.F
IGURE
3. Systematic investigation for second moments sums.
Below is a summary of the first and second moments for some families where we are able to prove closedform expressions for the first two moments. We give the proofs in the next section; the arguments there arerepresentative of the ones needed for all the families.
Family: y = 4 x + ax + bx + c + dt : • First moment: A ,ε ( p ) = 0 . • Second moment: A ,ε ( p ) = ( p − p − p · (cid:0) − p (cid:1) − p · (cid:0) a − bp (cid:1) if a − b = 0 p − p + p ( p − (cid:0) − p (cid:1) otherwise. (1.29) Family: y = 4 x + (4 m + 1) x + n · tx : • First moment: A ,ε ( p ) = 0 . • Second moment: A ,ε ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 . (1.30) Family: y = x − t x + t : • First moment: A ,ε ( p ) = − p . • Second moment: A ,ε ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) p (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) . (1.31)For families that we are not able to find closed-form expressions, we calculated the average bias of thesecond moment sums for the first primes; see Figure 4.In addition, we studied biases in the fourth and sixth moments, the first time such investigations have beendone. We calculated the average bias of the fourth and sixth moment sums for the first primes; seeFigure 4. For some families, fluctuations of larger lower order terms that (we believe) average to zerodrown out biases in even smaller lower order terms, and unfortunately make it impossible to see the biasesnumerically. We can however gather data supporting that the first lower order term averages to zero forfamilies with lower ranks, and we will discuss the statistical tests needed to glean that from our observations.Through numeric computations for these families, we believe that the rank of the elliptic curves might playa role in determining the bias. Our data suggests the following (for details see Sections 2 and 3). • The rank and rank families have negative biases in their second moment sums. However, it islikely that higher rank families ( rank( E ( Q )) ≥ ) have positive biases. • For the fourth moment, we also believe that the rank and rank families have negative biases.We see this in some families; in others we see the presence of terms of size p / whose behavior isconsistent with their averaging to zero. Thus the data at least suggests a weaker version of the biasconjecture: the first lower order term does not have a positive bias for lower rank families. On theother hand, higher rank families ( rank( E ( Q )) ≥ ) appear to have positive biases. IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 13 F IGURE
4. Numerical data for the average biases of 2nd, 4th and 6th moments sums. • The sixth moment results are similar to the fourth moment. The results are consistent with either anegative bias, or a leading term (now of size p / ) averaging to zero for lower rank families. Ourdata also suggests that higher rank families have positive biases. • For the odd moments, the coefficients of the leading term vary with the primes. Our data suggeststhat the average value of the main term for the k + 1 -th moment is − C k +1 rank( E ( Q )) p k +1 , where C n is the n-th term of the Catalan numbers2. B IASES IN SECOND MOMENTS OF ELLIPTIC CURVE FAMILIES
In this section, we explore the existence of negative bias in second moments sums through numerical evi-dence for general elliptic curve families and computation of the bias in closed form for several new families.All previous research on the negative bias conjecture of second moment sums only studied specific, specialfamilies where the resulting sums could be done in closed form. Therefore, it is possible that the nega-tive bias conjecture does not apply to all elliptic curves, and is an artifact of looking at carefully chosen,non-generic families.2.1.
Systematic investigation for the second moments sums.
We have systematically investigated gen-eral one-parameter families. For some families we are able to find polynomial expressions for the secondmoment expansions by separating the primes into different congruence classes, which suggests that there is aclosed-form expression for some families (from previous work of Miller [Mi3] we know however that thereare families where the p / term arises, and these are not polynomials). The following table summarizes thenumerical results for the second moment expansions of the families we studied; see Figure 5. F IGURE
5. Systematic investigation for second moments sums.For all the families that we are not able to find the second moments expansions, we cannot find formulasfor primes in congruence classes modulo · · , which strongly suggests that there isn’t a closed-form polynomial for their second moment sums. We chose to study this modulus as many properties ofelliptic curves depend on powers of 2 and 3. Further, in the analysis above the largest modulus needed was
24 = 2 · , suggesting that · · is a reasonably safe choice.In all the cases above where the numerics suggest closed-forms answers, we are able to prove it mathemati-cally.We now turn from numerical experimentation to theoretical calculation, and discuss some representativecalculations of ours. IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 15
First and second moments of the family y = 4 x + ax + bx + c + dt .Lemma 2.1. The first moment of the family y = 4 x + ax + bx + c + dt is 0.Proof. For all p > d , send t to d − t : Thus X t ( p ) (cid:18) dtp (cid:19) = X t ( p ) (cid:18) tp (cid:19) . (2.32)Therefore, A ,ε ( p ) = − X t ( p ) X x ( p ) (cid:18) x + ax + bx + 4 t + cp (cid:19) = − X x ( p ) X t ( p ) (cid:18) t + 4 x + ax + bx + cp (cid:19) . (2.33)As p when p = 2 , the t-sum vanishes by linear sum theorem, and A ,ε ( p ) = 0 . (2.34)By the Rosen-Silverman Theorem, this is a rank 0 family. (cid:4) Lemma 2.2.
The second moment of the family y = 4 x + ax + bx + c + dt is A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) a − bp (cid:19) if a − b = 0 p − p + p ( p − (cid:18) − p (cid:19) otherwise. (2.35) Proof.
We have A ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) x + ax + bx + 4 t + cp (cid:19)(cid:18) y + ay + by + 4 t + cp (cid:19) m ( x ) = 4 x + ax + bx + cn ( y ) = 4 y + ay + by + cA ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) t + 4( m + n ) t + mnp (cid:19) . (2.36)The discriminant of t + 4( m + n ) t + mn is ∆ t ( x, y ) = 16( m + n ) − mn = 16( m − n ) δ = ∆ t ( x, y ) δ = 4( m − n )= 4(4 x + ax + bx + c − y − ay − by − c )= 4( x − y )(4 x + 4 xy + 4 y + ax + ay + b ) . (2.37)If p | δ , then p | x − y or p | x +4 xy +4 y + ax + ay + b . x, y range from to p − , so p | x − y exactly p times. Bythe quadratic formula mod p , x + 4 xy + 4 y + ax + ay + b ≡ y + (4 x + a ) y + 4 x + ax + b ≡ p ) when y = − x − a ± p ∆ y ( ∆ y is the discriminant of the polynomial y + (4 x + a ) y + 4 x + ax + b interms of y): ∆ y = (4 x + a ) − · x + ax + b )= − x − ax + a − b. (2.38)If ∆ y is a non-zero square mod p , there are two solutions. If ∆ y is 0 mod p , there is one solution. If ∆ y isnot a square mod p , there is no solution.The number of pairs of x, y such that p | x + 4 xy + 4 y + ax + ay + b is X x ( p ) (cid:18) − x − ax + a − bp (cid:19) = p + X x ( p ) (cid:18) − x − ax + a − bp (cid:19) . (2.39)The discriminant of − x − ax + a − b is ∆ x = (8 a ) − · ( − a − b )= 256 a − b = 256( a − b ) . (2.40)We break into cases, depending on the value of the discriminant. Case 1: a − b = 0 : By the Quadratic Legendre Sum Theorem, if p a − b ) , X x ( p ) (cid:18) − x − ax + a − bp (cid:19) = − (cid:18) − p (cid:19) . (2.41)The number of pairs of x, y such that p | x + 4 xy + 4 y + ax + ay + b is p + X x ( p ) (cid:18) − x − ax + a − bp (cid:19) = p − (cid:18) − p (cid:19) . (2.42)The cases that we double count x = y and p | x + 4 xy + 4 y + ax + ay + b is x + 4 xy + 4 y + ax + ay + b ≡ y + 2 ay + b ≡ p ) . (2.43)The discriminant of y + 2 ay + b is ∆ y = (2 a ) − · b = 2 · ( a − b ) . (2.44)By the quadratic formula mod p , the number of solutions is computable, depending on a − b .The number of solutions to (2.43) is (cid:18) · ( a − b ) p (cid:19) = 1 + (cid:18) a − bp (cid:19) . (2.45)Therefore, the total number of times that p | ( x − y )(4 x + 4 xy + 4 y + ax + ay + b ) is the number of times p | x − y plus the number of times p | x + 4 xy + 4 y + ax + ay + b minus the cases that we double count.The number of times that p | ( x − y )(4 x + 4 xy + 4 y + ax + ay + b ) is p + p − (cid:18) − p (cid:19) − − (cid:18) a − bp (cid:19) = 2 p − − (cid:18) − p (cid:19) − (cid:18) a − bp (cid:19) . (2.46) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 17
The number of times that p ( x − y )(4 x + 4 xy + 4 y + x + y − is p − (cid:18) p − − (cid:18) − p (cid:19) − (cid:18) a − bp (cid:19)(cid:19) = p − p + 1 + (cid:18) − p (cid:19) + (cid:18) a − bp (cid:19) . (2.47)By Quadratic Legendre Sum Theorem, A ,E ( p ) = ( p − (cid:20) p − − (cid:18) − p (cid:19) − (cid:18) a − bp (cid:19)(cid:21) − (cid:20) p − p + 1 + (cid:18) − p (cid:19) + (cid:18) a − bp (cid:19)(cid:21) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) a − bp (cid:19) . (2.48) Case 2: a − b = 0 : By the Quadratic Legendre Sum Theorem, since p | , we have X x ( p ) (cid:18) − x − ax + a − bp (cid:19) = ( p − (cid:18) − p (cid:19) . (2.49)The number of pairs of x, y such that p | x + 4 xy + 4 y + ax + ay + b is p + X x ( p ) (cid:18) − x − ax + a − bp (cid:19) = p + ( p − (cid:18) − p (cid:19) . (2.50)The cases that we double count x = y and p | x + 4 xy + 4 y + ax + ay + b is x + 4 xy + 4 y + ax + ay + b ≡ y + 2 ay + b ≡ p ) . (2.51)The discriminant of y + 2 ay + b is ∆ y = (2 a ) − · b = 2 · ( a − b ) . (2.52)By the quadratic formula mod p , the number of solutions is computable, depending on a − b .The number of solutions to (2.51) is (cid:18) · ( a − b ) p (cid:19) = 1 . (2.53)Therefore, the total number of times that p | ( x − y )(4 x + 4 xy + 4 y + ax + ay + b ) is the number of times p | x − y plus the number of times p | x + 4 xy + 4 y + ax + ay + b minus the cases that we double count.The number of times that p | ( x − y )(4 x + 4 xy + 4 y + ax + ay + b ) is p + p + ( p − (cid:18) − p (cid:19) − p − p − (cid:18) − p (cid:19) . (2.54)The number of times that p ( x − y )(4 x + 4 xy + 4 y + x + y − is p − (cid:18) p − p − (cid:18) − p (cid:19)(cid:19) = p − p + 1 − ( p − (cid:18) − p (cid:19) . (2.55) A ,E ( p ) = ( p − (cid:20) p − p − (cid:18) − p (cid:19)(cid:21) − (cid:20) p − p + 1 − ( p − (cid:18) − p (cid:19)(cid:21) = p − p + p ( p − (cid:18) − p (cid:19) . (2.56) Thus we have shown A ,E ( p ) = ( p − p − p · (cid:0) − p (cid:1) − p · (cid:0) a − bp (cid:1) if a − b = 0 p − p + p ( p − (cid:0) − p (cid:1) otherwise. (2.57) (cid:4) Lemma 2.3.
When a − b is a non-zero square, the second moment of the family y = 4 x + ax + bx + c + dt is A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) (2.58)Nine of the families that we investigated through numerical computations are special cases of this ellip-tic curve family.2.2.1. y + xy = x − x + t . ( a = 1 a = 0 a = 0 a = − a = t ). y = 4 x + x − x + 4 t.a − b = 1 − · ( −
4) = 7 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.59)2.2.2. y + xy − y = x + x + t . ( a = 1 a = 0 a = − a = 1 a = t ). y = 4 x + x + 4 t + 4 .a − b = 1 − · (0) = 1 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.60)2.2.3. y + xy + y = x − x + t . ( a = 1 a = 0 a = 1 a = − a = t ). y = 4 x + x − x + 4 t + 1 .a − b = 1 − · ( −
2) = 5 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.61)2.2.4. y + xy − y = x + x + x + t . ( a = 1 a = 1 a = − a = 1 a = t ). y = 4 x + 5 x + 2 x + 4 t + 1 .a − b = 5 − · (2) = 1 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.62) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 19 y + xy − y = x + x + x + t . ( a = 1 a = 1 a = − a = 1 a = t ). y = 4 x + 5 x − x + 4 t + 9 .a − b = 5 − · ( −
2) = 7 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.63)2.2.6. y + xy − y = x + x + t . ( a = 1 a = 0 a = − a = 1 a = t ). y = 4 x + x − x + 4 t + 9 .a − b = 1 − · ( −
2) = 5 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.64)2.2.7. y + xy − y = x + x + x + t . ( a = 1 a = 1 a = − a = 1 a = t ). y = 4 x + 5 x + 4 t + 4 .a − b = 5 − · (0) = 5 .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) . (2.65)2.2.8. y + y = x + x + x + t . ( a = 0 a = 1 a = 1 a = 1 a = t ). y = 4 x + 4 x + 4 x + 4 t + 1 .a − b = 4 − · (4) = − .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) − p (cid:19) . (2.66)2.2.9. y + 3 y = x + x + x + t . ( a = 0 a = 1 a = 3 a = 1 a = t ). y = 4 x + 4 x + 4 x + 4 t + 9 .a − b = 4 − · (4) = − .A ,ε ( p ) = 0 .A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) − p (cid:19) . (2.67)2.3. First and second moments of the family y = 4 x + (4 m + 1) x + n · tx .Lemma 2.4. The first moment of the family y = 4 x + (4 m + 1) x + n · tx is 0. Proof.
For all p > n , send t to n − t . We have X t ( p ) (cid:18) ntp (cid:19) = X t ( p ) (cid:18) tp (cid:19) .A ,ε ( p ) = − X t ( p ) X x ( p ) (cid:18) x + (4 m + 1) x + 4 txp (cid:19) = − p − X x =1 X t ( p ) (cid:18) xt + 4 x + (4 m + 1) x p (cid:19) . (2.68)As p x when p = 2 , the t -sum vanishes by the linear Legendre sum theorem, and hence A ,ε ( p ) = 0 (2.69)as claimed. (cid:4) Lemma 2.5.
The second moment of the family y = 4 x + (4 m + 1) x + n · tx is A ,E ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 . (2.70) Proof. A ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) x + (4 m + 1) x + 4 txp (cid:19)(cid:18) y + (4 m + 1) y + 4 typ (cid:19) = X t ( p ) X x ( p ) X y ( p ) (cid:18) xyp (cid:19) · (cid:18) t + 4[4 x + (4 m + 1) x + 4 y + (4 m + 1) y ] t + [4 x + (4 m + 1) x ][4 y + (4 m + 1) y ] p (cid:19) .a ( x ) = 4 x + (4 m + 1) x.b ( y ) = 4 y + (4 m + 1) y.A ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) xy · t + 4 xy ( a + b ) · t + xy · abp (cid:19) . (2.71)We look at the discriminant of our polynomial in t : ∆ = 16 x y [( a + b ) − ab ]= 16 x y ( a − b ) . (2.72) δ = ∆ δ = 4 xy ( a − b )= 4 xy [4 x + (4 m + 1) x − y − (4 m + 1) y ]= 4 xy ( x − y )(4 x + 4 y + 4 m + 1) . (2.73)If p | δ , then xy = 0 or x = y or p | x + 4 y + 4 m + 1 . We can ignore the cases when xy = 0 because (cid:18) xyp (cid:19) = 0 , so it does not contribute to the second moment sum. IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 21
We need to count the number of pairs of x, y ranging from 0 to p − such that p | x + 4 y + 4 m + 1 . Thuswe must consider choices where ≤ x + 4 y + 1 ≤ p − .For p > m , ≤ x + 4 y + 4 m + 1 < p .If p | x + 4 y + 4 m + 1 , then x + 4 y + 4 m + 1 = 0 , p, p, ..., p, p .There is no solution for x + y + m ) + 1 = 0 , p, p, p, p . If x + y + m ) + 1 = p , Case 1: p = 4 k + 1 : We must solve x + y + m = k ( x = 0 y = k − m , ..., ( x = 0 y = k − m. (2.74)There are k − m + 1 solutions.If k and m are both even or both odd, there is one solution such that x = y . Case 2: p = 4 k + 3 : Nosolution. If x + y + m ) + 1 = 3 p , Case 1: p = 4 k + 1 : We have x + y + m ) = 12 k + 2 : no solution.Case 2: p = 4 k + 3 : We have to solve x + y + m = 3 k + 2 ( x = 0 y = 3 k + 2 − m , ... , ( x = 3 k + 2 − my = 0 . (2.75)There are k + 3 − m solutions.If k and m are both even or both odd, there is one solution such that x = y . If x + y + m ) + 1 = 5 p , Case 1: p = 4 k + 1 : We have to solve x + y + m = 5 k + 1( x, y < k + 1) (2.76) ( x = k + 1 − my = 4 k , ..., ( x = 4 ky = k + 1 − m. (2.77)There are k + m solutions.If k and m have opposite parity, there is one solution such that x = y .Case 2: p = 4 k + 3 : x + y + m ) = 20 k + 14 : no solution If x + y + m ) + 1 = 7 p , Case 1: p = 4 k + 1 : x + y + m ) = 28 k + 6 : no solution Case 2: p = 4 k + 3 : we have to solve x + y + m = 7 k + 5( x, y < k + 3) (2.78) ( x = 3 k + 3 − my = 4 k + 2 , ... , ( x = 4 k + 2 y = 3 k + 3 − m. (2.79)There are k + m solutions.If k and m have opposite parity, there is one solution such that x = y .Combining the above, the total number of solutions is • if p = 4 k + 1 : ( k − m + 1) + 3 k + m = 4 k + 1 = p , • if p = 4 k + 3 : (3 k + 3 − m ) + k + m = 4 k + 3 = p .There are always p times that p | x + 4 y + 4 m + 1 when ≤ x, y ≤ p − .There is always exactly one time such that p | x + 4 y + 4 m + 1 and x = y . Consider p = 4 k + 1 : Notice that the sum of x and y for each solution is k − m or p + k − m Suppose that y = k − m − x .By the Quadratic Legendre sum theorem, the contribution of a prime p | x + y ) + 1 to the sum is ( p − (cid:18) xyp (cid:19) . Thus we have ( p − · X x ( p ) (cid:18) x · ( k − m − x ) p (cid:19) = ( p − · X x ( p ) (cid:18) − x + ( k − m ) x ) p (cid:19) = ( p − · (cid:18) − (cid:18) − p (cid:19)(cid:19) . (2.80)As p = 4 k + 1 , (cid:18) − p (cid:19) = − , we find ( p − · X x ( p ) (cid:18) x · ( k − m − x ) p (cid:19) = p − . (2.81)There is always one time such that p | x + 4 y + 4 m + 1 and x = y , so there are p − times such that x = y , p x + 4 y + 4 m + 1 .The contribution of the cases x = y to the sum is ( p − · ( p − (cid:18) xyp (cid:19) = ( p − · ( p − (cid:18) x p (cid:19) = ( p − p − . (2.82)We know that X x ( p ) X y ( p ) (cid:18) xyp (cid:19) = 0 .For all the cases such that p | xy ( x − y )(4 x + 4 y + 4 m + 1) , the sum of (cid:18) xyp (cid:19) is p −
2) = p − . IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 23
Therefore, for all the cases such that p xy ( x − y )(4 x +4 y +4 m +1) , the sum of (cid:18) xyp (cid:19) is − ( p −
1) = 1 − p .The contribution of all the cases such that p xy ( x − y )(4 x + 4 y + 4 m + 1) to the second moment sum is − (cid:18) xyp (cid:19) , which is p − .For p = 4 k + 1 , A ,E ( p ) = p − p − · ( p −
1) + p − p − p. (2.83) Consider now p = 4 k + 3 . Notice that the sum of x and y for each solution is k + 2 − m or p + 3 k + 2 − m .Suppose that y = 3 k + 2 − m − x .By the Quadratic Legendre sum theorem, the contribution of a prime p | x + y + m ) + 1 to the sum is ( p − (cid:18) xyp (cid:19) = ( p − · X x ( p ) (cid:18) x · (3 k + 2 − m − x ) p (cid:19) = ( p − · X x ( p ) (cid:18) − x + (3 k + 2 − m ) xp (cid:19) = ( p − · (cid:18) − (cid:18) − p (cid:19)(cid:19) . (2.84)As p = 4 k + 3 , we have (cid:18) − p (cid:19) = 1 . ( p − · X x ( p ) (cid:18) x · (3 k + 2 − m − x ) p (cid:19) = 1 − p. (2.85)There is always one time such that p | x + 4 y + 4 m + 1 and x = y , so there are p − times such that x = y , p x + 4 y + 4 m + 1 .The contribution of the cases x = y to the sum is ( p − · ( p − (cid:18) xyp (cid:19) = ( p − · ( p − (cid:18) x p (cid:19) = ( p − p − . (2.86)We know that X x ( p ) X y ( p ) (cid:18) xyp (cid:19) = 0 .For all the cases such that p | xy ( x − y )(4 x + 4 y + 4 m + 1) , the sum of (cid:18) xyp (cid:19) is − p −
2) = p − .Therefore, for all the cases such that p xy ( x − y )(4 x +4 y +4 m +1) , the sum of (cid:18) xyp (cid:19) is − ( p −
3) = 3 − p .The contribution of all the cases such that p xy ( x − y )(4 x + 4 y + 4 m + 1) to the second moment sum isthe sum of − (cid:18) xyp (cid:19) , which is p − .For p = 4 k + 1 , A ,E ( p ) = 1 − p + ( p − · ( p −
1) + p − p − p. (2.87) Therefore, the above analysis has shown that A ,E ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 . (2.88) (cid:4) Three of the families that we investigated through numerical computations are special cases of this ellipticcurves family.2.3.1. y + xy = x + tx . ( a = 1 a = 0 a = 0 a = t a = 0 ). y = 4 x + x + 4 tx. (2.89) A ,E ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 (2.90)2.3.2. y + xy = x − x + tx . ( a = 1 a = − a = 0 a = t a = 0 ). We first put in Weierstrassform: y = 4 x − x + 4 tx. (2.91)We have A ,E ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 . (2.92)2.3.3. y + xy = x + x + tx . ( a = 1 a = 1 a = 0 a = t a = 0 ). Changing variables to have it inWeierstrass form gives y = 4 x + 5 x + 4 tx. (2.93)We find A ,E ( p ) = ( p − p if p = 4 k + 1 p − p if p = 4 k + 3 . (2.94)2.4. First and second moments of the family y = x − t x + t . ( a = 0 a = 0 a = 0 a = − t a = t ) Lemma 2.6.
The first moment of the family y = x − t x + t is − p . IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 25
Proof.
We have A ,ε ( p ) = − X t ( p ) X x ( p ) (cid:18) x − t x + t p (cid:19) = − X t ( p ) (cid:18) t p (cid:19) − X t ( p ) p − X x =1 (cid:18) x − t x + t p (cid:19) = − p + 1 − X t ( p ) p − X x =1 (cid:18) t x − t x + t p (cid:19) = − p + 1 − X t ( p ) p − X x =1 (cid:18) t p (cid:19)(cid:18) t + t ( x − x ) p (cid:19) = − p + 1 − p − X x =1 X t ( p ) (cid:18) t + t ( x − x ) p (cid:19) . (2.95)We compute the discriminant of the polynomial in t : ∆ = ( x − x ) = [( x − x ( x + 1)] (2.96)Note p | ∆ when x = 1 or p − , and p ∆ when x = 2 , , . . . , p − .By the Quadratic Legendre Sum Theorem, A ,ε ( p ) = − p + 1 − p −
1) + ( p −
3) = − p. (2.97) (cid:4) Lemma 2.7.
The second moment of the family y = x − t x + t is A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) p (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) (2.98) Proof.
We have A ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) x − t x + t p (cid:19)(cid:18) y − t y + t p (cid:19) = X t ( p ) X x ( p ) X y ( p ) (cid:18) t x − t x + t p (cid:19)(cid:18) t y − t y + t p (cid:19) = X t ( p ) X x ( p ) X y ( p ) (cid:18) t p (cid:19)(cid:18) x − x + tp (cid:19)(cid:18) y − y + tp (cid:19) = X t ( p ) X x ( p ) X y ( p ) (cid:18) t + ( x − x + y − y ) t + ( x − x )( y − y ) p (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) a ( x ) = x − xb ( y ) = y − yA ,E ( p ) = X t ( p ) X x ( p ) X y ( p ) (cid:18) t + ( a + b ) t + abp (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) . (2.99) The discriminant of t + ( a + b ) t + ab is ∆ t = ( a + b ) − ab = ( a − b ) (2.100) δ = ∆ t δ = ( a − b )= x − x − y + y = ( x − y )( x + xy + y − (2.101)If p | δ , then p | x − y or p | x + xy + y − x, y range from 0 to p-1, so p | x − y p times.When p | x + xy + y − ,By quadratic formula mod p , y + xy + x − ≡ p ) when y = − x ± p ∆ y ( ∆ y is the discriminant of the polynomial y + xy + x − in terms of y) ∆ y = x − · ( x − − x + 4 (2.102)If ∆ y is a non-zero square mod p , there are two solutions. If ∆ y is 0 mod p , there is one solution. If ∆ y isnot a square mod p , there is no solution.The number of pairs of x, y such that p | y + xy + x − is X x ( p ) (cid:18) − x + 4 p (cid:19) = p + X x ( p ) (cid:18) − x + 4 p (cid:19) . (2.103)The discriminant of − x + 4 is ∆ x = 0 − − ·
4= 48 (2.104)By the Quadratic Legendre Sum Theorem, if p , , (cid:18) − x + 4 p (cid:19) = − (cid:18) − p (cid:19) (2.105)The number of pairs of x, y such that p | y + xy + x − is p + X x ( p ) (cid:18) − x + 4 p (cid:19) = p − (cid:18) − p (cid:19) . (2.106)We need the number of cases that we double count x = y and p | y + xy + x − . If x = y , y + xy + x − ≡ y − ≡ p ) . (2.107)The discriminant of y − is ∆ y = 0 − · · ( − . (2.108)By the quadratic Legendre sum formula mod p , the number of solutions is (cid:18) p (cid:19) . Therefore, the totalnumber of times that p | ( x − y )( y + xy + x − is the number of times p | x − y plus the number of times IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 27 p | y + xy + x − minus the cases that we double count. p + p − (cid:18) − p (cid:19) − − (cid:18) p (cid:19) = 2 p − − (cid:18) − p (cid:19) − (cid:18) p (cid:19) . (2.109)The number of times that p ( x − y )(4 x + 4 xy + 4 y + x + y − is p − (2 p − − (cid:18) − p (cid:19) − (cid:18) p (cid:19) ) .Again by Quadratic Legendre Sum Theorem, A ,E ( p ) = ( p − (cid:20) p − − (cid:18) − p (cid:19) − (cid:18) p (cid:19)(cid:21) − (cid:20) p − (2 p − − (cid:18) − p (cid:19) − (cid:18) p (cid:19) ) (cid:21) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) p (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) . (2.110)Thus X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) = X x ( p ) (cid:18) x − xp (cid:19) X y ( p ) (cid:18) y − yp (cid:19) = X x ( p ) (cid:18) x − xp (cid:19) X y ( p ) (cid:18) ( y − y ( y + 1) p (cid:19) . (2.111)We know that (cid:18) − p (cid:19) = ( − p − .When p = 4 k + 3 , (cid:18) − p (cid:19) = − X y ( p ) (cid:18) ( y − y ( y + 1) p (cid:19) = k X y =2 (cid:18) ( y − y ( y + 1) p (cid:19) + k − X y =2 k +1 (cid:18) ( y − y ( y + 1) p (cid:19) = k X y =2 (cid:18) ( y − y ( y + 1) p (cid:19) + (cid:18) − p (cid:19)(cid:18) ( y − y ( y + 1) p (cid:19) = 0 . (2.112)In conclusion, A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) p (cid:19) − X x ( p ) X y ( p ) (cid:18) x − xp (cid:19)(cid:18) y − yp (cid:19) . (2.113)When p ≡ , A ,E ( p ) = p − p − p · (cid:18) − p (cid:19) − p · (cid:18) p (cid:19) . (2.114) (cid:4) Numerical data for second moment sums.
Now we report on families where we cannot find closed-form polynomials for their second moments sums. These are more generic families than the ones investi-gated both above and in previous work, and provide a new and stronger test of the bias conjecture.For these families, we have calculated the second moment sums for the first primes. By Michel’stheorem, we know that the main term of the sum is p , and lower order terms have size p / , p, p / or . From the data we have, we can tell if it is likely that the second moment has a p / term. If the value ofsecond moment − p p converges or stays bounded as the prime grows, then it is likely that the largest lowerorder term of the second moment sum is p , as if there were a p / term we would have fluctuations of size p / .By subtracting the main term ( p ) from the sum and then dividing by the largest lower term ( p / or p ), wecalculated the average bias; see Figure 6.F IGURE
6. Numerical data for the average biases of second moments sums.The data shows that all the families where we do not believe there is a p / term clearly have negative biases(around − ). When the p / exists, the bias unfortunately becomes impossible to see. The reason is that the p / term drowns it out; we now have to divide by p / . If that term averages to zero, then the term of size p , once we divide by p / , is of size /p / .Let’s investigate further the consequence of having a term of size p / . We divide the difference of theobserved second moment minus p (the expected value) by p / . We now have signed summands of size 1.By the Philosophy of Square-Root Cancellation, if we sum N such signed terms we expect a sum of size √ N . As we are computing the average of these second moments, we divide by N and have an expectedvalue of order / √ N . In other words, if the p / term is present and averages to zero, we expect sums overranges of primes to be about / √ N . If N = 1000 this means we expect sums on the order of .0316. Lookingat the data in Figure 6, what we see is consistent with this analysis. Thus, while we cannot determine if the IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 29 first lower order term that does not average to zero has a negative bias, we can at least show that the data isconsistent with the p / term averaging to zero for lower rank families.The table suggests a lot more. From the data, we can see that all the rank and rank families have negativebiases. However, all four rank families have shown positive biases from the first 1000 primes. Thus,we look further to see if it is likely the result of fluctuations, or if perhaps it is evidence against the biasconjecture.We divide the primes into groups of for further analysis. If the p / term averages to zero, wewould expect each of these groups to be positive and negative equally likely, and we can compare counts.We now expect each group to be on the order of / √ ≈ . . Thus we shouldn’t be surprised if it is a fewtimes .14 (positive or negative); remember we do not know the constant factor in the p / term and are justdoing estimates.For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown positive biases. Figure 7 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
7. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .We now further analyze our data by dividing the 1000 primes into 100 groups of 10 for this family. Asshown from the data, of the groups of primes have shown positive biases; however, this is stillconsistent with a term that averages to zero by the Philosophy of Square-Root Cancelation or the CentralLimit Theorem (if we have 100 outcomes that are positive half the time and negative half the time, theexpected number of positive outcomes is 50 and the standard deviation is 5; thus having 58 positive and 42negative groups is well-within two standard deviations). Figure 8 is a histogram plot of the distribution ofthe average biases among the groups. These numbers are consistent with the p / term averaging tozero.For the rank family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t , of the groups of primes haveshown positive biases. Figure 9 is a histogram plot of the distribution of the average biases among the 20groups. Again the results are consistent with the p / term averaging to zero.From the primes that we have analyzed, the two families above appear to have positive biases more fre-quently than negative biases, but not by a statistically significant margin . Now we compare them to somefamilies that have shown negative biases in the first 1000 primes.For the rank family a = 1 , a = 0 , a = 0 , a = 1 , a = t , all of the groups of primes have F IGURE
8. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .F IGURE
9. Distribution of average biases in the first 1000 primes for family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t .shown negative biases. Figure 10 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
10. Distribution of average biases in the first 1000 primes for family a = 1 , a = 0 , a = 0 , a = 1 , a = t .We now further analyze our data by dividing the 1000 primes into 100 groups of 10 for this family. Asshown by the data, all of the groups of primes have shown negative biases, which strongly indicates that IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 31 the negative bias exists in this family. Figure 11 is a histogram plot of the distribution of the average biasesamong the groups.F
IGURE
11. Distribution of average biases in the first 1000 primes for family a = 1 , a = 0 , a = 0 , a = 1 , a = t For the rank family a = 1 , a = 1 , a = − , a = t , a = 0 , of the groups of primes have shownnegative biases. Figure 12 is a histogram plot of the distribution of the average biases among the 20 groups.F IGURE
12. Distribution of average biases in the first 1000 primes for family a = 1 , a = 1 , a = − , a = t , a = 0 .For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown negative biases. Figure 13 is a histogram plot of the distribution of the average biases among the 20groups.From these data, the rank and rank families have negative biases more frequently, but we are workingwith small data sets and must be careful in how much weight we assign such results. In the rank family, all the groups of primes have negative biases. In both of the rank families, of the groups of theprimes have negative biases, though this percentage is not statistically significant. On the other hand, two ofthe three rank families appear to have positive biases in the first primes, but again this value is notstatistically significant. We believe that the rank of the families might play a role in determining the bias.Therefore, it is possible that the negative bias conjecture does not hold for some families with larger rank.To further examine the biases in families with larger ranks, we investigate the rank 6 family a = 0 , a =2(16660111104 t )+811365140824616222208 , a = 0 , a = [2( − t − t +2 t − , a = [2(2149908480000) t + 343107594345448813363200]( t + 2 t − . Our data suggests that there is a positive bias in this family. The average F IGURE
13. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 bias of second moments sums for the first primes is . . Figure 14 is a histogram plot of thedistribution of the average biases among the groups of primes. of the groups of primes havepositive biases, which suggests that it is likely that the second moment of this family has a positive p / term. F IGURE
14. Distribution of average biases in the first 1000 primes for a rank 6 family.Therefore, our data for the rank and rank families has shown that it is likely that higher rank families( rank( E ( Q )) ≥ ) have positive biases.3. B IASES IN FOURTH AND SIX MOMENTS OF ELLIPTIC CURVE FAMILIES
We now explore, for the first time, the higher moments of the Dirichlet coefficients of the elliptic curve L -functions to see if biases we found in the first and second moments persist. Unfortunately existing techniqueson analyzing the second moment sums do not apply to the higher moments, even if we choose nice families.If we switch orders of the moments’ sums and sum over t , we are going to get a cubic or higher degreespolynomials. Therefore, we can only try to predict or observe the biases through numerical evidence. Wecalculated the 4th and 6th moment sums for the first primes. From Section B.2, we know that themain term of the fourth moment sum is p , and the largest possible lower order terms have size p / . FromSection B.3, we know that the main term of the sixth moment sum is p , and the largest possible lowerorder terms have size p / . From the data we have gathered, all the 4th moments of these families have p / terms, and all the 6th moments have p / terms. By subtracting the main term ( p ) from the fourth momentsum and then dividing by the size of the largest lower term ( p / ), we calculated the average bias for the IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 33 fourth moment of the first primes. Similarly, we subtracted p from the sixth moment sum and thendivided by p / to calculate the average bias for the sixth moment of the first primes; See Figure 15.F IGURE
15. Numerical data for the average biases of 2nd, 4th and 6th moments sums.3.1.
Biases in fourth moment sums.
From the data, we can see that all the biases for lower rank families inthe fourth moment are relatively small (smaller than . ), which indicates that the p / term likely averagesto . By the Philosophy of Square-Root Cancellation, we expect the order of the size of fluctuation to bearound √ / ≈ . . Therefore, if the bias is between − . and . , we would expect p to be thelargest lower order term.Note that for out of families, the bias in fourth moments appear to be similar to the bias in secondmoments (families that have negative bias in second moments also seem to have negative bias in fourthmoments, and vice versa), though much smaller magnitudes likely due to the presence of a p / term thatis averaging to zero. We now explore a few families whose 4-th moment biases have different scales inmagnitudes.For the rank family a = 1 , a = 0 , a = 0 , a = 2 , a = t , of the groups of primes have shownnegative biases. Figure 16 is a histogram plot of the distribution of the average biases among the 20 groups.For the rank family a = 1 , a = 1 , a = 1 , a = 1 , a = t , of the groups of primes have shownnegative biases. Figure 17 is a histogram plot of the distribution of the average biases among the 20 groups.For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown negative biases. Figure 18 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
16. Distribution of average biases in the first 1000 primes for family a = 1 , a = 0 , a = 0 , a = 2 , a = t .F IGURE
17. Distribution of average biases in the first 1000 primes for family a = 1 , a = 1 , a = 1 , a = 1 , a = t .F IGURE
18. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .We further analyze our data by dividing the 1000 primes into 100 groups of 10 for this family. As shownin Figure 19, of the groups of primes have shown negative biases. The probability of having 17 or IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 35 more negatives than positives (or 17 or more positives than negatives) in 100 tosses of a fair coin (so headsis positive and tails is negative) is about 1.2%. While unlikely, this is not exceptionally unlikely.F
IGURE
19. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown negative biases. Figure 20 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
20. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .We further analyzes our data by dividing the 1000 primes into 100 groups of 10 for this family. As shownin Figure 21, of the groups of primes have shown negative biases.For the rank family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t , of the groups of primes haveshown negative biases. Figure 22 is a histogram plot of the distribution of the average biases among the 20groups.Despite the fluctuations, all the rank and rank families seem to have negative biases more frequently inthe first primes, which suggests that it is possible that negative bias exists in the fourth moments of allrank and rank families. Similar to the second moment sums, the fourth moment sums of families withlarger rank appear to have positive biases for the first 1000 primes, but this might due to the fluctuations ofthe p / term as we are working with small data set.To further examine the biases in families with larger ranks, we investigate the rank 6 family a = 0 , a =2(16660111104 t )+811365140824616222208 , a = 0 , a = [2( − t − t + F IGURE
21. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .F IGURE
22. Distribution of average biases in the first 1000 primes for family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t . t − , a = [2(2149908480000) t + 343107594345448813363200]( t + 2 t − . Our data suggests that there is a positive bias in this family. The averagebias of the fourth moments sums for the first primes is . . Figure 23 is a histogram plot of thedistribution of the average biases among the groups of primes. of the groups of primes havepositive biases, which suggests that it is likely that the fourth moment of this family has a positive p / term.F IGURE
23. Distribution of average biases in the first 1000 primes for a rank 6 family.
IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 37
Similar to the second moment, our data for the rank and rank families has shown that it is likely thathigher rank families ( rank( E ( Q )) ≥ ) have positive biases.3.2. Biases in sixth moment sums.
We now explore the 6th moment biases for these families.For the rank family a = 1 , a = 0 , a = 0 , a = 2 , a = t , of the groups of primes have shownnegative biases. Figure 24 is a histogram plot of the distribution of the average biases among the 20 groups.F IGURE
24. Distribution of average biases in the first 1000 primes for family a = 1 , a = 0 , a = 0 , a = 2 , a = t .For the rank family a = 1 , a = 1 , a = 1 , a = 1 , a = t , of the groups of primes have shownnegative biases. Figure 25 is a histogram plot of the distribution of the average biases among the 20 groups.F IGURE
25. Distribution of average biases in the first 1000 primes for family a = 1 , a = 1 , a = 1 , a = 1 , a = t .For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown negative biases. Figure 26 is a histogram plot of the distribution of the average biases among the 20groups.We further analyze our data by dividing the 1000 primes into 100 groups of 10 for this family. As shown inFigure 27, of the groups of primes have shown negative biases. F IGURE
26. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .F IGURE
27. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 .For the rank family a = 1 , a = t , a = − , a = − t − , a = 0 , of the groups of primes haveshown negative biases. Figure 28 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
28. Distribution of average biases in the first 1000 primes for family a = 1 , a = t , a = − , a = − t − , a = 0 . IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 39
For the rank family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t , of the groups of primes haveshown negative biases. Figure 29 is a histogram plot of the distribution of the average biases among the 20groups. F IGURE
29. Distribution of average biases in the first 1000 primes for family a = 0 , a = 5 , a = 0 , a = − t , a = 64 t .As shown from the data, for most families, it is inconclusive whether the 6th moments have negative biases.For three of the five families, of the groups of primes have negative biases, which strongly suggeststhat the p / term averages to , and the p term is drowned out by the fluctuations of p / term.For the family a = 1 , a = t , a = − , a = − t − , a = 0 and family a = 1 , a = 1 , a = − , a = t , a = 0 , it is likely that the p / term of their 6th moment sums have negative biases(around − . and − . by Figure 15).For the rank 6 family a = 0 , a = 2(16660111104 t ) + 811365140824616222208 , a = 0 , a =[2( − t − t +2 t − , a = [2(2149908480000) t +343107594345448813363200]( t + 2 t − , the average bias of the sixth mo-ments sums for the first primes is . . Figure 30 is a histogram plot of the distribution of the averagebiases among the groups of primes. of the groups of primes have positive biases, whichsuggests that it is likely that the sixth moment of this family has a positive p / term. - F IGURE
30. Distribution of average biases in the first 1000 primes for a rank 6 family.To sum up, while we are not able to tell if the negative bias exists in the higher even moments, the data is atleast consistent with the first lower order term averaging to zero or negative for families with smaller ranks.
Thus our numerics support a weaker form of the bias conjecture: the first lower order term does not have apositive bias for smaller rank families. For families with rank( E ( Q )) ≥ , it is likely that the negative biasconjecture does not hold.4. B IASES IN THE T HIRD , F
IFTH , AND S EVENTH M OMENTS
We now explore the third, fifth, and seventh moments of the Dirichlet coefficients of elliptic curve L -functions. By the Philosophy of Square-Root Cancellation, p times the third moment, p times the fifthmoment, and p times the seventh moment should have size p , p , p respectively (we are multiplying by p to remove the /p averaging). For example, the third moment is a sum of p terms, each of size √ p . Thusas these are signed quantities, we expect the size to be on the order of √ p · p / .We believe that there are bounded functions c E, ( p ) , c ,E ( p ) , and c ,E ( p ) such that pA ,E ( p ) = c ,E ( p ) p + O ( p / ) , A ,E ( p ) = c ,E ( p ) p + O ( p / ) , A ,E ( p ) = c ,E ( p ) p + O ( p / ); (4.115)our data supports these conjectures. Unlike the second, fourth, and sixth moments, the coefficient of theleading term can vary with the prime in the third, fifth, and seventh moments. We calculated the averagevalues of c ,E ( p ) , c ,E ( p ) , and c ,E ( p ) for each elliptic curve family by dividing the size of the main term( p for third moment, p for fifth moment, and p for the seventh moment); see Figure 31.F IGURE
31. Numerical data for the average constant for the main term of 3rd and 5thmoments sums.
IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 41
Our data suggests an interesting relationship between the average constant value for the main term and therank of elliptic families for these odd moments.
Conjecture 4.1.
The average value of the main term of the 3rd moment is − E ( Q )) p . Conjecture 4.2.
The average value of the main term of the 5th moment is − E ( Q )) p . F IGURE
32. Numerical data for the average constant for the main term of 7th moments sums.
Conjecture 4.3.
The average value of the main term of the 7th moment is − E ( Q )) p . Conjecture 4.4.
Let C n be the n-th term of the Catalan numbers. For k ∈ Z + , the average value of themain term of the k + 1 th moment is − C k +1 rank( E ( Q )) p k +1 . We can try to analyze the third, fifth and seventh moments the same way as we did the fourth and sixth.In doing so, we would obtain expansions that do have terms related to the first moment (and hence by theRosen-Silverman theorem the rank of the group of rational solutions); unfortunately there are other termsthat arise now, due to the odd degree, that are not present in the even moments and which we cannot controlas easily. We thus leave a further study of these odd moments as a future project.5. F
UTURE WORK
Natural future questions are to continue investigating the second moment bias conjecture in more and morefamilies, theoretically if possible, numerically otherwise. Since the bias in the second moments doesn’t im-ply biases in higher moments, we can also explore whether there is a corresponding negative bias conjecturefor the higher even moments. As these will involve quartic or higher in t Legendre sums, it is unlikely thatwe will be able to compute these in closed form, and thus will have to resort to analyzing data, or a newapproach through algebraic geometry and cohomology theory (Michel proved that the lower order terms arerelated to cohomological quantities associated to the elliptic curve).Any numerical exploration will unfortunately be quite difficult in general, as there is often a term of size p / which we believe averages to zero for some families, but as it is √ p larger than the next lower orderterm, it completely drowns out that term and makes it hard to see the bias.For the odd moments, our numerical explorations suggest that the bias in the first moment, which is re-sponsible for the rank of the elliptic curve over Q ( T ) , persists. A natural future project is to try to extendMichel’s work to prove our conjectured main term formulas for the odd moments.
6. A
CKNOWLEDGEMENT
I would like to thank my mentor, Professor Steven J. Miller, for guiding me throughout the research process.Without his guidance, I would not have been able to learn the material and complete the research in thisshort period of time.I am also grateful to my parents and friends for their unwavering support.7. D
ECLARATION OF ACADEMIC HONESTY
I hereby confirm that the paper is the result of my own independent scholarly work under the guidance of theinstructor, and that in all cases material from the work of others (in books, articles, essays, dissertations, andon the internet) is acknowledged, and quotations and paraphrases are clearly indicated. No material otherthan that listed has been used.A
PPENDIX
A. L
INEAR AND Q UADRATIC L EGENDRE S UMS
The Dirichlet coefficients of elliptic curve L -functions can be written as cubic Legendre sums; while wedo not have closed form expressions for these in general, we do for linear and quadratic sums. The proofsbelow are standard computations; see for example [BEW, Mi1]. We include them for completeness.A.1. Linear Legendre Sums.Lemma A.1.
We have S ( n ) := p − X x =0 (cid:18) ax + bp (cid:19) = ( p · (cid:0) bp (cid:1) if p | a otherwise. (A.1) Proof.
When p | a , we have S ( n ) = p − X x =0 (cid:18) bp (cid:19) = p · (cid:18) bp (cid:19) . (A.2)When p ∤ a , gcd( p, a ) = 1 and we can send x to a − x , which yields S ( n ) = p − X x =0 (cid:18) x + bp (cid:19) = p − X x =0 (cid:18) xp (cid:19) = 0 . (A.3) Ξ A.2.
Factorizable Quadratics in Sums of Legendre Symbols.
Before analyzing the most general qua-dratic Legendre sum, we first do an important special case. The following proof is directly copied from[Mi1], and is included for the convenience of the reader.
Lemma A.2.
For p > S ( n ) = p − X x =0 (cid:18) n + xp (cid:19)(cid:18) n + xp (cid:19) = ( p − if p | n − n − otherwise. (A.4) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 43
Proof.
Shifting x by − n , we need only prove the lemma when n = 0 . Assume ( n, p ) = 1 as otherwisethe result is trivial. For ( a, p ) = 1 we have S ( n ) = p − X x =0 (cid:18) n + xp (cid:19)(cid:18) xp (cid:19) = p − X x =0 (cid:18) n + a − xp (cid:19)(cid:18) a − xp (cid:19) = p − X x =0 (cid:18) an + xp (cid:19)(cid:18) xp (cid:19) = S ( an ) . (A.5)Hence S ( n ) = 1 p − p − X a =1 p − X x =0 (cid:18) an + xp (cid:19)(cid:18) xp (cid:19) = 1 p − p − X a =0 p − X x =0 (cid:18) an + xp (cid:19)(cid:18) xp (cid:19) − p − p − X x =0 (cid:18) xp (cid:19) = 1 p − p − X x =0 (cid:18) xp (cid:19) p − X a =0 (cid:18) an + xp (cid:19) −
1= 0 − − . (A.6) Ξ Where do we use p > ? We used p − X a =0 (cid:18) an + xp (cid:19) = 0 for ( n, p ) = 1 . This is true for all odd primes (asthere are p − quadratic residues, p − non-residues, and ); for p = 2 , there is one quadratic residue, nonon-residues, and . As we never need to use this lemma for p = 2 , this complication will not affect any ofour proofs.A.3. General Quadratics in Sums of Legendre Symbols.
The following proof is directly copied from[Mi1], and is included for the convenience of the reader.
Lemma A.3.
Assume a and b are not both zero mod p and p > . Then p − X t =0 (cid:18) at + bt + cp (cid:19) = ( ( p − (cid:0) ap (cid:1) if p | b − ac − (cid:0) ap (cid:1) otherwise . (A.7) Proof.
Assume a p ) as otherwise the proof is trivial. Let δ = 4 − ( b − ac ) . Then p − X t =0 (cid:18) at + bt + cp (cid:19) = p − X t =0 (cid:18) a − p (cid:19)(cid:18) a t + bat + acp (cid:19) = p − X t =0 (cid:18) ap (cid:19)(cid:18) t + bt + acp (cid:19) = p − X t =0 (cid:18) ap (cid:19)(cid:18) t + bt + 4 − b + ac − − b p (cid:19) = p − X t =0 (cid:18) ap (cid:19)(cid:18) ( t + 2 − b ) − − ( b − ac ) p (cid:19) = p − X t =0 (cid:18) ap (cid:19)(cid:18) t − δp (cid:19) = (cid:18) ap (cid:19) p − X t =0 (cid:18) t − δp (cid:19) . (A.8)If δ ≡ p ) we get p − . If δ = η , η = 0 , then by Lemma A.2 p − X t =0 (cid:18) t − δp (cid:19) = p − X t =0 (cid:18) t − ηp (cid:19)(cid:18) t + ηp (cid:19) = − . (A.9)We note that p − X t =0 (cid:18) t − δp (cid:19) is the same for all non-square δ ’s (let g be a generator of the multiplicative group, δ = g k +1 , change variables by t → g k t ). Denote this sum by S , the set of non-zero squares by R , and thenon-squares by N . Since p − X δ =0 (cid:18) t − δp (cid:19) = 0 we have p − X δ =0 p − X t =0 (cid:18) t − δp (cid:19) = p − X t =0 (cid:18) t p (cid:19) + X δ ∈R p − X t =0 (cid:18) t − δp (cid:19) + X δ ∈N p − X t =0 (cid:18) t − δp (cid:19) = ( p −
1) + p −
12 ( −
1) + p − S = 0 (A.10)Hence S = − , proving the lemma. Ξ A PPENDIX
B. F
ORMS OF TH AND TH MOMENTS SUMS
B.1.
Tools for higher moments calculations.
The Dirichlet Coefficients of the elliptic curve L -functioncan be written as a t ( p ) = √ p ( e iθ t ( p ) + e − iθ t ( p ) ) = 2 √ p cos( θ t ( p )) , (B.11)with θ t ( p ) real; this expansion exists by Hasse’s theorem, which states | a t ( p ) | ≤ √ p . Define sym k ( θ ) := sin(( k + 1) θ )sin θ . (B.12) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 45
By the angle addition formula for sine, sym k ( θ ) = sym k − ( θ ) cos θ + cos( kθ ) (B.13)When k = 1 , we have sym ( θ ) = 2 cos θ (B.14)Michel [Mic] proved that X t ( p ) sym k ( θ t ( p )) = O ( √ p ) , (B.15)where the big-Oh constant depends only on the elliptic curve and k ; thus while we should have a k subscriptin the implied constant, as k is fixed in our investigations we omit it for notational simplicity.B.2. Form of 4th moments sums.
A lot is known about the moments of the a t ( p ) for a fixed elliptic curve E t . However, as we are only concerned with averages over one-parameter families, we do not need toappeal to any results towards the Sato-Tate distribution, and instead we can directly prove convergence ofthe moments on average to the moments of the semicircle. In particular, the average of the m -th momentshas main term m + 1 (cid:18) mm (cid:19) p m − . The coefficients m + 1 (cid:18) mm (cid:19) are the Catalan numbers, and the firstfew main terms of the even moments are p, p , p and p . Lemma B.1.
The average fourth moment of an elliptic surface with j ( T ) non-constant has main term p : X t ( p ) a t ( p ) = 2 p + O ( p ) . (B.16) Proof.
We have to compute a t ( p ) = 16 p cos θ t ( p ) . (B.17)We first collect some useful trigonometry identities: cos(2 θ ) = 2 cos ( θ ) − ( θ ) = 12 cos(2 θ ) + 12 . (B.18)We use these to re-write cos θ in terms of quantities we can compute: cos ( θ ) = 14 cos (2 θ ) + 12 cos(2 θ ) + 14= 18 cos(4 θ ) + 12 cos(2 θ ) + 38= 18 [sym ( θ ) − sym ( θ ) cos θ ] + 12 cos(2 θ ) + 38 . (B.19) The following expression will arise in our expansion, so we analyze it first: −
18 sym ( θ ) cos θ = −
18 sin(4 θ )sin θ cos θ = −
18 2 sin(2 θ ) cos(2 θ )sin θ cos θ = −
18 2 · θ cos θ cos(2 θ )sin θ cos θ = −
12 cos θ cos(2 θ )= −
12 cos θ (2 cos θ − − cos θ + 12 cos θ p · (cid:18) −
18 sym ( θ ) cos θ (cid:19) = − p cos θ + 8 p cos θ = − p cos θ + 2 p · a t ( p ) . (B.20)Thus p cos θ = 2 p sym θ − p cos θ + 2 p · a t ( p ) + 4 p · a t ( p ) − p · (16 p cos θ ) = 2 p sym θ + 6 p · a t ( p ) − p X t ( p ) (16 p cos θ ) = p X t ( p ) sym θ + 3 p X t ( p ) a t ( p ) − p X t ( p ) a t ( p ) = p · O ( √ p ) + 3 p ( p + O ( p )) − p = 2 p + O ( p ) , (B.21)as claimed. Ξ B.3.
Form of 6th moments sums.Lemma B.2.
The average sixth moment of an elliptic surface with j ( T ) non-constant has main term p : X t ( p ) a t ( p ) = 5 p + O ( p ) . (B.22) Proof.
We have a t ( p ) = 64 p cos θ t ( p )cos(3 θ ) = 4 cos θ − θ cos θ = cos(3 θ ) + 3 cos θ . (B.23) IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 47
We first expand cos θ : cos θ = cos (3 θ ) + 9 cos θ + 6 cos θ cos(3 θ )16= cos(6 θ ) + + 9[ cos(2 θ ) + ] + 6 cos θ [4 cos θ − θ ]16= 10 + cos(6 θ ) + 9 cos(2 θ ) + 48 cos θ −
36 cos θ
32= 10 + cos(6 θ ) + 9 cos(2 θ ) + 48[ cos(4 θ ) + cos(2 θ ) + ] −
36 cos θ
32= 10 + cos(6 θ ) + 9 cos(2 θ ) + 48[ cos(4 θ ) + cos(2 θ ) + ] −
18 cos(2 θ ) − θ ) + 6 cos(4 θ ) + 15 cos(2 θ )32= cos(6 θ )32 + 10 + 6 cos(4 θ ) + 15 cos(2 θ )32= sym ( θ ) − sym ( θ ) cos θ
32 + 10 + 6 cos(4 θ ) + 15 cos(2 θ )32 . (B.24)Next we find a formula for the symmetric function that will appear: −
132 sym ( θ ) cos θ = −
132 ( sin(6 θ )sin θ ) cos θ = −
132 cos θ ( 3 sin(2 θ ) − (2 θ )sin θ )= −
132 cos θ ( 6 sin θ cos θ −
32 sin θ cos θ sin θ )= −
132 cos θ [6 cos θ −
32 sin θ cos θ ]= −
316 cos θ + (1 − cos θ ) cos θ = −
316 cos θ + cos θ − cos θ p ( −
132 sym ( θ ) cos θ ) = − p cos θ + 64 p cos θ − p cos θ = − p cos θ + 4 pa t ( p ) − p a t ( p ) . (B.25)Thus p cos θ = 2 p sym ( θ ) − p cos θ + 4 pa t ( p ) − p a t ( p ) + 12 p cos(4 θ ) + 30 p cos(2 θ ) + 20 p p cos θ = p sym ( θ ) + 2 pa t ( p ) − p a t ( p ) + 6 p cos(4 θ ) + 15 p cos(2 θ ) + 10 p . (B.26)We can re-express some of the terms above in a more convenient form: p cos(2 θ ) = 15 p (2 cos θ − p cos θ − p = 152 p a t ( p ) − p (B.27) and p cos(4 θ ) = 6 p [sym ( θ ) − sym ( θ ) cos θ ]= 6 p [sym ( θ ) − θ + 4 cos θ ]= 6 p sym ( θ ) − pa t ( p ) + 6 p a t ( p ) . (B.28)Thus p cos θ = p sym ( θ ) + 2 pa t ( p ) − p a t ( p ) + 6 p sym ( θ ) − pa t ( p ) + 6 p a t ( p )+ 152 p a t ( p ) − p + 10 p X t ( p ) p cos θ = X t ( p ) [ p sym ( θ ) + 2 pa t ( p ) − p a t ( p ) + 6 p sym ( θ ) − pa t ( p ) + 6 p a t ( p )+ 152 p a t ( p ) − p + 10 p ] . (B.29)Therefore X t ( p ) a t ( p ) = p X t ( p ) sym ( θ ) + 6 p X t ( p ) sym ( θ ) − p X t ( p ) a t ( p ) + 12 p X t ( p ) a t ( p ) − p = p O ( √ p ) + 6 p O ( √ p ) − p (2 p + O ( p )) + 12 p ( p + O ( p )) − p = 5 p + O ( p ) , (B.30)completing the proof. Ξ A PPENDIX
C. M
ATHEMATICAL CODE
C.1.
Finding formulas for second moment sums.
Below is an example of mathematica code to find theformulas for the elliptic curve family y + xy + y = x + x + x + t . It would take four to five days to run1000 primes for a family on an old laptop which was devoted to this problem. To use one chooses the valuesof the input polynomials a , a , a , a , a and the file name (which for convenience we take to be related tothe coefficients).a1 [ t _ ] : = 1 ;a2 [ t _ ] : = 1 ;a3 [ t _ ] : = 1 ;a4 [ t _ ] : = 1 ;a6 [ t _ ] : = t ;sums [ a1_ , a2_ , a3_ , a4_ , a6_ , p s t a r t _ , pend_ , p o w e r o f 2 _ , p o w e r o f 3 _ ,f a m i l y n a m e _ ] : = Module [ { } ,( * d e f i n e s t h e p o l y n o m i a l s we w i l l u s e * )( * num moments i s t h e number o f moments we do * )( * p s t a r t and pend i s i n d e x o f f i r s t and l a s t p r i m e s t u d i e d * )( * we w i l l f i n d t h e f o r m u l a f o r t h e s e c o n d moment sum a c c o r d i n g t ot h e c o n g r u e n c e c l a s s e s o f 2^ p o w e r o f 2 *3^ p o w e r o f 3 * )b2 [ t _ ] : = a1 [ t ] ^ 2 + 4* a2 [ t ] ;b4 [ t _ ] : = 2* a4 [ t ] + a1 [ t ] * a3 [ t ] ;b6 [ t _ ] : = a3 [ t ] ^ 2 + 4* a6 [ t ] ; IASES IN MOMENTS OF DIRICHLET COEFFICIENTS OF ELLIPTIC CURVE FAMILIES 49 c4 [ t _ ] : = b2 [ t ] ^ 2 − 24* b4 [ t ] ;c6 [ t _ ] : = −b2 [ t ] ^ 3 + 36* b2 [ t ] * b4 [ t ] − 216* b6 [ t ] ;( * p r i m e t o n [ n ] =m, t h e mth p r i m e i s n * )numdo = 1 0 0 0 ;F o r [ n = 1 , n <= P r i m e [ numdo ] , n ++ , p r i m e t o n [ n ] = 0 ] ;F o r [ n = 1 , n <= numdo , n ++ , p r i m e t o n [ P r i m e [ n ] ] = n ] ;( * c r e a t e s l i s t s t o p u t d i f f e r e n t p r i m e s i n t o d i f f e r e n t p r i m ec o n g r u e n c e c l a s s * )F o r [m = 1 , m <= 2^ p o w e r o f 2 *3^ p o w e r o f 3 , m++ ,{ primeModgroup [m] = { } ;} ] ;F o r [m = 3 , m <= pend , m++ ,{ p = P r i m e [m ] ;j = Mod [ p , 2^ p o w e r o f 2 *3^ p o w e r o f 3 ] ;primeModgroup [ j ] = AppendTo [ primeModgroup [ j ] , p ] ;} ] ;( * i n i t i a l i z e s moment l i s t s t o empty * )moment = { } ;( * l o o p s o v e r t h e p r i m e s we s t u d y * )F o r [ k = p s t a r t , k <= pend , k ++ ,{ p = P r i m e [ k ] ;momenttemp = 0 ;F o r [ t = 0 , t <= p − 1 , t ++ ,{ a e p t = Sum [ J a c o b i S y m b o l [ x ^3 − 27* c4 [ t ] * x − 54* c6 [ t ] , p ] ,{ x , 0 , p − 1 } ] ;c u r r e n t v a l u e = 1 ;c u r r e n t v a l u e = c u r r e n t v a l u e * a e p t * a e p t ;momenttemp = momenttemp + c u r r e n t v a l u e ;} ] ; ( * end o f t l o o p t o c o m p u t e a_E ( t ) * )moment = AppendTo [ moment , { p , momenttemp } ] ;P r i n t [ " Working w i t h p r i m e p = " , p , " . " ] ;P r i n t [ " The s e c o n d moment sum i s " , moment [ [ k − p s t a r t + 1 , 2 ] ] ] ;P r i n t [ " " ] ;} ] ;s a v e n a m e = T o S t r i n g [ f a m i l y n a m e ] ;P r i n t [ " We a r e s a v i n g t h e f i l e t o " , f a m i l y n a m e ] ;S e t D i r e c t o r y [ " " ] ;
P u t [ s a v e l i s t , s a v e n a m e ] ;P u t [ { moment [ 2 ] } , s a v e n a m e ] ;( * S a v i n g t h e d a t a o f s e c o n d moment sums on t h e c o m p u t e r * )F o r [m = 1 , m <= 2^ p o w e r o f 2 *3^ p o w e r o f 3 , m++ ,{ I f [ L e n g t h [ primeModgroup [m] ] ! = 0 , {I f [ L e n g t h [ primeModgroup [m] ] < 4 ,P r i n t [ " p r i m e mod" , 2^ p o w e r o f 2 *3^ p o w e r o f 3 , " : " , m," T h e r e a r e n o t e n o u g h p r i m e s " ] && C o n t i n u e [ ] ] ;p r i m e L i s t = primeModgroup [m ] ;F o r [ i = 2 , i <= L e n g t h [ p r i m e L i s t ] , i ++ ,{ p r i m e [ i ] = p r i m e L i s t [ [ i ] ] ; ( * f i n d t h e i − t h p r i m e * )n t h P r i m e [ i ] = p r i m e t o n [ p r i m e [ i ] ] ;sum [ i ] = moment [ [ n t h P r i m e [ i ] − p s t a r t + 1 , 2 ] ] − ( p r i m e [ i ] ) ^ 2 ;( * s e c o n d moment sum=p ^2+ ap +b , sum [ i ] = ap +b * )} ] ;a = ( sum [ 3 ] − sum [ 2 ] ) / ( p r i m e [ 3 ] − p r i m e [ 2 ] ) ;b = sum [ 2 ] − a * p r i m e [ 2 ] ;p o l y w o r k = 1 ;F o r [ i = 4 , i <= L e n g t h [ p r i m e L i s t ] , i ++ ,{ I f [ sum [ i ] ! = a * p r i m e [ i ] + b ,{ p o l y w o r k = 0 ;i = L e n g t h [ p r i m e l i s t ] + 1 0 0 ;} ] ;} ] ; ( * end o f i f s t a t e m e n t * ) ( * I f p o l y w o r k d o e s n o t work f o rt h e i − t h p r i m e , s e t p o l y w o r k t o 0 * )I f [ p o l y w o r k == 1 ,P r i n t [ " p r i m e mod " , 2^ p o w e r o f 2 *3^ p o w e r o f 3 , " : " , m," s e c o n d moment sum=p ^2 + ( " , a , " ) p + " , b ] ,P r i n t [ " p r i m e mod" , 2^ p o w e r o f 2 *3^ p o w e r o f 3 , " : " , m,"We can ’ t f i n d t h e f o r m u l a . " ] ] ;} ] ;} ] ;] ; ( * end o f module * )sums [ a1 , a2 , a3 , a4 , a6 , 1 , 4 4 0 , 3 , 3 , " 1 1 1 1 t . d a t " ]
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