AA REFINEMENT OF SATO–TATE CONJECTURE
TARO KIMURA 木村 太 郎 Abstract.
We propose a refined version of the Sato–Tate conjecture about the spacing distribution of theangle determined for each prime number. We also discuss its implications on L -function associated withelliptic curves in the relation to random matrix theory. Introduction
The Sato–Tate (ST) conjecture, independently proposed by M. Sato and J. Tate [Tat65], then provedby Clozel–Harris–Shepherd-Barron–Taylor [CHT08, Tay08, HSBT10] after 40 years, is about a statisticalproperty of the number of points on the elliptic curve over the finite field. Let E ( F p ) be the number ofpoints on the elliptic curve E over the finite field F p with p a prime number, and N the conductor of E . Foreach pair ( p, E ), we define a p = p + 1 − E ( F p ) for p (cid:45) N and a p = p − E ( F p ) for p | N . Hasse’s theorem(also equivalent to the Ramanujan conjecture in some cases) on elliptic curves states that (cid:12)(cid:12)(cid:12)(cid:12) a p √ p (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (1)from which we define the angle θ p ∈ [0 , π ] for a pair ( E, p ),cos θ p = a p √ p . (2) Theorem (The Sato–Tate conjecture. Proof by Clozel–Harris–Shepherd-Barron–Taylor) . Suppose that theelliptic curve E has no complex multiplication. Then, the probability such that the angle θ p is found in theinterval [ a, b ] ⊆ [0 , π ] is given by P [ a ≤ θ p ≤ b ] := lim X →∞ P X [ a ≤ θ p ≤ b ] = (cid:90) ba d θ ρ ( θ ) , (3) where P X [ a ≤ θ p ≤ b ] = { a ≤ θ p ≤ b, p ≤ X } { p ≤ X } (4) with the density function given by ρ ( θ ) = 2 π sin θ . (5)In fact, the density function discussed here is interpreted as a one-point correlation function of the angledistributed on [0 , π ]. We may obtain it from the probability (3) in the following limit,lim d θ → ε (cid:28) P [ θ ≤ θ p ≤ θ + d θ ] = ρ ( θ ) d θ . (6)From this point of view, it is natural to consider more generic statistical quantities to characterize thedistribution of the angles. For example, we may consider the n -point generalization of the probability (3) Date : January 14, 2021. a r X i v : . [ m a t h . N T ] J a n TARO KIMURA 木村 太 郎 for 0 ≤ a i ≤ b i ≤ π , i = 1 , . . . , n , P [ a i ≤ θ p i ≤ b i , i = 1 , . . . , n ] := lim X →∞ P X [ a i ≤ θ p i ≤ b i , i = 1 , . . . , n ]= lim X →∞ { a i ≤ θ p i ≤ b i , i = 1 , . . . , n, p i ≤ X } { p ≤ X } . (7)Then, the n -point function is similarly obtained through the limitlim d θ i → ε (cid:28) P [ θ i ≤ θ p i ≤ θ i + d θ i , i = 1 , . . . , n ] = ρ n ( θ , . . . , θ n ) d θ · · · d θ n . (8)In this paper, we do not directly consider such n -point functions, but instead study another statisticalquantity about spacings between the angles. We define an integral map from [0 , π ] to [0 , θ ) = (cid:90) θ d θ (cid:48) ρ ( θ (cid:48) ) . (9)Since dΘ = ρ ( θ ) d θ , the variable Θ is uniformly distributed on [0 , θ is distributed on[0 , π ] with the density profile ρ ( θ ). We introduce a set of variables (Θ( θ p )) p ∈ primes = (Θ ≤ Θ ≤ · · · ), thatwe call unfolded variables. Then, we have the following conjecture based on numerical computations of theunfolded variables shown in Section 3. Conjecture (Spacing distribution of the unfolded angles) . For the elliptic curve E , for which the STconjecture holds, the next k nearest neighbor spacing distribution of the unfolded variables is given by thePoisson distribution, p k ( s ) := lim X →∞ P X (cid:20) Θ i + k +1 − Θ i = s { p ≤ X } (cid:21) = s k e − s k ! . (10) The spacing distribution is normalized (cid:90) ∞ d s p k ( s ) = 1 , (cid:90) ∞ d s s p k ( s ) = k + 1 . (11)This is a refined version of the ST conjecture in a sense that it concerns more detailed statistical propertyof the angles beyond the density function addressed in the ordinary ST conjecture.The Poisson distribution appears as a spacing distribution of uncorrelated random variables. See, forexample, [Meh04]. Therefore, this conjecture implies that the angles θ p and the unfolded variables Θ i statistically behave as uncorrelated random variables. From this point of view, it is speculated that the n -point function (8) would be simply given as a product of the density functions, ρ n ( θ , . . . , θ n ) = n (cid:89) i =1 ρ ( θ i ) . (12)We remark that a similar behavior is observed for prime numbers [Lib98, Tim06, TT07, Wol14]: The spacingdistribution of prime numbers is given by the Poisson distribution. Compared to these results, one can moreclearly see that the spacing distribution of the angles is given by the Poisson distribution since the anglesare distributed on the finite interval [0 , π ], whereas the prime numbers are distributed on the infinite interval[0 , ∞ ]. Acknowledgments.
I would like to thank S. Koyama and N. Kurokawa for valuable discussions and usefulcomments on the draft. In fact, this study was motivated by Kurokawa-sensei’s seminar explaining a sim-ilarity between the Sato–Tate conjecture and Wigner’s semi-circle law of random matrices. I’m gratefulfor his exposition. This work has been supported in part by “Investissements d’Avenir” program, ProjectISITE-BFC (No. ANR-15-IDEX-0003), and EIPHI Graduate School (No. ANR-17-EURE-0002). Nearest neighbor ( k = 0), Next nearest neighbor ( k = 1), Next-next nearest neighbor ( k = 2), etc. REFINEMENT OF SATO–TATE CONJECTURE 3 L -function for elliptic curves We address an implication of the refined ST conjecture in the context of the L -function associated withelliptic curves.2.1. Global zero and local zero.
In addition to the algebraic formulation shown above, the numbers a p also appear on the analytic side as the Fourier coefficients of the cusp form associated with the elliptic curve E (See, for example, [Kob93]), f ( q ) = ∞ (cid:88) n =1 a n q n . (13)We define the L -function associated with the elliptic curve E , a.k.a., the Hasse–Weil L -function, from theFourier coefficients of the cusp form, which has the Euler product form due to the multiplicative propertyof a p ’s, L ( s ; E ) = ∞ (cid:88) n =1 a n n s = (cid:89) p (cid:45) N (cid:0) − a p p − s + p − s (cid:1) − (cid:89) p | N (cid:0) − p − s (cid:1) − . (14)We then define the local zeta function for a pair ( p, E ), which shows the following factorization, Z p ( s ; E ) = 1 − a p p − s + p − s = (1 − e + iθ p / − s )(1 − e − iθ p / − s ) . (15)In other words, the relation (1) is equivalent to the local version of Riemann Hypothesis: The zeros of thelocal zeta function, that we call the local zeros, are found on the critical line Re( s ) = . This is for a cuspform of weight two, and a similar formulation is also possible for higher weight cases. From this point of view, the ST conjecture about the distribution of the angles describes how the imag-inary part of the local zero is distributed on the critical line. In particular, our conjecture implies that theimaginary part of the local zero behaves as an uncorrelated random number. This is in contrast to thestatistical behavior of zeros of the L -function, that we instead call the global zeros. It has been known thatthe statistical property of the nontrivial zeros of the zeta function agrees with eigenvalue statistics of randommatrices [Mon73, Odl87]. This is true also for the L -function associated with elliptic curves [BK13]. Then, wehave a question: What is a counterpart of the local zero in random matrix theory? In the context of Gaussianrandom matrix theory, we typically consider statistical properties of eigenvalues exhibiting nontrivial corre-lation, rather than uncorrelated matrix elements which are independently distributed. Namely, eigenvaluesare correlated, whereas matrix elements are not. This observation leads to the following correspondence:Correlation Random matrices Zeta functionCorrelated Eigenvalue Global zeroUncorrelated Matrix element Local zeroRecalling the Euler product of the L -function (14), the local zeta function is a building block of the total L -function. A similar interpretation is possible for the ordinary zeta function in the Euler product form forRe( s ) > ζ ( s ) = ∞ (cid:88) n =1 n s = (cid:89) p ∈ primes (cid:0) − p − s (cid:1) − , (16) The local zeta function for the cusp form of weight k is Z p ( s ; E ) = 1 − a p p − s + p k − − s , so that the critical line is givenby Re( s ) = ( k − /
2. In Section 3, we will discuss the modular discriminant, which is a cusp form of weight 12.
TARO KIMURA 木村 太 郎 because the spacing distribution of prime numbers is also the Poisson distribution as mentioned earlier. Thisseems analogous to the relation between matrix elements and eigenvalues of random matrices: Correlatedvariables are constructed from uncorrelated variables.2.2. Group structure and higher genus/rank generalization.
We remark that, introducing U ( θ ) =diag (cid:0) e + iθ , e − iθ (cid:1) ∈ SU(2), we may write the local zeta function (15) as a characteristic polynomial, Z p ( s ; E ) = det (cid:16) − U ( θ p ) p / − s (cid:17) . (17)The local zeta function depends only on the conjugacy class parametrized by the angle θ p since it is invariantunder the SU(2) transform, U ( θ p ) → g − U ( θ p ) g for ∀ g ∈ SU(2). In fact, the measure ρ ( θ ) d θ agrees with theHaar measure on SU(2). This factor is also interpreted as follows: Introducing another variable, x = cos x ,the measure is given by ρ ( θ ) d θ = π √ − x d x , which is obtained by the projection onto the interval x ∈ [ − ,
1] from a three sphere S = { x + y + z + w = 1 } , identified with SU(2). From this point of view,our conjecture implies that we have an uncorrelated homogeneous map from prime numbers to SU(2).A similar interpretation may be possible for higher genus cases (hyperelliptic curves). It has been proposedthat a higher rank group G = USp(2 g ) corresponds to genus g curves, which is reduced to SU(2) for g = 1elliptic curves [KS99]. A naive expectation is that we would similarly have uncorrelated homogeneous mapfrom prime numbers to G . However, in this case, we should use g angles for each pair ( p, E ), ( θ ( i ) p ) i =1 ,...,g ,so that statistical behaviors of the angles would be more involved because of a correlation among them dueto the Haar measure on G . Further examination is required in this case.3. Numerical computations
We demonstrate the statistical property of the angles with numerical computations. We consider thefollowing cusp forms as examples: f ( q ) N { p ≤ X } (a) η ( q ) η ( q ) ( E : y + y = x − x ) 11 2000(b) η ( q ) η ( q ) ( E : y = x + x − x ) 2, 5 2000(c) η ( q ) (modular discriminant ∆) − η ( q ) = q / ∞ (cid:89) n =1 (1 − q n ) . (18)The right most column shows the number of primes that we use in the computations. We remark that,since the weight of the modular discriminant (example (c)) is 12, the angle is defined as cos θ p = a p / p / ,instead of the previous definition (2). In this case, the coefficient a p is also denoted by τ ( p ), which is calledthe Ramanujan tau function. Similarly, the local zeta function is given by Z p ( s, ∆) = 1 − a p p − s + p − s .Fig. 1 shows numerical computation of the density function of the angles θ p for three examples. We seethat it agrees with the curve ρ ( θ ) = 2 π sin θ as stated by the ST conjecture. Fig. 2 shows the unfoldedvariables (Θ i ) i =1 ,..., { p ≤ X } , which are uniformly distributed on the interval [0 , k nearest neighbor spacing distribution p k ( s ) for k = 0 , ,
2. We seethe agreement with the Poisson distribution s k e − s /k !, which provides a numerical evidence of the conjecture. REFINEMENT OF SATO–TATE CONJECTURE 5 (a) θρ ( θ ) (b) θρ ( θ ) (c) θρ ( θ ) Figure 1.
Numerical computation of the density function ρ ( θ ). The red line shows thecurve 2 π sin θ . (a)Θ i i (b)Θ i i (c)Θ i i Figure 2.
The unfolded variables Θ i for i = 1 , . . . , { p ≤ X } .(a) sp ( s ) (b) sp ( s ) (c) sp ( s ) Figure 3.
Nearest neighbor spacing distribution p ( s ). The red line shows the curve e − s .(a) sp ( s ) (b) sp ( s ) (c) sp ( s ) Figure 4.
Next nearest neighbor spacing distribution p ( s ). The red line shows the curve s e − s . TARO KIMURA 木村 太 郎 (a) sp ( s ) (b) sp ( s ) (c) sp ( s ) Figure 5.
Next-next nearest neighbor spacing distribution p ( s ). The red line shows thecurve s e − s / References [BK13] P. Bourgade and J. P. Keating,
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