Sato-Tate Conjecture Research Articles

Patterns of primes in joint Sato–Tate distributions

For j=1,2, let fj(z)=∑n=1∞aj(n)e2πinz be a holomorphic, non-CM cuspidal newform of even weight kj≥2 with trivial nebentypus. For each prime p, let θj(p)∈[0,π] be the angle such that aj(p)=2p(k−1)/2cos⁡θj(p). The now-proven Sato–Tate conjecture states that the angles (θj(p)) equidistribute with respect to the measure dμST=2πsin2⁡θdθ. We show that, if f1 is not a character twist of f2, then for subintervals I1,I2⊆[0,π], there exist infinitely many bounded gaps between the primes p such that θ1(p)∈I1 and θ2(p)∈I2. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.

Effective Sato–Tate conjecture for abelian varieties and applications

From the generalized Riemann hypothesis for motivic L -functions, we derive an effective version of the Sato–Tate conjecture for an abelian variety A defined over a number field k with connected Sato–Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A . We discuss three applications of this conditional result. First, for an abelian variety defined over k , we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A' defined over k with no common factors up to k -isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A' have opposite sign.

Arithmetic progressions in certain subsets of finite fields

In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp={t2:t∈Fp} and Cp={t3:t∈Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k≥3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given byp22k+R, where|R|≤(k−24−k−22k−1)⋅p32+ck⋅p and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length ℓ in the set Sp when ℓ∈{3,4,5} and p is an odd prime number. For ℓ=4,5, our formulas are based on the number of points on certain elliptic curves, and the error term is best possible due to the Sato-Tate conjecture.

Blown-up toric surfaces with non-polyhedral effective cone

Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space M ¯ 0 , n \overline{M}_{0,n} of stable rational curves is not polyhedral for n ≥ 10 n\geq 10 . These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density.

The Sato-Tate conjecture and Nagao's conjecture

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized to smooth irreducible projective surfaces. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for surfaces which form families of quadratic twists of fixed elliptic curves or fixed hyperelliptic curves of genus g≥1.

An Unconditional Explicit Bound on the Error Term in the Sato–tate Conjecture

AbstractLet $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states that the angles θp are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric power L-functions due to Newton and Thorne, we explicitly bound the error term in the Sato–Tate conjecture when f corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when f has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid N, \theta_p\in I\}$ and $\pi(x) := \# \{p \leq x \}$, we prove the following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.

The Lang–Trotter conjecture for products of non-CM elliptic curves

Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over \({\mathbb {Q}}\) and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over \({\mathbb {Q}}\) that are not \({\overline{{\mathbb {Q}}}}\)-isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.

Sato-Tate distributions of y2 = xp − 1 and y2 = x2p − 1

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y2=xp−1 and y2=x2p−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.

The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture of Banaszak and Kedlaya

The algebraic Sato-Tate conjecture was initially introduced by Serre and then discussed by Banaszak and Kedlaya. This note shows that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture.

Machine-learning the Sato–Tate conjecture

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently.

Tate module tensor decompositions and the Sato–Tate conjecture for certain abelian varieties potentially of $$\mathrm {GL}_2$$-type

We introduce a tensor decomposition of the $$\ell $$ -adic Tate module of an abelian variety $$A_0$$ defined over a number field which is geometrically isotypic. If $$A_0$$ is potentially of $$\mathrm {GL}_2$$ -type and defined over a totally real number field, we use this decomposition to describe its Sato–Tate group and to prove the Sato–Tate conjecture in certain cases.

New frontiers in Langlands reciprocity

In this survey, I discuss some recent developments at the crossroads of arithmetic geometry and the Langlands programme. The emphasis is on recent progress on the Ramanujan–Petersson and Sato–Tate conjectures. This relies on new results about Shimura varieties and torsion in the cohomology of locally symmetric spaces.

The explicit Sato–Tate conjecture for primes in arithmetic progressions

Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.

Almost all primes satisfy the Atkin–Serre conjecture and are not extremal

Let \(f(z)=\sum _{n=1}^{\infty } a_f(n)e^{2\pi i n z}\) be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight \(k\ge 2\). Deligne’s proof of the Weil conjectures shows that \(|a_f(p)|\le 2p^{\frac{k-1}{2}}\) for all primes p. We prove for 100% of primes p that \( 2p^{\frac{k-1}{2}}{\log \log p}/{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor .\) Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that \(|a_f(p)|\) is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

Effective forms of the Sato–Tate conjecture

We prove effective forms of the Sato-Tate conjecture for holomorphic cuspidal newforms which improve on the author's previous work (solo and joint with Lemke Oliver). We also prove an effective form of the joint Sato-Tate distribution for two twist-inequivalent newforms. Our results are unconditional because of recent work of Newton and Thorne.

Sato-Tate distributions on Abelian surfaces

We prove a few new cases of the Sato-Tate conjecture for abelian surfaces, using a new automorphy theorem of Allen et al. Then in the unproven cases, we use partial results to describe nontrivial asymptotics on the trace of Frobenius, and prove their optimality given current knowledge.

Patterns of primes in the Sato–Tate conjecture

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $\mu_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1, 1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with $\mu_{ST}(I)\ge 0.36$. Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.

Exceptional splitting of reductions of abelian surfaces

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to Charles' theorem on exceptional isogeny of reductions of a given pair of elliptic curves and Elkies' theorem on supersingular reductions of a given elliptic curve, our theorem shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

Central limit theorems for elliptic curves and modular forms with smooth weight functions

In [11], the second and third-named authors established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in [3] by the first and second-named authors. In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a Central Limit Theorem for much smaller families of modular forms, and we succeed in proving a theorem of this type for families of elliptic curves under the Riemann Hypothesis for L-functions associated to Hecke eigenforms for the full modular group.

Primitive divisors of sequences associated to elliptic curves

Let {nP+Q}n≥0 be a sequence of points on an elliptic curve defined over a number field K. In this paper, we study the denominators of the x-coordinates of this sequence. We prove that, if Q is a torsion point of prime order, then for n large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and a Lang-Trotter conjecture. Indeed, given two points P and Q on the elliptic curve, we prove a lower bound for the number of primes p such that P is in the orbit of Q modulo p.