Trace Of Frobenius Research Articles

P-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

Murmurations of Elliptic Curves

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that ${\mathbb {L}}$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when $E=\mathbb {Q}$, then ${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

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.

Lower bounds on the maximal number of rational points on curves over finite fields

AbstractFor a given genus $g \geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$ . As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$ . Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series F2⁎ and F4⁎ in terms of F34-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series F2 and F4. We also deduce an expression for the finite field Appell series F4⁎ in terms of F23-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series F4⁎.

Non-CM elliptic curves with infinitely many almost prime Frobenius traces

Let E/Q be a non-CM elliptic curve. Write p+1−ap(E) for the number of Fp-points of the reduction of E modulo a prime p of good reduction. We prove the following: (i) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), the number of primes p<x with |ap(E)| a prime is bounded above by C1(E)x(logx)2; (ii) under GRH, the number of primes p<x with |ap(E)| the product of at most 4 distinct primes, counted without multiplicity, is bounded below by C2(E)x(logx)2; (iii) under GRH, the number of primes p<x with |ap(E)| the product of at most 5 distinct primes, counted with multiplicity, is bounded below by C3(E)x(logx)2; (iv) under GRH, Artin's Holomorphy Conjecture, and a Pair Correlation Conjecture for Artin L-functions, the number of primes p<x with |ap(E)| the product of at most 2 distinct primes, counted with multiplicity, is bounded below by C4(E)x(logx)2. The constants Ci(E), 1≤i≤4, are factors of an explicit constant C(E) that appears in the conjecture #{p<x:|ap(E)|is prime}∼C(E)x(logx)2.

Explicit Sato–Tate type distribution for a family of K ⁢ 3 K3 surfaces

Abstract In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato–Tate for non-CM elliptic curves). In analogy with Birch’s result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces A λ ⁢ ( p ) {A_{\lambda}(p)} of a certain family of K ⁢ 3 {K3} surfaces X λ {X_{\lambda}} with generic Picard rank 19 is the O ⁢ ( 3 ) {O(3)} distribution. This distribution, which we denote by 1 4 ⁢ π ⁢ f ⁢ ( t ) {\frac{1}{4\pi}f(t)} , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] {[-3,3]} and has vertical asymptotes at t = ± 1 {t=\pm 1} . Here we make this result explicit. We prove that if p ≥ 5 {p\geq 5} is prime and - 3 ≤ a &lt; b ≤ 3 {-3\leq a&lt;b\leq 3} , then | # ⁢ { λ ∈ 𝔽 p : A λ ⁢ ( p ) ∈ [ a , b ] } p - 1 4 ⁢ π ⁢ ∫ a b f ⁢ ( t ) ⁢ 𝑑 t | ≤ 98.28 p 1 / 4 . \biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-% \frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field 𝔽 p {\mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near t = ± 1 {t=\pm 1} . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms.

Functional graphs of families of quadratic polynomials

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.

Sums of multiplicative coefficients twisted by Frobenius traces

Sums of multiplicative coefficients twisted by Frobenius traces

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T

We study the moments of ${\rm Tr}(\Theta _\chi)$ as $\chi $ runs over Dirichlet characters defined over $\mathbb F_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behaves lik

Airy sheaves for reductive groups

We construct a class of ℓ $\ell$ -adic local systems on A 1 $\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y ′ ′ ( z ) = z y ( z ) $y^{\prime \prime }(z)=zy(z)$ . We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GL n $\mathrm{GL}_n$ , we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞ $\infty$ . These conjectures, in particular, imply cohomological rigidity of Airy sheaves.

$${}_{4}F_{3}$$-Gaussian hypergeometric series and traces of Frobenius for elliptic curves

$${}_{4}F_{3}$$-Gaussian hypergeometric series and traces of Frobenius for elliptic curves

Lower order terms for expected value of traces of Frobenius of a family of cyclic covers of ℙ𝔽q1 and one-level densities

We consider the expected value of [Formula: see text] where [Formula: see text] runs over a thin family of [Formula: see text]-cyclic covers of [Formula: see text] for any [Formula: see text]. We obtain many lower order terms dependent on the divisors of [Formula: see text]. We use these results to calculate the one-level density of the family and hypothesize a refined one-level density result.

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.

Equidistribution of αpθ$\alpha p^{\theta }$ with a Chebotarev condition and applications to extremal primes

We establish a joint distribution result concerning the fractional part of α p θ $\alpha p^\theta$ for θ ∈ ( 0 , 1 ) , α > 0 $\theta \in (0,1), \ \alpha >0$ , where p is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over Q $\mathbb {Q}$ . As an application, for a fixed non-CM elliptic curve E / Q $E/\mathbb {Q}$ , an asymptotic formula is given for the number of primes at the extremes of the Sato–Tate measure modulo a large prime ℓ. These are precisely the primes p for which the Frobenius trace a p ( E ) $a_p(E)$ satisfies the congruence a p ( E ) ≡ [ 2 p ] mod ℓ $a_p(E)\equiv [2\sqrt {p}] \bmod \ell$ . We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.

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.

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

AbstractLet $g \geq 1$ be an integer and let $A/\mathbb Q$ be an abelian variety that is isogenous over $\mathbb Q$ to a product of g elliptic curves defined over $\mathbb Q$ , pairwise non-isogenous over $\overline {\mathbb Q}$ and each without complex multiplication. For an integer t and a positive real number x, denote by $\pi _A(x, t)$ the number of primes $p \leq x$ , of good reduction for A, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$ and $\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$ if $t \neq 0$ . These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $\operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|&gt;p^{\frac {1}{3 g + 1} - \varepsilon }$ for any fixed $\varepsilon&gt;0$ .

A note on the trace of Frobenius for curves of the form y² = x³ + dx

An explicit description of the trace of Frobenius is given for any elliptic curve over ℚ of the form y² = x³ + dx This description leads to an algorithm which computes the trace at a cost of one modular exponentiation.

Hypergeometric L-functions in average polynomial time

We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.