Trace Of Frobenius Research Articles

Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.

Galois representations attached to moments of Kloosterman sums and conjectures of Evans

AbstractKloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

P-adic Gamma function and the polynomials [formula omitted] and [formula omitted] over [formula omitted

In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. For d⩾2, we establish four different expressions for the number of distinct zeros of the polynomials xd+ax+b and xd+axd−1+b over Fq in terms of special values of the function Gd−1d−1[⋯].

P-Adic gamma function and the trace of Frobenius of elliptic curves

In [12], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function G22[⋯]. As a result, we obtain two transformation formulas of the function G22[⋯] with different parameters.

Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

The Frobenius traces distribution for modular Abelian surfaces

We prove that there exist at least 15 pairs of measures for which the two normalized Frobenius traces on modular Abelian surfaces are equidistributed. Under an assumption about stochastic independence, we prove that there are exactly 18 such pairs of measures. Moreover, we describe all of them by giving their moments and distributions.

Hypergeometric functions over 𝔽_{𝕢} and traces of Frobenius for elliptic curves

We present explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2 F 1 {_{2}}F_1 -hypergeometric functions over F q \mathbb {F}_q for q ≡ 1 ( mod 6 ) q \equiv 1 ( \text {mod}~6) and q ≡ 1 ( mod 4 ) q \equiv 1 ( \text {mod}~4) .

Elliptic curves and special values of Gaussian hypergeometric series

We express the trace of Frobenius of certain families of elliptic curves in terms of Gaussian hypergeometric functions. We also find some special values of F12 Gaussian hypergeometric functions containing characters of order 4 as parameters.

The trace of Frobenius of elliptic curves and thep-adic gamma function

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p >3, that the trace of Frobenius of any elliptic curve over Fp, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p 1 .mod 12/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define

Local models of Shimura varieties and a conjecture of Kottwitz

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

Traces of Hecke operators in level 1 and Gaussian hypergeometric functions

We provide formulas for traces of p t h p^{th} Hecke operators in level 1 in terms of values of finite field 2 F 1 _2F_1 -hypergeometric functions, extending previous work of the author to all odd primes p p instead of only those p ≡ 1 ( mod 12 ) p \equiv 1 \pmod {12} . We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p > 3 p > 3 .

Rationality of trace and norm L-functions

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of the local Frobenius traces of F over the points of G(k_r) (where k_r is the extension of degree r of k) with norm t. For G the one-dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.

Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures

We extend Kobayashiʼs formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case ap≠0, where ap is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions Lp♯ and Lp♭ with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when ap=0 and p is odd. We then generalize Kobayashiʼs methods to define two Selmer groups Sel♯ and Sel♭ and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our p-adic L-functions Lp♯ and Lp♭. We then use results by Kato to prove a divisibility statement. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators T k ( p ) for all p > 3 on S k ( Γ 0 ( 3 ) ) and S k ( Γ 0 ( 9 ) ) , the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over F q , when q ≡ 1 ( mod 3 ) . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η ( 3 z ) 8 , the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.

A Note on lower bounds for Frobenius traces

A Note on lower bounds for Frobenius traces

Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve E over F q satisfying j(E) ≠ 0,1728 and q ≡ 1 (mod 12) in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced by Fuselier for one-parameter families of curves to express the trace of Frobenius of E as a function of its j-invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.

3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

2, 3, 5, Legendre: ± Trace Ratios in Families of Elliptic Curves

For a given integer A and various families of elliptic curves over finite fields, we compare the number of occurrences of A with the number of occurrences of —A as the trace of Frobenius in the family.

Corrigendum to ``Distribution of the traces of Frobenius on elliptic curves over function fields'' (Acta Arith. 106 (2003), 255–263)

Corrigendum to ``Distribution of the traces of Frobenius on elliptic curves over function fields'' (Acta Arith. 106 (2003), 255–263)

Hypergeometric functions over ${\mathbb {F}_p}$ and relations to elliptic curves and modular forms

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.