FINITE FIELD ELEMENTS OF HIGH ORDER ARISING FROM MODULAR CURVES (APPEARED IN DESIGNS, CODES, AND CRYPTOGRAPHY)

Abstract. In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result. 1

High Order Elements in Finite Fields Arising from Recursive Towers

We provide a recipe to construct towers of fields producing high order elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot 3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n} = v(x_{n - 1})$, for odd $q$, or $x_{n}^3 + x_{n} = v(x_{n - 1})$, for $q=2$, where $v(x)$ is a polynomial of small degree over the prime field $\mathrm{GF}(q,1)$ and $x_n$ belongs to the finite field extension $\mathrm{GF}(q,2^n)$, for $q$ odd, or to $\mathrm{GF}(2,2\cdot 3^n)$. Several examples are carried out and analysed numerically. The lower bounds of the orders of the groups generated by $x_n$, or by the discriminant $\delta_n$ of the polynomial, are similar to the ones obtained in [BCG+09], but we get better numerical results in some cases

On Elliptic Curves, Modular Forms, and the Distribution of Primes

In this thesis, we present four problems related to elliptic curves, modular forms, the distribution of primes, or some combination of the three. The first chapter surveys the relevant background material necessary for understanding the remainder of the thesis. The four following chapters present our problems of interest and their solutions. In the final chapter, we present our conclusions as well as a few possible directions for future research. Hurwitz class numbers are known to have connections to many areas of number theory. In particular, they are intimately connected to the theory of binary quadratic forms, the structure of imaginary quadratic number fields, the theory of elliptic curves, and the theory of modular forms. Hurwitz class number identities of a certain type are studied in Chapter 2. To prove these identities, we demonstrate three different techniques. The first method involves a relation between the Hurwitz class number and elliptic curves, while the second and third methods involve connections to modular forms. In Chapter 3, we explore the construction of finite field elements of high multiplicative order arising from modular curves. The field elements are constructed recursively using the equations that Elkies discovered to describe explicit modular towers. Using elementary techniques, we prove lower bounds for the orders of these elements. Prime distribution has been a central theme in number theory for hundreds of years. Mean square error estimates for the Chebotarëv Density Theorem are proved in Chapter 4. These estimates are related to the classical Barban-Davenport-Halberstam Theorem and will prove to be indispensable for our work in Chapter 5, where we take up the study of the Lang-Trotter Conjecture \u27on average\u27 for elliptic curves defined over number fields. We begin Chapter 4 by proving upper bounds on the mean square error in Chebotarëv\u27s theorem. It is this upper bound which features as a key ingredient in Chapter 5. As another application of this upper bound, we continue in Chapter 4 to prove an asymptotic formula for the mean square error. In Chapter 5, we turn to the discussion of the Lang-Trotter Conjecture for number fields \u27on average.\u27 The Lang-Trotter Conjecture is an important conjecture purporting to give information about the arithmetic of elliptic curves, the distribution of primes, and GL(2)-representations of the absolute Galois group. In this chapter, we present some results in support of the conjecture. In particular, we show that the conjecture holds in an average sense when one averages over all elliptic curves defined over a given number field