Ethan Smith

Finite field elements of high order arising from modular curves

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.

Sums of Euler products and statistics of elliptic curves

We present several results related to statistics for elliptic curves over a finite field

A Cohen-Lenstra phenomenon for elliptic curves

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Group Structures of Elliptic Curves Over Finite Fields

Fp as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang–Trotter conjecture, the average Koblitz conjecture, and the vertical Sato–Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring’s theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

A generalization of the Barban–Davenport–Halberstam Theorem to number fields

Letting p vary over all primes and E vary over all elliptic curves over the nite eld F p , we study the frequency to which a given group G arises as a group of points E(F p ). It is well-known that the only permissible groups are of the form Gm;k := Z =mZ � Z =mkZ . Given such a candidate group, we let M(Gm;k ) be the frequency to which the group Gm;k arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for M(Gm;k ) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(Gm;k ), pointwise and on average. In particular, we show that M(Gm;k ) is bounded above by a constant multiple of the expected quantity when mk A and that the conjectured asymptotic for M(Gm;k ) holds for almost all groups Gm;k when mk 1= 4 ϵ . We also apply our methods to study the frequency to which a given integer N arises as the group order #E(F p ).

Corrigendum: Elliptic curves with a given number of points over finite fields

Given an elliptic curve E and a finite Abelian group G, we consider the problem of counting the number of primes p for which the group of points modulo p is isomorphic to G. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves.

Averages of the number of points on elliptic curves

Let H(N) denote the Hurwitz class number. It is known that if p is a prime, then ∑|r|<2pH(4p−r2)=2p. In this paper, we investigate the behavior of this sum with the additional condition r≡c(modm). Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler–Selberg trace formula for Hecke operators to obtain Hurwitz class number relations. The cases m=2,3 and 4 are treated in full. Partial results, as well as several conjectures, are given for m=5 and 7.

A VARIANT OF THE BARBAN-DAVENPORT-HALBERSTAM THEOREM

It is well-known that if