Order of reductions of elliptic curves in arithmetic progression

Official URL: https://risc01.sabanciuniv.edu/record=b2766804_(Table of contents)

Let E be an elliptic curve defined over a number field K with ring of integers R. We consider the set S of all the orders of reductions of E modulo the primes of R. Given an integer m > 1, one may ask how many residue classes modulo m have an intersection of positive density with S. Using results of Serre and Katz, we show that there are at least two such residue classes; except for explicit families of elliptic curves and corresponding values of m. We then describe this exceptional set of elliptic curves and list the values of m when K is of degree at most 3; or K is Galois of degree 4. We also consider the following divisibility question on orders of elliptic surfaces over finite fields. Given an integer m≥ 2 and a finite field k, is there an elliptic curve Et over k[t] such that for all k-rational values of t the order of Et(k) is divisible by m ? We suggest a method to construct such elliptic curves. Consequently, when m = 3, we provide two such elliptic curves over k[t] whenever k is of prime order congruent to 1 mod 3. Finally, we discuss how the growth of the order of the torsion subgroup of an elliptic curve E over K after a base change is linked to the divisibility of the orders of reductions of E modulo the primes of R. In particular, we provide examples of elliptic curves over the rational field for which we can list all the possible congruence classes of the orders of the reductions modulo a certain integer m≥ 2; together with the density of appearance of these congruence classes.