A pair elliptic curves E/Q and E’/Q are isogenous if and only if they have the same number of points mod p for every (good) prime p. A conjecture of Frey and Mazur predicts that E and E’ are isogenous if and only if they are N-congruent for any sufficiently large integer N > N_0 (i.e., #E(F_p) = #E’(F_p) mod N for all good p).

Congruences appear quite naturally in applications, for example:

- In isogeny-based cryptography (an abelian surface being (N,N)-split implies that the corresponding pair of elliptic curves are N-congruent).

- In Diophantine problems (e.g., Fermat’s last theorem),

- In descent obstructions (via Mazur’s notion of “visible elements of Sha”).

Despite the Frey—Mazur conjecture, it is not known for which integers there exist non-isogenous N-congruent elliptic curves: what is N_0? I will discuss progress towards refining the Frey—Mazur conjecture by studying the geometry of “Humbert surfaces” which parametrise N-congruences.