Let ℓ be a prime number. Katz established a local-global principle for elliptic curves over a number field that have nontrivial ℓ-torsion locally everywhere. Sutherland gave an analogous local-global principle for elliptic curves that admit a rational ℓ-isogeny locally everywhere. By analyzing the subgroups of GL 2 (F ℓ ), we show that a failure of either of these "locally everywhere" conditions must be rather significant. More specifically, we prove that if an elliptic curve over a number field fails one of the above "locally everywhere" conditions, then its reductions at no more than 75% of primes ideals may satisfy the underlying local condition. In the appendix, we give for (conjecturally) all elliptic curve over the rationals without complex multiplication, the densities of prime that satisfy the local conditions mentioned above.

Let E/Q be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod ℓ Galois representation of E is surjective for each prime number ℓ that is sufficiently large. Under GRH, we give an upper bound on the largest non-surjective prime, in terms of the conductor of E, that makes effective a bound of the same asymptotic quality due to Larson-Vaintrob.

A curious number is a palindromic number whose base ten representation has the form a . . . ab . . . ba . . . a. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for such numbers to several single variable families. From here, we complete the proof in two different ways. The first approach is elementary, though somewhat ad hoc. The second entails studying integral points on elliptic curves and is more systematic.

Let A be a principally polarized abelian variety of dimension g over a number field K. Assume that the image of the adelic Galois representation of A is open. Then there exists a positive integer m so that the Galois image of A is the full preimage of its reduction modulo m. The least m with this property, denoted m A , is called the image conductor of A. A recent paper [2] established an upper bound for m A , in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. In this note, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.