In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

1 Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication. For any positive integer k, let φk be the Galois representation associated to the k-division points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which φl is not surjective. 1

A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serre’s theorem. More precisely, we obtain a uniform version of Serre’s theorem for non-constant elliptic curves defined over function fields, and a uniform version of Serre’s theorem for one-parameter families of elliptic curves defined over Q.

Abstract. In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) ̸ = 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols. 1.

Let us recall the following theorem of J-P. Serre ([20], result (7)). Theorem 1 (Serre [20]). — Let E be an elliptic curve without complex multiplication over a number field K. There exists a number B(E, K) such that for any prime number p> B(E, K), the image GE,p of Gal ( ¯ K/K) in the group Aut(E[p]) ≃ GL2(Fp) of

1.1. Background. L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the