We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over Q. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.Date: May 19, 2015.

An eta-quotient of level N is a modular form of the shape f (z) = δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ 0 (N ) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f (z) is a holomorphic modular form that is nonvanishing on the upper half plane and has integer Fourier coefficients at infinity, then f (z) is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of J 0 (2 k )(Q).

Abstract. Given an abelian algebraic group A over a global field F , α ∈ A(F ), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F ) with α ∈ 2A(F (A[4])), then the density of p with (α mod p) having odd order is 11/21.

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Abstract. A cusp form f (z) of weight k for SL 2 (Z) is determined uniquely by its first ℓ := dim S k Fourier coefficients. We derive an explicit bound on the nth coefficient of f in terms of its first ℓ coefficients. We use this result to study the nonnegativity of the coefficients of the unique modular form of weight k with Fourier expansionIn particular, we show that k = 81632 is the largest weight for which all the coefficients of F k,0 (z) are non-negative. This result has applications to the theory of extremal lattices.