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### Abstract

This thesis is made up of $3$ separate pieces of work in two themes. In the first half, we prove a few cases of the Sato-Tate conjecture, which says that for an abelian surface $A$ over a totally real field $F$, the Frobenius elements $\Frob_{\lambda}$ acting on the dual of the $\ell$-adic Tate module can be formed into a compatible system of elements over all $\ell$, viewed (up to twist) as lying in a compact subgroup of $\GL_4(\C)$, and have traces that are equidistributed according to the smallest such compact subgroup possible. To do so, we use a result of Allen et al which proves automorphy of certain $\ell$-adic representations, and in another case we construct a new decomposition of the $\ell$-adic Tate module representation as a tensor product of a finite-image representation and a $2$-dimensional representation easily handled by earlier methods. Then we consider the final remaining cases and prove some partial results on the distribution of the traces of the Frobenii, and conversely explain precisely why we can't say more without further automorphy theorems.
In the latter half of this thesis, we consider the question of how the odd-power coefficients of a modular form control the even-power coefficients in the space of modular forms of weight $2$ level $\Gamma_0(N)$ with $N$ prime, from two different angles. We first study a question of Kedlaya and Medvedovsky about the number of modular lifts of a mod $2$ dihedral representation, and give lower bounds for the number of such lifts depending on $N\bmod8$ and whether the representation is totally real. We use multiple different methods to construct lifts: in some cases, we are able to use the connectedness of the real points of the Jacobian $J_0(N)$ of the modular curve $X_0(N)$ to double the dimension; in other cases, we are able to use the class group of the fixed field of the representation to manually construct weight $1$ forms that can be multiplied by a lift of the Hasse Invariant to give weight $2$ forms of the correct level and Nebentypus.
We then prove that the difference between the anemic Hecke algebra that excludes $T_2$ and the full Hecke algebra including $T_2$ is exactly described by the space of Katz forms in characteristic $2$, weight $1$ and level $\Gamma_0(N)$. We prove first that the difference is encompassed in the space of mod $2$ forms with only even-power terms, which then arise from weight $1$ forms by squaring. We then prove that there are no weight $2$ level $\Gamma_0(N)$ Katz forms, so every form arising from weight $1$ is a classical form, completing the bijection between the Katz forms in weight $1$ and the weight $2$ forms with only even-power monomials, and hence with the quotient $\T/\T^{\an}$. Finally, we end with questions about the proportion of primes $N$ for which $\T$ of level $N$ is equal to $\T^{\an}$; if $N\equiv3\bmod4$ there are only finitely many examples, but for $N\equiv1\bmod4$ we observe that it's probable there are a positive proportion of such primes.

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