In this thesis, we use the Serre-Tate deformation theory for ordinary abelian varieties to study its associated p-adic Galois representations. As applications, we study two types of questions. The first is to determine the indecomposability of the Galois representations restricted to the p-decomposition group attached to a non CM nearly ordinary weight two Hilbert modular form over a totally real ¯eld. Then second is to study the Mumford-Tate conjecture for absolutely simple abelian fourfolds with trivial endomorphism algebras.

The dissertation is a study of the modal accounts of knowledge. It is a platitude in epistemology that knowledge is incompatible with a belief’s being true as a matter of luck or coincidence. In order to eliminate luckily/coincidentally true beliefs from the realm of knowledge, an anti-luck condition or non-coincidental condition on knowledge is needed. Once the condition is unpacked, we will have, at least, a partial analysis of knowledge. An influential view in epistemology today is that the condition should be cashed out in modal notions. Roughly speaking, S’s belief in p is not true as a matter of luck or coincidence only if there is a modal connection, e.g., safety or sensitivity, between S’s belief in p and p, viz., S’s belief in p is true not only in the actual world but also in a relevant set of possible worlds. I argue that the modal connection is both too strong and too weak as the anti-luck condition no matter whether the modal account is restricted to beliefs in contingent truths or strengthened to accommodate beliefs in necessary truths. There are cases where the belief is luckily true despite exhibiting the modal connection as well as cases where the belief is non-luckily true despite not exhibiting the modal connection. In addition, it does not help to tweak the modal connection in terms of its modal strength. Therefore, the modal account is extensionally inadequate. What’s worse, the modal account fails to capture the plausible idea that we can always extend our knowledge by competently deducing consequences from what is already known. In a word, whether a belief is non-luckily or non-coincidentally true is orthogonal to whether it exhibits the modal connection which, in turn, makes the modal account of knowledge extensionally inadequate. The modal account fails because it does not capture a key feature of coincidence, i.e., a coincidence is a concurrence without an appropriate explanatory connection. This illuminates another approach that makes use of the notion of explanation. I then sketch the explanationist account of knowledge according to which, in order to know, one’s forming a belief in the target proposition and the target proposition’s being true should be appropriately explanatorily connected. Otherwise, we would have a case of epistemic coincidence which falls short of knowledge.