Levin, Brandon2025-05-302025-05-302025-052025-04-25May 2025https://hdl.handle.net/1911/118523Let K be a finite unramified extension of Q_p with p ≥ 5. In the first part of this thesis, we study the local geometry of the irreducible components in the reduced part of the Emerton–Gee stack for GL_2, which serves as a moduli space for two-dimensional mod p representations of Gal(K/K). We determine precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We prove that the normalizations of these components admit smooth–local covers by Cohen-Macaulay and resolution-rational varieties, which are generally not Gorenstein. Finally, we determine the singular loci in the components, providing insights which up- date expectations about the conjectural categorical p–adic Langlands correspondence. In the second part of this thesis, we introduce a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve E_b : y^2 = x^3 + bx defined over Q(i), where b ∈ Z[i] and φ is a degree 2 isogeny of E_b. We begin by associating a weighted graph G_b to each curve E_b, whose vertices correspond to the odd Gaussian primes dividing b. The weights on the edges connecting these vertices are determined by the quartic residue symbols between these primes. We then establish a bijection between the elements of the φ-Selmer group of E_b and certain partitions of the graph G_b. This correspondence provides a linear-algebraic interpretation of the φ-Selmer group through the Laplacian matrix of G_b. Using our algorithm, we explicitly construct several subfamilies of elliptic curves E_b over Q(i) with trivial Mordell–Weil rank. Furthermore, by combining our method with Tao’s Constellation Theorem for Gaussian primes, we prove the existence of infinitely many elliptic curves E_b over Q(i) with rank exactly 2. Additionally, we show that for each pair of rational twin primes (p, q), the curve E_{pq} considered over Q(i) has rank either 2 or 4, with the rank exactly 2 when p ≡ 5 mod 8. Lastly, we show that for each rational prime of the form p = a^2 + c^4 (of which there are infinitely many), the elliptic curve E_{-p} over Q(i) has rank either 2 or 4, with rank exactly 2 if p ≡ 5 or 9 mod 16.application/pdfenNumber TheoryArithmetic Geometryp-adic Hodge TheoryLanglands Programp-adic LanglandsElliptic CurvesSelmer GroupsGraph TheoryThesis2025-05-30