Browsing by Author "Levin, Brandon"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Components of the Emerton-Gee Moduli Stack of Galois Representations for GL2 and A Graph-Theoretic Approach to Computing Selmer Groups of Elliptic Curves over Q(i)
    (2025-04-25) Savoie, Ben; Levin, Brandon
    Let 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.
  • Thumbnail Image
    Item
    G-valued crystalline deformation rings in the Fontaine–Laffaille range
    (Cambridge University Press, 2023) Booher, Jeremy; Levin, Brandon; Mathematics
    Let G be a split reductive group over the ring of integers in a ppp-adic field with residue field FF\mathbf {F}. Fix a representation ¯ρρ¯¯¯\overline {\rho } of the absolute Galois group of an unramified extension of QpQp\mathbf {Q}_p, valued in G(F)G(F)G(\mathbf {F}). We study the crystalline deformation ring for ¯ρρ¯¯¯\overline {\rho } with a fixed ppp-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for GGG-valued representations. In particular, we give a root theoretic condition on the ppp-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.
  • Thumbnail Image
    Item
    Serre weights for three-dimensional wildly ramified Galois representations
    (MSP, 2024) Le, Daniel; Le Hung, Bao V.; Levin, Brandon; Morra, Stefano; Mathematics
    We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod p Galois representations under a genericity condition when the field is unramified at p. This removes the assumption made previously that the representation be tamely ramified at p. We also prove a version of Breuil’s lattice conjecture and a mod p multiplicity one result for the cohomology of U(3)-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023).