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Showing 1 to 8 of 8 for “"Selmer group"”.
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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)
… 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 …
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Selmer groups as flat cohomology groups
… prime number p, Bloch and Kato showed how the p Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm1-Selmer group Selpmn A need not be determined by the mod pm Galois representation A[pm]; we show, however, that this is the case …
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Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction
… Tate curve extensions F[infinity], with Galois group G = Gal(F[infinity]/Q). I show how the p[infinity]-Selmer group of E over F[infinity] controls the p[infinity]-Selmer rank growth within the false Tate curve extension, and how it is connected to the root numbers of E twisted by absolutely …
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Iwasawa theory of p-adic Lie extensions
The Iwasawa theory of p-adic Lie groups investigates arithmetic objects above infinite field extensions of a number field k whose Galois group is a p-adic analytic group. The most prominent example (due to Serre) is produced by adjoining the p-torsion points of an elliptic curve defined over k …
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Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves
… on computing the Cassels-Tate pairing on the 2-Selmer group of J. We start by studying the Cassels-Tate pairing when J admits a Richelot isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute the Cassels-Tate pairing ⟨ , ⟩CT on Selφ(J)×Selφ(J) following the Weil …
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Iwasawa theory for tensor products of Hilbert modular forms
… critical values of the complex L-series, and the Selmer group which is an important object used to control the growth of arithmetic data. In this thesis, we explore some questions that arise organically from the Iwasawa Main Conjecture, applied to Hilbert modular forms and their tensor products. …
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Capitulation Discriminants of Genus One Curves
… to non-trivial elements of the Tate-Shafarevich group of the Jacobian curve E of C. Such a curve C will always admit a point over a degree n eld extension F of Q, and we say C capitulates over F, and that the discriminant of F is a capitulation discriminant for C. Our aim will be to obtain a …