In this paper we show how one can use an inner product formula of Heim giving the inner product of a pullback Eisenstein series from Sp10 to Sp2×Sp4×Sp4 with a newform on GL2 and a SaitoKurokawa lift to produce congruences between Saito-Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp4×GL2. The congruence is then used to produce nontrivial torsion...

While the literature on private sector expatriates is burgeoning, research on public sector expatriates is scant, despite their increasing numbers. This is unfortunate, as these two groups of expatriates may face different work conditions at their foreign assignment. The target group for this study was the US Department of Defense administrators located at US embassies around the world. Results...

Let $E$ be an elliptic curve over a number field $K$ defined by monic irreducible cubic polynomial $F(x)$. When is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms the $2$-rank modified ideal class group $L=K[x]/{(F(x))}$, which call \textit{semi-narrow group} $L$. We then provide several sufficient conditions for being nice prime. As application, when real quadr...

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zp-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasawa ...

Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.

Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic represent...

Let κ ≥ 6 be an even integer, M an odd square-free integer, and f ∈ S2κ−2(Γ0(M)) a newform. We prove that under some reasonable assumptions that half of the λ-part of the Bloch-Kato conjecture for the near central critical value L(κ, f) is true. We do this by bounding the l-valuation of the order of the appropriate Bloch-Kato Selmer group below by the l-valuation of algebraic part of L(κ, f). W...

We show that the parity conjecture for Selmer groups is invariant under deformation in p-adic families of self-dual pure Galois representations satisfying Pančǐskin’s condition at all primes above p. 2000 Mathematics Subject Classification: 11G40 11R23

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we ...