Daniel Delbourgo

Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Let E/Q be a semistable elliptic curve, and p 6= 2 a prime of bad multiplicative reduction. For each Lie extension QFT /Q with Galois group G∞ ∼= ZpoZp , we construct p-adic L-functions interpolating Artin twists of the Hasse-Weil L-series of the curve E. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern-Yang Lee’s results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the MH(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

Transition formulae for ranks of abelian varieties

Let A/k denote an abelian variety defined over a number field k with good ordinary reduction at all primes above p, and let K∞ = ⋃ n≥1Kn be a p-adic Lie extension of k containing the cyclotomic Zp-extension. We use K-theory to find recurrence relations for the λ-invariant at each σ-component of the Selmer group over K∞, where σ : Gk → GL(V ). This provides upper bounds on the Mordell-Weil rank for A(Kn) as n → ∞ whenever G∞ = Gal(K∞/k) has dimension at most 3.

HIGHER ORDER CONGRUENCES AMONGST HASSE–WEIL

For the

L

(d+1)

-VALUES

-dimensional Lie group

EXCEPTIONAL ZEROES OF P-ADIC L-FUNCTIONS OVER NON-ABELIAN FIELD EXTENSIONS

G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}

IMPROVING THE CHEN AND CHEN RESULT FOR ODD PERFECT NUMBERS

, we determine through the use of

A Dirichlet series expansion for the p-adic zeta-function

p

Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

-power congruences a necessary and sufficient set of conditions whereby a collection of abelian

Algebraic invariants arising from the chromatic polynomials of theta graphs

L