For a two-dimensional p-adic Galois representation V associated to a p-ordinary Hecke eigen cusp form f of weight k > 1, we identify the L-invariant (of R. Greenberg) of the (three dimensional) adjoint square Ad(V) of V with the derivative of the p-coefficient of the Lambda-adic lift of f. By this result, for a given p-adic analytic family of ordinary Hecke eigenforms, the L-invariant does not vanish for almost all members in the p-adic family (as expected).

In this thesis, we study the structure of various arithmetic cohomology groups as Iwasawa

modules, made out of two different $p$-adic variations. In the first part, for an abelian variety

over a number field and a $\mathbb{Z}_p$-extension, we study the relation between structure of

the Mordell-Weil, Selmer and Tate-Shafarevich groups over the $\mathbb{Z}_p$-extension as Iwasawa

modules.

In the second part, we consider the tower of modular curves, which is an analogue of the

$\mathbb{Z}_p$-extension in the first part. We study the structure of the ordinary parts of the arithmetic

cohomology groups of modular Jacobians made out of this tower. We prove that the ordinary

parts of $\Lambda$-adic Selmer groups coming from one chosen tower and its dual tower have almost

the same $\Lambda$-module structures. This relation of the two Iwasawa modules explains well the

functional equation of the corresponding $p$-adic $L$-function. We also prove the cotorsionness

of $\Lambda$-adic Tate-Shafarevich group under mild assumptions.

We examine the $p$-adically completed Mordell-Weil groups $\pJ(K)$ and $\calpJQ(K)$, where $K$ is a $p$-adic number field and $\pJ$ and $\calpJQ$ arise as Hecke components of certain $p$-adic completions of Jacobians of either a suitable elliptic modular curve or a suitable Shimura curve. The local ring $\mathbb{T}$ is an appropriate component of the Hecke algebra of a given level $N$ and $\mathbb{T}_Q$ is the corresponding local ring with level augmented by primes in $Q$. For elliptic modular curves, $Q$ refers to a finite set of primes $\{q_1,\dots,q_r\}$ at which the residual Galois representation determined by $\mathbb{T}$ is $q$-distinguished (and analogously for Shimura curves). The $\calpJQ$ carry a natural $\Delta_Q$-action, where $\Delta_Q$ is the $p$-Sylow subgroup of $(\mathbb{Z}/\prod_{i=1}^r q_i\mathbb{Z})^\times$. We study the action of $\Delta_Q$ on $\calpJQ$ and show that we can form a Taylor-Wiles system $(\mathbb{T}_Q,\calpJQ(K)^\ast)_{Q \in X}$, where $X$ ranges over an infinite set of finite sets $Q$ consisting of $q$-distinguished primes and $\ast$ denotes the $\mathcal{O}$-dual for $\mathcal{O}$ a discrete valuation ring, finite and flat, over $\mathbb{Z}_p$. As a consequence, we show that $\pJ(K)$ is free of finite rank $[K:\mathbb{Q}_p]$ over the local ring $\mathbb{T}$ of the Hecke algebra.

In this thesis, we use the Serre-Tate deformation theory for ordinary abelian varieties to study its associated p-adic Galois representations. As applications, we study two types of questions. The first is to determine the indecomposability of the Galois representations restricted to the p-decomposition group attached to a non CM nearly ordinary weight two Hilbert modular form over a totally real ¯eld. Then second is to study the Mumford-Tate conjecture for absolutely simple abelian fourfolds with trivial endomorphism algebras.

Arithmetic invariants are often naturally associated to motives over number fields.

One of the basic questions is the non-triviality of the invariants.

One typically expects generic non-triviality of the invariants as the motive varies in a family.

For a prime $p$, the invariants can often be normalised to be $p$-integral.

One can thus further ask for the generic non-triviality of the invariants modulo $p$.

The invariants can often be expressed in terms of modular forms.

Accordingly, one can try to recast the non-triviality as a modular phenomenon.

If the phenomena can be proven, the non-triviality typically follows in turn.

This principle can be found in the work of Hida and Vatsal among a few others.\

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We have been trying to explore a strategy initiated by Hida in the case of central criticial Hecke L-values over the $\Z_p$-anticyclotomic extension of a CM-field.

The strategy crucially relies on a linear indepedence of mod $p$ Hilbert modular forms.

Several arithmetic invariants seem to admit modular expression analogous to the case of Hecke L-values.

This includes the case of Katz $p$-adic L-function, its cyclotomic derivative and $p$-adic Abel-Jacobi image of generalised Heegner cycles.

We approach the non-triviality of these invariants based on the independence.

An analysis of the zero set of the invariants suggests finer versions of the independence.

We approach the versions based on Chai's theory of Hecke stable subvarieties of a mod $p$ Shimura variety.

We formulate a conjecture regarding the analogue of the independence for mod $p$ modular forms on other Shimura varieties.

We prove the analogue in the case of quaternionic Shimura varieties over a totally real field.