Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $g$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $g$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $g$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Let $R$ be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over $R$.

We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings of a homomorphism $\varphi :\,R\,\to \,S$ of finite flat dimension.

We prove that, for any prime ℓ and any even integer n, there are infinitely many exponents k for which appears as a Galois group over . This generalizes a result of Wiese from 2006, which inspired this paper.

We define here an analogue, for a semi-stable group scheme whose generic fiber is an abelian variety, of M. J. Taylor's class-invariant homomorphism (defined for abelian schemes), and we give a geometric description of it. Then we extend a result of Taylor, Srivastav, Agboola and Pappas concerning the kernel of this homomorphism in the case of an elliptic curve.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Rédei's triple symbol.