Hi all, I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function attached to Hida families of modular …

Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta_{p^n})$ (where $\zeta_{p^n}$ is a primitive root of unity of exact order $p^n$ …

I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v_p(B_{p-i}) = v_p(|A_i|)$, where $A_i$ denotes the $i$-th eigenspace of the Galoi …

If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in a …

In Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1985), 76–91, Roland Gillard shows the following result ( …

[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely. As the title says, I would …

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functi …

One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-the …

In http://mathoverflow.net/questions/60326 it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is no …

Motivation: Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyc …

Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring. Let G be a connected p-divisible group which is finite-dimen …

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$. If $E / \m …

In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that "in a cyclotomic $\mathbf{Z}_p$-extension only finitely many prime …

Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Mor …

I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k_1 \equiv k_2 \pmod {p^{n}(p-1)}$ for some integer …