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I think this question is based on a misconception. “Being an eigenvariety” isn’t a rigorously defined property of a space (or map of spaces) which you could prove to hold or to not hold. It’s more lik …

You might want to study the work of Darmon--Lauder--Rotger, notably this paper: https://web.mat.upc.edu/victor.rotger/docs/DLR1.pdf They study cases of the equivariant BSD conjecture where $\rho$ is a …

Are you absolutely sure you want to compute $p$-adic etale cohomology for a smooth proper $\mathbb{Z}[1/S]$-scheme with $p \notin S$, so $p$ is not invertible on $X$? This will be painful, and I stron …

Even for 0-dimensional varieties over number fields, the statement of Poitou–Tate duality is much more subtle than this: it's not enough just to compare the kernels of base-extension to $\overline{K}$ …

I think your question already contains its own answer. In classical, "vertical" Iwasawa theory one studies class groups, or other arithetic widgets like elliptic curve Selmer groups, in a limit over $ …

(1) Borel's article in the Corvallis proceedings does this slightly differently: he chooses a specific Frobenius element $\sigma$, and then looks at the subset $\widehat{G} \times \{\sigma\}$ of ${}^L …

Yes, this is true. It works with arbitrary algebraic varieties, no need to be specific to Shimura varieties. Let $X \to^{\pi} Spec(K)$ be an algebraic variety, $\mathcal{F}$ an etale sheaf on $X$, and …

All of your questions are undermined by the same fundamental issue: you cannot talk about "the" p-adic $L$-function in this generality, because there is no sensible definition of what a $p$-adic $L$-f …

This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts. One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ …

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question). Your $S(N)$ is naturally a scheme over $\mathbb …

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a definitio …

Symplectic case: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}_{2n}$ is more convenient to work with than $\operatorname{Sp}_{2n}$. Firstly: there is no Shimura datum w …

I remember discussing this with Emmanuel Kowalski not long ago. The short answer is that generalising the result to $J_1(N)$ is an open problem, and seems to be very difficult.

This is essentially the same as your other recent CM-theory question, in a mild disguise; for both questions the point is that $\psi(I)$ is a generator of $I$. This follows easily from the fact that $ …

Let $n$ be a prime number which has remainder 5 or 7 on division by 8. Then there exists a right-angled triangle with integer side lengths whose area is $n$ times a square. This is a theorem of Paul M …