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By "norm 1 element", you mean "determinant 1 element", right? I guess also that your embedding $K \to M_2(\mathbf{Q})$ is chosen to send $O_K$ into $M_2(\mathbf{Z})$. Then $\Gamma$ is visibly a subgr …

I typed a comment but the formatting wouldn't come out right, so here it is as an answer! I cannot work out why you expect the "Plancherel or Parseval type" formula to work. Does it not bother you a …

I think this is false for cardinality reasons. Take some exceedingly thin, but infinite, set of primes P. Then you can take any valid Chebotarev datum and change it arbitrarily at each prime in P, and …

I believe Sug Woo Shin has done some work on a mod p theta correspondence; he has given a few talks with that title. I don't know if he's written anything up for public distribution though.

In order to get $\mathbb{C}_p$-valued functions, you need to choose a basis for the vector space $V_f$. If $f$ corresponds to an elliptic curve, there is a reasonably canonical way of doing this (usin …

This follows from well-known results (although perhaps not in a totally straightforward way). Let us write $x(a) = a^{-1} + p y(a)$, where $a^{-1}$ denotes the inverse of $a$ modulo $p^2$, and $y(a) …

This is a nice exercise. You start by making some reduction steps. Firstly, the BK Selmer group for $T$ is by definition saturated (i.e. it is the preimage in $H^1(K, T)$ of a $\mathbf{Q}_p$-subspace …

No. The Gauss sum is not a $p$-adic measure. One cheap way to see this is as follows: if $\chi$ has conductor $p^n$, the $p$-adic valuation of $G(\chi)$ is $n/2$. But if $\mu$ is a measure, the asympt …

If $H$ is definite, then the group of units of $H$ is finite. If $H$ is indefinite, then the group of units is a pretty chunky group; it embeds as a cocompact discrete subgroup of $SL(2, R)$, and the …

As Francois has very clearly explained, the Bloch--Kato dual exponential for $\mathbf{Q}_p(1)$ is indeed zero. Just as a footnote, let me say a few words about how one can "get at" the one-dimensional …

Let $K$ be a number field with $[K : \mathbf{Q}] = d$, and let $p$ be a prime. Let $\sigma_1, \dots, \sigma_d$ be the embeddings of $K$ into $\mathbf{C}_p$. Let $u_1, \dots, u_k$ be a basis of $\mathc …

This can't possibly work. Many p-adic Galois representations are not semisimple. For instance, $\mathbb{Z}_p$ occurs as a quotient of the Galois group of $\mathbb{Q}$ (as the Galois group of the cyclo …

Are you familiar with the paper "Overconvergent Eichler--Shimura isomorphisms" by Andreatta--Iovita--Stevens? In section 3 of this paper they give a variant of the "modular symbol" construction of the …

You are slipping up because $i$ does not generate the ring of integers of $L$ as an $\mathcal{O}_K$-algebra: we have $\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$, but $\mathcal{O}_L = \mathbb{Z}\left[i, \f …

Your question strongly suggests that you are trying to understand the set of integers that can be expressed as $a(b^2 - c^2) - b$. But this is rather easy: any integer has this form. Given any $N$, ju …