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This could be hard (though I admit all analytic number theory questions look hard to me). Indeed, we know quite a lot about the endomorphism algebra $X$ (built from cocycles attached to extra twists) …

This was getting too long for a comment I'll post it as an answer. Though the set of $K$-special points on the canonical model does not depend on the choice of an embedding of $K\hookrightarrow\opera …

My comments are getting too long, so here is a tentative answer. First, a general statement: conjectures predicting special values of $L$-functions are formulated for all motives over number fields. …

It seems to me that the proof you describe is fine except maybe at the prime 2, the problem being that the argument you are sketching (presumably) relies on expressing the missing operators via traces …

If by $f_1\equiv f_2$ modulo $p$, you mean that $a_n(f_1)\equiv a_{n}(f_2)$ modulo $p$ for all $n\in\mathbb N$ or maybe for all except finitely many, then this theorem cannot be true. Let's start with …

This is an answer of rather low quality, but let me report that in several discussions about this topic with (many) experts over the span of many years, I have neither met anyone knowing of such an ex …

This hinges on what you mean exactly by explicit description. Here is what is happening. Let me write $N_f$ for the conductor of $f$. Fontaine defined a number of so-called period rings to study $p$- …

In the ordinary case, the argument is simple so let me recall it here. The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^ …

The answer to the question in the title is yes, as explained in the last paragraph below. However, under a literal interpretation of "can" (implying actual feasibility), I believe the answer to the q …

By a famous theorem of Manin, one can define $\Omega^{\pm}$ such that $L(f\otimes\chi,j)\in \Omega^{\epsilon}_{f}\mathbb Q$ with $\chi(-1)(-1)^{j}=\epsilon$. So the period depends on $\chi$ only insof …

(1) The answer is yes. In fact this was done by Ohta himself essentially simultaneously as the other cases. The relevant publication is here (the proof in the case $k\equiv 2$ is around 569/570). (2) …

Everything you wish for is true for modular forms over $\mathbb Q$ even at $p=2$, as it follows from refined forms of Serre's conjecture; here I am assuming of course that $\bar{\rho}$ is absolutely i …

No. A supercuspidal representation, a Steinberg twisted by a ramified character and a principal series ramified at both characters at $p$ will all have zero $a_{p}$. A reference for this is Jacquet- …

Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This …

In fact much more than the equality of conductor is true: the local Galois representation $\rho_{F,\lambda}|G_{\mathbb Q_{p}}$ obtained by restricting $\rho_{F,\lambda}$ to the decomposition group at …