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Could you explain me what are concretely the formulas in those cases? These formulas are equalities, so their content is that the left-hand side is equal to the right-hand; a tautology for sure, …

The usual normalizations for Galois representation attached to an eigencusp form $f$ is either to take the étale cohomology or the usual Tate module of the abelian variety cut out in the Jacobian of t …

UPDATE: I have updated this answer slightly to take into account Victor's remark. I think that the precise questions being asked admit a straightforward answer. At the moment, no such formula is know …

Euler systems and Kolyvagin systems are closely related but quite different beasts nevertheless. According to their respective definitions, Euler systems are systems of classes $\{c(n)\in H^1(G_{\mat …

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 …

Just on top of my head, I can think of the following themes, all having in common that they can be approached with a minimal knowledge and that there exists a continuous path from them to current rese …

There is an article of J.Parson and B.Gross (On the local divisibility of Heegner points) from which I think some very very special instance of r=2, d=2 can be deduced. The argument is a combination o …

First a short answer. I don't think one can say that the commutative analytic side is known, as you do. It is fully known only in the cyclotomic $\mathbb{Z}_{p}$ situation, assuming the ETNC and in th …

There is a very close analogy but to unravel it requires some work. So take $X$ a smooth curve over $\mathbb F_{\ell}$ (more generally you could take $X$ a scheme over $\mathbb F_{\ell}$) and let $\m …

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. …

As stated, the answer to your question is certainly no. For instance, an elliptic curve $E/\mathbb Q$ with split multiplicative ordinary reduction at $p$ will have unbounded Tamagawa number at $p$ i …

The quadratic reciprocity law is a special case of the product formula for Hilbert's symbol: for all $(a,b)\in\mathbb{Q}^{\times}$ \begin{equation} \underset{v}{\prod}(a,b)_{v}=1 \end{equation} where …

As Will Sawin points out, the following is mostly an answer to the question "Is it true that the set of primes such that $\rho_p$ is reducible is finite?" which is related but strictly stronger than t …

A small correction to Hunter Brooks's answer: Kato's Euler system is unrelated to Heegner points but comes from Beilinson's elements on modular curves. The fact that Heegner points form an Euler syste …