4 problems

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $F/\mathbb{Q}$ be a finite real abelian extension with Galois group $G$, let $r={\rm rank}(E(\mathbb{Q}))$, and let $I_F$ be the…

Let $E/\mathbb{Q}$ be an elliptic curve, let $M$ be a positive integer, and let $\chi$ be an even Dirichlet character of conductor $M$. Let $\theta_{E,M}$ be the modular element, a…

Assume the paper's Hypothesis and set $r=r_{\rm alg}$. Let $F/\mathbb Q$, $G$, $I_F$, ${}_{c,d}z_F$, $\mathcal N_{F/\mathbb Q}$, $\iota_F$, ${\rm Boc}_{F,\bm{x}}$, and…

Mazur–Tate conjecture. For any $\psi\in\operatorname{Symb}_{\Gamma_0}(R)$ such that $T_q\psi=a_q\cdot\psi$ for all primes $q$, one has in $R$