Watkins has conjectured that if $R$ is the rank of the group of rational points of an elliptic curve $E$ over the rationals, then $2^R$ divides the modular parametrisation degree. We show, for a certain class of $E$, chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain $2$-adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others,...

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ be a linear form with nonzero integer coefficients $u_1,...,u_h,v.$ Let $𝒜=(A_1,...,A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\varphi$ and the sets $𝒜$ and $B$ as follows : $$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$ If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are...

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map $T=T_{L/K}:L\to K$ and the $O_K$-ideal $T\left(O_L\right)\subseteq O_K$. By I(L/K) we denote the group indexof $T\left(O_L\right)$ in $O_K$ (i.e., the norm of $T\left(O_L\right)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T\left(O_L\right)$ (Theorem 1)....

Let $p$ be a prime and let $K$ be a number field. Let ${\rho }_{E,p}:{\mathrm{G}}_K\to \mathrm{Aut}\left(T_{p}E\right)\cong {\mathrm{GL}}_2\left({\mathbb{Z}}_p\right)$ be the Galois representation given by the Galois action on the $p$-adic Tate module of an elliptic curve $E$ over $K$. Serre showed that the image of ${\rho }_{E,p}$ is open if $E$ has no complex multiplication. For an elliptic curve $E$ over $K$ whose $j$-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ${\rho }_{E,p}$.

In this paper we give a characterization of $\sigma$-order continuity of modular function spaces $L_{\varrho }$ in terms of the existence of best approximants by elements of order closed sublattices of $L_{\varrho }$. We consider separately the case of Musielak–Orlicz spaces generated by non-$\sigma$-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.