Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell> C_E$. Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$-adic representations attached to two elliptic curves.

By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.

We prove that the Drinfeld center $\mathcal {Z}(\operatorname {Vec}^{\omega }_{A_5})$ of the pointed category associated with the alternating group $A_5$ is the unique example of a perfect weakly group-theoretical modular category of Frobenius–Perron dimension less than $14400$.

We characterize all algebraic numbers $\alpha $ of degree $d\in \{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $\alpha _1$, $\alpha _2$, $\alpha _3$, and $\alpha _4$ of $\alpha $ satisfying the relation $\alpha _{1}+\alpha _{2}=\alpha _{3}+\alpha _{4}$. In particular, we prove that an algebraic number $\alpha $ of degree 6 satisfies this relation with $\alpha _{1}+\alpha _{2}\notin \mathbb {Q}$ if and only if $\alpha $ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of $\mathbb {Q}(\alpha )$ for such algebraic numbers $\alpha $. We also consider similar relations $\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}=0$ and $\alpha _{1}+\alpha _{2}+\alpha _{3}=\alpha _{4}$ for algebraic numbers of degree up to 7.

Let $[a_1(x),a_2(x),a_3(x),\dots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. Denote by $S_{n}(x):=\sum _{k=1}^{n} a_{k}(x)$ the sum of partial quotients of x. From the results of Khintchine (1935), Diamond and Vaaler (1986), and Philipp (1988), it follows that for Lebesgue almost every $x \in (0,1)$,

$$\begin{align*}\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\frac{1}{\log 2} \quad \text {and} \quad \limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\infty. \end{align*}$$

Given positive Radon measures, $\mu $ and $\lambda $, on the complex unit circle, we show that absolute continuity of $\mu $ with respect to $\lambda $ is equivalent to their reproducing kernel Hilbert spaces of “analytic Cauchy transforms” in the complex unit disk having dense intersection in the space of $\mu $-Cauchy transforms.

Let $M_\mu $ be the uncentered Hardy–Littlewood maximal operator with a Borel measure $\mu $ on $\mathbb {R}$. In this note, we verify that the norm of $M_\mu $ on $L^p(\mathbb {R},\mu )$ with $p\in (1,\infty )$ is just the upper bound $\theta _p$ obtained by Grafakos and Kinnunen and reobtain the norm of $M_\mu $ from $L^1(\mathbb {R},\mu )$ to $L^{1,\infty }(\mathbb {R},\mu )$. Moreover, the norm of the “strong” maximal operator $N_{\vec {\mu }}^{n}$ on $L^p(\mathbb {R}^n, \vec {\mu })$ is also given.