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Register a new userModularity of $operatorname{PGL}_2(mathbb{F}_p)$-representations over totally real fields
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We study an analogue of Serres modularity conjecture for projective representations $overline{rho}: operatorname{Gal}(overline{K} / K) rightarrow operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.
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Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d e0$ and $c^2-4d e0$, we show that there are infinitely many odd primes $p$ such that $$detbigg[left(frac{i^2+cij+dj^2}{p}right)bigg]_{0le i,jle p-1}=0,$$ where $(frac{cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
Let $mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $mathcal{R}=mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_p+u^{3}mathbb{F}_{p}+cdots+u^{k}mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $mathbb{Q}(sqrt{p})$ in all positive characteristic $p otequiv 1pmod{24}$.
In this paper, we examine how far a polynomial in $mathbb{F}_2[x]$ can be from a squarefree polynomial. For any $epsilon>0$, we prove that for any polynomial $f(x)inmathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial $g(x)inmathbb{F}_2[x]$ such that $mathrm{deg} (g) le n$ and $L_{2}(f-g)<(ln n)^{2ln(2)+epsilon}$ (where $L_{2}$ is a norm to be defined). As a consequence, the analogous result holds for polynomials $f(x)$ and $g(x)$ in $mathbb{Z}[x]$.
By work of Belyi, the absolute Galois group $G_{mathbb{Q}}=mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q})$ of the field $mathbb{Q}$ of rational numbers can be embedded into $A=mathrm{Aut}(widehat{F_2})$, the automorphism group of the free profinite group $widehat{F_2}$ on two generators. The image of $G_{mathbb{Q}}$ lies inside $widehat{GT}$, the Grothendieck-Teichmuller group. While it is known that every abelian representation of $G_{mathbb{Q}}$ can be extended to $widehat{GT}$, Lochak and Schneps put forward the challenge of constructing irreducible non-abelian representations of $widehat{GT}$. We do this virtually, namely by showing that a rich class of arithmetically defined representations of $G_{mathbb{Q}}$ can be extended to finite index subgroups of $widehat{GT}$. This is achieved, in fact, by extending these representations all the way to finite index subgroups of $A=mathrm{Aut}(widehat{F_2})$. We do this by developing a profinite version of the work of Grunewald and Lubotzky, which provided a rich collection of representations for the discrete group $mathrm{Aut}(F_d)$.
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