Let $N$ be a square-free, positive integer and let $p\ge 5$ be a prime dividing $N$. In this work, we construct semistable models of Atkin--Lehner quotients of $X_0(N)$ over the Witt ring $W(\overline{\mathbb{F}}_p)$ by taking the corresponding quotients of the Deligne--Rapoport model of $X_0(N)$. We give an explicit description of the special fiber of the models and obtain genus formulae for the Atkin--Lehner quotients of $X_0(N)$.
Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the \'etale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable \'etale cohomology. We also reprove,...
Let $p$ be a prime $\ge 5$. We determine the semisimplifications of the mod $p$ reductions of irreducible two-dimensional crystalline representations of the Galois group of ${\mathbb Q}_p$ of slope $2$ and weights $k\ge3p+4$ with $k \not \equiv 6 \bmod (p-1)$. The proof relies on the compatibility between the $p$-adic...
Let $\mathcal G=H\rtimes\Gamma$ be a semidirect product of a finite group $H$ with $\Gamma=\mathbb Z_p^d$, the direct product of finitely many copies of the additive group of the $p$-adic integers. In the case $d=1$, Iwasawa algebras of such profinite groups were studied by Ritter--Weiss, Lau, Nickel, and the author. W...
Let $E$ be an elliptic curve define over $\mathbb{Q}$. Further assume $E$ has good ordinary reduction at $p=2$. In this article, we prove Greenberg's conjecture on the value of the algebraic Iwasawa $\mu$-invariant when $E$ has a rational isogeny of degree 2, and obtain a sharp absolute upper bound of the $2$-adic $\mu...
Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitl...
Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a...
Pinxian Bie, Zheng-Jie Yu· 0 citations
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