Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moret\'o conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moret\'o conjecture for principal blocks.
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 $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Let $G$ be a finite solvable group such that, for any prime $p$ and any quotient $Q$ of $G$, there are two Sylow $p$-subgroups of $Q$ intersecting in $O_p(Q)$. Then, for every family $(P_i)_{i=1}^n$ of Sylow subgroups of $G$ for distinct primes $p_1,\ldots,p_n$, there exists $x \in G$ such that $P_i \cap P_i^x = O_{p_i...
Hong-Yi Huang, Francesca Lisi, A. Rizzoli et al.· 0 citations
Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of $\mathbb{F}_p$ with $p>3$. Generalizing results of Ghate--Le--Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use t...
Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}...
Zi-Chao Lin, Mu-Lun Yin· 2 citations
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