Non-equivalence of pro-$p$-Iwahori invariants

Kolyvagin's conjecture at non-ordinary primes

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...

On the saturability of $p$-adic Lie groups

Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-v...

Quasi-$F$-split primitive symplectic varieties in positive characteristic

Let $X$ be the good reduction of a projective hyperk\"ahler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof...

A COMPARISON

Homological branching laws for $p$-adic and mod $p$ representations of ${\rm GL}_2$ over a $p$-adic field

Let $F$ be a finite extension of $\mathbb Q_p$ and $p\geq 5$. In this paper, we establish homological branching laws for mod $p$ representations of $\mathrm{GL}_2(F)$ restricted to split and nonsplit maximal tori, treating principal and special series and, when $F=\mathbb Q_p$, supersingular representations. We determi...

Nonvanishing of cohomology for groups

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...