For $n\ge2$, let $F_n$ be the free group of rank $n$, and let $\IA(F_n)$ be the group of automorphisms of $F_n$ acting trivially on its abelianization. Two natural filtrations of $\IA(F_n)$ are its lower central series $\gamma_c(\IA(F_n))$ and the Andreadakis filtration $\mathrm{I}_{c+1}\mathrm{A}(F_n)$, consisting of automorphisms acting trivially on $F_n/\gamma_{c+1}(F_n)$, $c \geq 1$. Andreadakis conjectured that these filtrations coincide. We prove that the conjectured equality fails for all $n\ge3$ and $c\ge4$.
Let $R_n = \bigvee_{i=1}^{n} S^1$ be a finite connected one-dimensional CW complex with $n \geq 2$, and let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$, where $\pi_1(R_n)\cong F_n$. Let $p_k:\tilde{R}_n\rightarrow R_n$ be the standard $k$-sheeted regular cyclic cover of $R_n$ induced by...
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 $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $\mu$-i...
Let $G$ be a finite group, let $\rho(G)$ denote the number of distinct prime divisors of $|G|$, and let $\sigma(G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $\rho(n)=\max\{\rho(G) \mid G \text{ solvable}, \sigma(G)=n\}$. We prove that $\rho(n)\le 4n-1+\pi(2n)$ for all $n$,...
If $F$ is a number field and $F = F_0 \subset F_1 \subset F_2\subset \cdots$ is a tower of Galois extensions with $\mathrm{Gal}(F_n/F)$ cyclic of order $p^n$, K. Iwasawa gave a closed formula for the growth of the $p$-part of the class number of $F_n$ whenever $n$ is large enough. We present a new proof of this formula...
Let $F_n$ be the Fermat curve of degree $n$ over $\mathbb Q$, and let $D_n=F_n/S_3$ be its quotient by coordinate permutations. We prove that $D_n$ has trivial geometric automorphism group for every $n\ge 11$ with $\gcd(n,6)=1$. Its Jacobian has complex multiplication. For $n=11$, this Jacobian is absolutely simple of...
Saeed Tafazolian· 0 citations
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