We show that all of the nice behavior for Tamagawa numbers, Tate–Shafarevich sets, and other arithmetic invariants of pseudoreductive groups over global function fields, proved in another work, fails in general for noncommutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over global function fields.

Let $E∕\mathbb{Q}$ be an elliptic curve of conductor $N$, let be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate–Shafarevich group of $E$ over the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$ in terms of Heegner points.

In this paper, we give a proof of Perrin-Riou’s conjecture under mild hypotheses. Our proof builds on Howard’s theory of bipartite Euler systems and Wei Zhang’s work on Kolyvagin’s conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa–Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini, Darmon and Prasanna.

Nous construisons un modèle local pour les schémas de Hilbert–Siegel de niveau ${\mathrm{\Gamma }}_1\left(p\right)$, lorsque $p$ est non-ramifié dans le corps totalement réel. Notre outil clé est une variante du complexe de Lie anneau-équivariant défini par Illusie.

We construct a local model for Hilbert–Siegel moduli schemes with ${\mathrm{\Gamma }}_1\left(p\right)$-level structures, when $p$ is unramified in the totally real field. Our key tool is a variant of the ring-equivariant Lie complex defined by Illusie.

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f\left({\phi }^n\left(x_0\right)\right)$, where $\phi :X--\to X$ and $f:X--\to {\mathbb{P}}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\phi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $\phi$ orbit of $x$ is Zariski dense in $X$ then there is a multiplicative torus ${\mathbb{𝔾}}_m^d$ and maps $\mathrm{\Psi }:{\mathbb{𝔾}}_m^d\to {\mathbb{𝔾}}_m^d$ and $g:{\mathbb{𝔾}}_m^d\to {\mathbb{𝔾}}_m$ such that $a_n=\left(g\circ {\mathrm{\Psi }}^n\right)\left(y\right)$ for some $y\in {\mathbb{𝔾}}_m^d$. We then obtain results about the coefficients of $D$-finite power series using these facts.

We prove that a certain genuine Hecke algebra $\mathscr{H}$ on the nonlinear double cover of a simple, simply laced, simply connected, Chevalley group $G$ over ${\mathbb{Q}}_2$ admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to $\mathscr{H}$-mod. Second, $\mathcal{\mathscr{H}}$ is isomorphic to the Iwahori–Hecke algebra of the linear group $G∕Z_2$, where $Z_2$ is the $2$-torsion of the center of $G$. This isomorphism of Hecke algebras provides a correspondence between genuine unramified principal series of the double cover of $G$ and the Iwahori-unramified representations of the group $G∕Z_2$.

Let $G$ be a finite primitive permutation group on a set $\mathrm{\Omega }$ with point stabiliser $H$. Recall that a subset of $\mathrm{\Omega }$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b\left(G,H\right)$, to be the minimal size of a base for $G$. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that $b\left(G,H\right)\le 4$ if $G$ is soluble. In this paper we extend Seress’s result by proving that $b\left(G,H\right)\le 5$ for all finite primitive groups $G$ with a soluble point stabiliser $H$. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that $4$ random elements in $\mathrm{\Omega }$ form a base tends to $1$ as $\left|G\right|$ tends to infinity.

We study fields of values of the restriction to Sylow subgroups of irreducible characters of the normalizers of Sylow subgroups in symmetric and alternating groups. As an application, we show that these classes of groups admit a McKay bijection that respects fields of values and restriction to Sylow subgroups.

Let $K$ be a number field, and let $E∕K$ be an elliptic curve over $K$. The Mordell–Weil theorem asserts that the $K$-rational points $E\left(K\right)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E\left(K\right)$ for $K$ a cubic number field.

To do so, we determine the cubic points on the modular curves $X_1\left(N\right)$ for

$$N=21,22,24,25,26,28,30,32,33,35,36,39,45,65,121.$$

As part of our analysis, we determine the complete lists of $N$ for which $J_0\left(N\right)$, $J_1\left(N\right)$, and $J_1\left(2,2N\right)$ have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1\left(N\right)\left(\mathbb{Q}\right)$ is generated by $Gal(\bar{\mathbb{Q}}∕\mathbb{Q})$-orbits of cusps of $X_1{\left(N\right)}_{\bar{\mathbb{Q}}}$ for $N\le 55$, $N\ne 54$.