For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.
Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\textrm{Tr}(z)) = (2,2,1)$, there exist conjugacy classes $X, Y \subset M_n(\mathbb{F}_q)$ such that the characteristic polynomial of $X$ is irreducible of degree $n$ and the characteristic polynomial of $Y$ is of the form $ (T- \lambda) h(T)$, where $\lambda \in \mathbb{F}_q$, $h$ is irreducible of degree $n-1$ and $h(\lambda) \neq 0$. These classes can be chosen so that $\textrm{Tr}(z) = \textrm{Tr}(X) + \textrm{Tr}(Y)$. For such $X$ and $Y$, let $$ N_{X,Y}(z) = \# \{ (x,y) \in X \times Y : x + y = z \}. $$ We prove the following estimate: $$ \left| N_{X,Y}(z) - q^{(n-1)^2} \right| \leq 42 q^{(n-1)^2 -1}. $$ Thus, for nonscalar matrices with matching trace, the number of such additive decompositions is $approximately$ the same, namely $q^{(n-1)^2}$, with an absolute error constant independent of $n$ and $q$.
Let $C$ be a smooth projective geometrically connected curve over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian variety of $C$. For an integer $r\geq 1$ and a positive integer $n$ prime to the characteristic of $k$, we prove that the Galois symbol map \[ K(k;J,\mathbb{G}_{m},\ldots,\mathbb{G}_{m})/n \to H_{\mathrm{\acute et}}^{r+1}\bigl(k,J[n]\otimes \mu_n^{\otimes r}\bigr) \] is injective, where the multiplicative group $\mathbb{G}_{m}$ occurs $r$ times. The proof uses Akhtar's description of higher Chow groups of zero-cycles and the Beilinson--Lichtenbaum theorem. The case $r=1$ recovers a theorem of Spiess.
Let $X$ be an $n$-dimensional generalized Grassmannian not isomorphic to $\mathbb{P}^n$. We prove that $k(X)\le n-1$, where $k(X)$ denotes the maximal integer such that every uniform bundle on $X$ of rank at most $k(X)$ is homogeneous. In particular, for smooth quadrics $\mathbb{Q}^n$, we have $k(\mathbb{Q}^n)=n-1$ for odd $n$, and $n-2\le k(\mathbb{Q}^n)\le n-1$ for even $n$. We classify uniform rank $n$ bundles on $\mathbb{Q}^{n}$ for $n=3$, $5$. Furthermore, we characterize projective spaces among generalized Grassmannians in terms of uniform bundles.
Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $\Gamma\backslash\mathscr{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^\epsilon$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{\chi(\Gamma\backslash\mathscr{B})}L(\Gamma,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^\epsilon(\Gamma\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.
Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.
Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than $n-1$ elements. In this article, we prove that the equality $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension $t_n(\mathbb{F})$.
Clément de Seguins Pazzis· 0 citations
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