Related papers: Ihara's lemma for imaginary quadratic fields

Let $k$ be a field of characteristic zero and I an ideal defining an arrangement of linear subspaces in the affine space $A^n_k$. We compute the D-module theoretic characteristic cycle of the local cohomology modules $H^r_I(k[x_1,...,x_n])$…

The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of $\operatorname{Cl}_K[p^\infty]$ whne $K$ runs over $\Gamma$-fields and $p\nmid|\Gamma|$. In this paper, we prove several results on the distribution of ideal…

We prove the quantum ergodicity of Eisenstein series on the arithmetic hyperbolic 3-manifold $\operatorname{PSL}_2(\mathcal{O}_F)\backslash \mathbb{H}^3$, where $F$ is an imaginary quadratic field with ring of integers $\mathcal{O}_F$ and…

The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym*…

KLR algebras of affine ADE types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the strata of this stratification reduces to semicuspidal cases, which split…

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal…

Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian…

Let $X$ be a compact K\"{a}hler manifold, and let $L$ be a line bundle on $X.$ Define $I_k(L)$ to be the kernel of the multiplication map $ Sym^k H^0 (L) \to H^0 (L^k).$ For all $h \leq k,$ we define a map $\rho : I_k(L) \to Hom (H^{p,q}…

In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a…

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of…

Let $K$ be a number field with the discriminant $D_K$ and the class number $h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove that every ideal class of $K$ contains a prime ideal with norm less than…

Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume…

We introduce an invariant, associated to a coherent sheaf over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of…

In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture,…

Kurihara established a refinement of the minus-part of the Iwasawa main conjecture for totally real number fields using the higher Fitting ideals. In this paper, by using Kurihara's methods and Mazur-Rubin theory, we study the higher…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. We call $(P, f)$ a (locally)projective $I$-cover of $M$ if $f$ is an epimorphism from $P$ to $M$, $P$ is (locally)projective, $Kerf\subseteq IP$, and whenever $P=Kerf+X$,…

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we…

Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a…

We study interpretable sets in henselian and sigma-henselian valued fields with value group elementarily equivalent to Q or Z. Our first result is an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely ramified…