相关论文: Ihara's lemma for imaginary quadratic fields

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…

Let $\mathbb{K}$ be an imaginary quadratic field such that $2$ splits into two primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$. Let $\mathbb{K}_{\infty}$ be the unique $\mathbb{Z}_2$-extension of $\mathbb{K}$ unramified outside…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…

We study the arithmetic of Eisenstein cohomology classes (in the sense of G. Harder) for symmetric spaces associated to GL_2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of a special…

In this paper, we prove the cohomological Lichtenbaum conjecture of abelian extensions of imaginary quadratic fields up to a finite set of bad primes.

In this paper, we obtain an asymptotic formula for the number of imaginary quadratic fields with prime discriminant and class number up to $H$, as $H\to \infty$. Previously, such an asymptotic was only known under the assumption of the…

Let T be a triangulated category, A a graded abelian category and h: T -> A a homology theory on T with values in A. If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an…

We investigate the number ${\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\Cal F}(h)$ and…

The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings…

We construct an Euler system attached to general-type cohomological cuspidal automorphic representations of $\mathrm{GSp}(4)$ twisted by a Groessencharacter of an imaginary quadratic field. We then use this to bound strict Selmer groups…

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

We prove that for any given positive integer $\ell$ there are infinitely many imaginary quadratic fields with 2-class group of type $(2,2^\ell)$, and provide a lower bound for the number of such groups with bounded discriminant for…

Let X be a smooth projectibe curve over a finite field. We consider the Hall algebra H whose basis is formed by isomorphism classes of coherent sheaves on X and whose typical structure constant is the number of subsheaves in a given sheaf…

The Eichler-Shimura isomorphism realizes the automorphic representation generated by an automorphic newform in certain cohomology of an arithmetic group. In this short note, we give a cohomological interpretation of the Eichler-Shimura…

The global analogue of a Henselian local ring is a Henselian pair-a ring R and an ideal I which satisfy a condition resembling Hensel's lemma regarding lifting coprime factorizations of monic polynomials over R/I to factorizations over R.…

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit…

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and…

We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal…

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…