Related papers: Endomorphisms of ordinary superelliptic jacobians

Fix a prime number $\ell \geq 5$. Let $K = \mathbb{F}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q \equiv 1 \text{ mod } \ell$. Let $C:y^\ell = F(x)$ be a non-isotrivial superelliptic curve over $K$ such…

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial…

Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\sum_{n=1}^\infty a_f(n)q^n\in S_k\cap O_L[[q]]$ (note: $q := e^{2\pi iz}$) is a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbb{Z})$. We say…

Let $K_{m}=\Bbb{Q}(\zeta_{m})$ where $\zeta_{m}$ is a primitive $m$th root of unity. Let $p>2$ be prime and let $C_{p}$ denote the group of order $p.$ The ring of algebraic integers of $K_{m}$ is $\Cal{O}_{m}=\Bbb{Z}[\zeta_{m}].$ Let…

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The…

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Progress in Math. 195 (2001), 473--490; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431; Proc. Amer. Math. Soc. 131 (2003), no. 1, 95--102) the…

We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe…

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V…

Let $J$ be an abelian variety over a number field such that the center of its endomorphism ring is equal to the ring of integers. If the endomorphism ring splits at a prime number $l$, then the $l$-adic representation is defined by the…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\tau: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every…

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that…

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic…

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel-Voevodsky motivic stable homotopy category than in Voevodsky's derived category of motives. In…

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy…

Let $q$ be an odd prime power and $n$ be a positive integer. Let $\ell\in \mathbb F_{q^n}[x]$ be a $q$-linearised $t$-scattered polynomial of linearized degree $r$. Let $d=\max\{t,r\}$ be an odd prime number. In this paper we show that…

Let $f$ be a positive definite integral quadratic form in $d$ variables. In the present paper, we establish a direct link between the genus representation number of $f$ and the order of higher even $K$-groups of the ring of integers of real…

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic…

There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction.…