相关论文: Ovoidal fibrations in $PG(3,q), q$ even

Let $\ell$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $\mathbb{Q}$. If $E$ has an $\ell$-torsion point, then there has been significant work done on characterizing the $\ell$-divisibility of the…

We construct a family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$, the parabolic quadric of $\textup{PG}(4,q)$, for $q\equiv 3\pmod 4$. The existence of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$ was only known for $q=3, 7,$ or $11$. Our construction…

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…

In this paper we characterize the non-singular Hermitian variety ${\mathcal H}(6,q^2)$ of $\mathrm{PG}(6, q^2)$, $q\neq2$ among the irreducible hypersurfaces of degree $q+1$ in $\mathrm{PG}(6, q^2)$ not containing solids by the number of…

An $m$-ovoid of a finite polar space $\mathcal{P}$ is a set $\mathcal{O}$ of points such that every maximal subspace of $\mathcal{P}$ contains exactly $m$ points of $\mathcal{O}$. In the case when $\mathcal{P}$ is an elliptic quadric…

We study a family of hypergeometric motives $H(\alpha,\beta|t)$ attached to a pair of tuples $\alpha=(1/4,1/2,3/4)$, $\beta=(0,0,0)$. To each such motive we can attach a system of $\ell$--adic realisations with the trace of geometric…

Let $p \geq 3$ be a prime. Let $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ be elliptic curves with isomorphic $p$-torsion modules $E[p]$ and $E'[p]$. Assume further that either (i) every $G_\mathbb{Q}$-modules isomorphism $\phi : E[p] \to E'[p]$…

Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(\mu_\ell)$ be the…

Let $(S,L)$ be a general polarized Enriques surface, with $L$ not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible $\delta$-nodal curves in the linear system $|L|$, with $0\leq…

Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $\O_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $\O_K / p \O_K$ determine lattices $\Lambda_\ell (C) $ over $K$. If $ p…

We classify nets of quadrics in P^3 which give rise to elliptic fibrations of Mordell-Weil rank zero.

We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be…

Let T be a general bidegree (2,2) divisor in the product of two projective planes. Recently A.Verra proved that the existence of two conic bundle structures (c.b.s.) on T implies a new counterexample to the Torelli theorem for Prym…

In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the…

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of…

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimple for almost all $n$. Jones showed that there is a canonical symmetric bilinear form on $TL_n(q)$, whose radical $R_n(q)$ is generated by a…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

Let $E$ be a set of solids (hyperplanes) in $PG(4,q)$, $q$ even, $q>2$, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3-q^2)$ solids of $E$, and every plane of $PG(4,q)$ lies in either $0$, $\frac12q$ or…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider…

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of…