Related papers: On a conjecture of Deligne

Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$…

Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only…

Let Z be an affine algebraic variety and ED(Z)= max(2 dim Z+1, dim TZ). Let X be a smooth algebraic variety isomorphic to a semi-simple linear algebraic group whose Lie algebra is a sum of special linear Lie algebras. We show that if dim X…

We define a function in terms of quotients of the $p$-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the $p$-adic setting. We prove, for primes $p > 3$, that the…

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 $\phi : E[p] \to E'[p]$…

For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum…

We construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative unital algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated…

In this paper we give smoothness criterions for a good quotient Y of a smooth variety X by a reductive group G. Our results partially answer a question raised by J. Fogarty in the case where G is a finite group. They also give a converse to…

For a curve C and a reductive group G in prime characteristic, we relate the de Rham moduli of logarithmic G-connections on C to the Dolbeault moduli of logarithmic G-Higgs bundles on the Frobenius twist of C. We name this result the…

Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $\theta : F_* \O_X \to \O_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $\theta$. In…

Let A be a finite dimensional, unital, and associative algebra which is endowed with a non-degenerate and invariant inner product. We give an explicit description of an action of cyclic Sullivan chord diagrams on the normalized Hochschild…

Let $X$ be the germ of a smooth complex variety at a given point $x\in {\mathbbb P}^N$ with regular osculation at order $q$ and suppose that, for any direction $v\in {\mathbbb P}T_xX$, there exists a rationnal normal curve locally contained…

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this…

Proofs that an arbitrary field has a separable closure are necessarily non-constructive, and separable closures are unique only up to non-canonical isomorphism. This means that the absolute Galois group of a field is defined only up to…

Let $R$ be a commutative chain ring. We use a variation of Gr\"obner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]/I$ Frobenius? When is…

Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be the local ring at the vertex of the affine cone for some embedding $\iota: X\hookrightarrow \mathbb{P}^n_k$. G. Lyubeznik asked (in \cite{l2}) whether the integers…

Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample,…

We prove that if q is a power of an odd prime then there is no genus-2 curve over F_q whose Jacobian has characteristic polynomial of Frobenius equal to x^4 + (2-2q)x^2 + q^2. Our proof uses the Brauer relations in a biquadratic extension…

In this paper we provide a full account of the Weil conjectures including Deligne's proof of the conjecture about the eigenvalues of the Frobenius endomorphism. Section 1 is an introduction into the subject. Our exposition heavily relies on…

Let $A/F$ be an abelian variety over a field. Does there exist a smooth projective $F$-variety $X$, such that $A$ is isomorphic to the automorphism group scheme of $X/F$? We show that the answer is positive, if and only if $A$ has only…