Related papers: The Andr\'e-Oort Conjecture for Drinfeld Modular V…

We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over…

Let $X_1$ and $X_2$ be deformation equivalent projective hyperk\"ahler manifolds. We prove that the Andr\'e motive of $X_1$ is abelian if and only if the Andr\'e motive of $X_2$ is abelian. Applying this to manifolds of $\mbox{K3}^{[n]}$,…

By means of fixed point index theory for multi-valued maps, we provide an analogue of the classical Birkhoff--Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general…

The present paper deals with Atkin-Lehner theory for Drinfeld modular forms. We provide an equivalent definition of $\mathfrak{p}$-newforms (which makes computations easier) and commutativity results between Hecke operators and Atkin-Lehner…

We extend the classical Poincar\'e-Birkhoff-Witt theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is…

We study algebraic points of bounded degree on polarized projective varieties. To do so, we refine further the filtration construction and Subspace Theorem approach, for the study of integral points, which has origins in the work of…

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel [Ha] proved that generalized tilting…

The article provides a sufficient condition for a locally finite module over the absolute Galois group of a finite field F to satisfy the Riemann Hypothesis Analogue with respect to the projective line. The condition holds for all smooth…

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a…

We show that every component of the locus of smooth supersingular curves of genus $4$ in characteristic $p>2$ has a trivial generic automorphism group. As a result, we prove Oort's conjecture about automorphism groups of supersingular…

Let $F_\infty=\mathbb{F}_q(\!(1/T)\!)$ be the completion of $\mathbb{F}_q(T)$ at $1/T$. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat-Tits building of $\mathrm{PGL}_r(F_\infty)$, $r\geq 2$,…

We obtain an extended Reich fixed point theorem for the setting of generalized cone rectangular metric spaces without assuming the normality of the underlying cone. Our work is a generalization of the main result in \cite{AAB} and…

In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to…

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting…

In this paper we describe the compactification of the Drinfeld modular curve. This compactification is analogous to the compactification of the classical modular curve given by Katz and Mazur. We show how the Weil pairing on Drinfeld…

Let p be an odd prime and let L/k be a Galois extension of number fields whose Galois group is isomorphic to the dihedral group of order 2p. Let S be a finite set of primes of L which is stable under the action of Gal(L/k). The Lichtenbaum…

We prove the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila-Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier…

We study expansions of Drinfeld modular forms of rank \(r \geq 2\) along the boundary of moduli varieties. Product formulas for the discriminant forms \(\Delta_{\mathfrak{n}}\) are developed, which are analogous with Jacobi's formula for…

In this Note we present an expository account for \dieu modules and revisit supersingular abelian varieties. We give a simple proof of the uniqueness of products of two or more supersingular elliptic curves (a theorem due to Deligne, Ogus…

It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has…