Related papers: Local-global divisibility on algebraic tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R…

In this article we obtain lower and upper bounds for global dimensions of a class of artinian algebras in terms of global dimensions of a finite subset of their artinian subalgebras. Finding these bounds for the global dimension of an…

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form $\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients that…

We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a…

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain…

A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in…

In this paper, we prove a version of the logarithmic Sobolev inequality of fractional order on noncommutative $n$-tori for any dimension $n\geq 2$.

Fix an integer $d>0$. In 2008, David and Weston showed that, on average, an elliptic curve over $\mathbf{Q}$ picks up a nontrivial $p$-torsion point defined over a finite extension $K$ of the $p$-adics of degree at most $d$ for only…

We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that its translations by $g_1,...,g_r$ sum up to…

Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $T \to X$ is locally trivial in the Zariski topology for every reduced algebraic variety $X$ defined over $k$. In…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution…

We give a function F(d,n,p) such that if K/Q_p is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the prime-to-p…

We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1…

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic…

The conjectural theory of local newofmrs for the split $p$-adic group ${\rm SO}_{2n+1}$, proposed by Gross, predicts that the space of local newforms in a generic representation is one-dimensional. In this note, we prove that this space is…

Let n be either 2, or an odd integer greater than 1, and fix a prime p > 2(n + 1). Under standard "adequate image" assumptions, we show that the set of components of n-dimensional p-adic potentially semistable local Galois deformation rings…

We show that a sufficiently general hypersurface of degree $d$ in $n$-dimensional projective space admits a toric Gr\"obner degeneration if and only if $d\leq 2n-1$.

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori. More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at…

Let R\_n be the ring of Laurent polynomials in n variables over a field k of characteristic zero and let K\_n be its fraction field.Given a linear algebraic k-group $G$, we show that a K\_n-torsor under G which is unramified with respect to…