Related papers: Higher Descent on Pell Conics III. The First 2-Des…
Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let…
We write explicit contour integral formulas for probability distributions of the multi-species q-TAZRP and the multi-species ASEP starting with q-exchangeable initial conditions. The formulas are equal to the corresponding explicit contour…
In this paper, we study convex elements in a (twisted) Weyl group introduced by Ivanov and the first named author. We show that each conjugacy class of the twisted Weyl group contains a convex element, and moreover, the Steinberg…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Recently it was shown that the Diophantine equations describing such a cuboid…
In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups $G=G(2m,m,2)$, $G_{12}$,…
Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive…
We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also…
We investigate the quadratic descent of totally decomposable algebras with involution of orthogonal type in characteristic two. Both separable and inseparable extensions are included.
We investigate the characteristic numbers of Del Pezzo surfaces using degenerations.
We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the…
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an…
We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties…
Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has…
We explicitly compute the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve using the Albanese-Albanese definition of the pairing given by Poonen and Stoll. This leads to a new proof that a pairing defined by Cassels on the…
The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that second…
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\bar\kappa$ type definable…
The aim of this paper is to study certain family of elliptic curves $\{\mathscr{X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$.…
Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper we give numerical evidence that suggests some conjectures on the 2-divisibility of certain sums of Heenger points on E of…
Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin's Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree $1$ whose geometric…
The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving…