James Stankewicz

James Stankewicz
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Mathematics - Number Theory (10)
 
Mathematics - Algebraic Geometry (2)
 
Mathematics - Combinatorics (1)

Publications Authored By James Stankewicz

We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker and Norine are planar. We give a notion of a bielliptic graph and show that each of these must embed into a closed orientable surface of genus one. Read More

Let $D > 546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $l/\mathbb Q$ such that the twist of the Shimura curve $X^D$ by the main Atkin-Lehner involution $w_D$ and $l/\mathbb Q$ violates the Hasse Principle over $\mathbb Q$. Read More

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may think of this as a higher-dimensional version of the Gauss Class Number problem. Before now, no one has ruled out quaternion algebras over $\mathbf{Q}$ not already ruled out by the Albert classification. Read More

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. Read More

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the descent (or equivalently \'etale Brauer-Manin) obstruction to rational points applied to \'etale coverings coming from the level structure. More precisely, we find infinitely many quadratic fields defined using congruence conditions such that a twist of a related Shimura curve by each of those fields violates the Hasse principle. Read More

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation. Read More

Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livn\'e on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. Read More

We study a family of semiample divisors on the moduli space $\bar{M}_{0,n}$ that come from the theory of conformal blocks for the Lie algebra $sl_n$ and level 1. The divisors we study are invariant under the action of $S_n$ on $\bar{M}_{0,n}$. We compute their classes and prove that they generate extremal rays in the cone of symmetric nef divisors on $\bar{M}_{0,n}$. Read More

Let g_j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x_1, ... Read More

We consider a generalization of the Frobenius Problem where the object of interest is the greatest integer which has exactly $j$ representations by a collection of positive relatively prime integers. We prove an analogue of a theorem of Brauer and Shockley and show how it can be used for computation. Read More

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. Read More