arXiv subjects

Owen Patashnick

Publications and source records attributed to Owen Patashnick.

6 recordsLinked to original sources

arXiv2025

Modular Symbols with Values in Beilinson-Kato Distributions

For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bmξ_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\bmμ$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bmξ_n$; we define $\bmξ_n$ essentially by taking the $n$-Steinberg product of $\bmμ$. The most non-trivial part of this construction is the cocycle property of $\bmξ_n$; we prove it by using an induction on $n$ based on the first two cases $\bmξ_1$ and $\bmξ_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations.

math.NT↗

arXiv2023

The automorphisms of certain Kummer surfaces act on the points mod $p$ by the alternating group

For the family of Kummer surfaces of the square of an elliptic curve over the prime field $\mathbb{F}_p$ with $p$ odd, we show that if the automorphism group acts transitively on the set of points then the action is at least alternating. This is analogous to a result of Meiri, Puder, and Chen arXiv:1702.08358 on the points mod $p$ of the Markoff surface.

math.NT↗

arXiv2021

Explicit coverings of families of elliptic surfaces by squares of curves

We show that, for each $n>0$, there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1$, and whose Hodge structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case $n=3$, we show that one particular family of K3 surfaces are covered by the square of genus $7$. Using this, we construct a correspondence between the square of a curve of genus $7$ and a general K3 surface in ${\mathbb P}^4$ with $15$ ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces isogenous to them, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.

math.AG↗

arXiv2016

Rational Mixed Tate Motivic Graphs

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of [12].

math.AG↗

arXiv2013

A Candidate for the abelian category of mixed elliptic motives

In this paper we suggest a definition for the category of mixed motives generated by the motive h^1(E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soule conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Sym^{n}h^1(E))(-1) we construct families of nontrivial motives whose highest associated weight graded piece is $Sym^{n}h^1(E))(-1). This paper was essentially written in the late 1990's whilst the author was at the University of Chicago. The author apologizes for the tardiness of this posting, and hopes the reader will still find the content interesting.

math.AG↗

arXiv2005

Energy-Momentum of a regular MMaS-class black hole

We compute the energy and momentum of a regular black hole of type defined by Mars, Martin-Prats, and Senovilla using the Einstein and Papapetrou definitions for energy-momentum density. Some other definitions of energy-momentum density are shown to give mutually contradictory and less reasonable results. Results support the Cooperstock hypothesis.

gr-qc↗