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Christopher Lyons

Publications and source records attributed to Christopher Lyons.

5 recordsLinked to original sources
arXiv2017

A simple formula for the Picard number of K3 surfaces of BHK type

The BHK mirror symmetry construction stems from work Berglund and Huebsch, and applies to certain types of Calabi-Yau varieties that are birational to finite quotients of Fermat varieties. Their definition involves a matrix $A$ and a certain finite abelian group $G$, and we denote the corresponding Calabi-Yau variety by $Z_{A,G}$. The transpose matrix $A^T$ and the so-called dual group $G^T$ give rise to the BHK mirror variety $Z_{A^T,G^T}$. In the case of dimension 2, the surface $Z_{A,G}$ is a K3 surface of BHK type. Let $Z_{A,G}$ be a K3 surface of BHK type, with BHK mirror $Z_{A^T,G^T}$. Using work of Shioda, Kelly shows that the geometric Picard number of $Z_{A,G}$ may be expressed in terms of a certain subset of the dual group $G^T$. We simplify this formula significantly to show that this Picard number depends only upon the degree of the mirror polynomial $F_{A^T}$.

math.AG↗
arXiv2015

Some results on surfaces with p_g=q=1 and K^2=2

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with p_g=q=1 and K^2=2. Our first main result provides an explicit surface surface X with these invariants defined over Q that has Picard number 2, which is the smallest possible for these surfaces. This is done by giving equations for the double cover Y of X, calculating the zeta function of the reduction of Y to F_3, and extracting from this the zeta function of the reduction of X to F_3; the basic idea used in this process may also be of independent interest. Our second main result is a big monodromy theorem for a family that contains all surfaces with p_g=q=1, K^2=2, and K is ample. It follows from this that a certain Hodge correspondence of Kuga and Satake, between such a surface and an abelian variety, is motivated (and hence absolute Hodge). This allows us to deduce our third main result, which is that the Tate Conjecture in characteristic zero holds for all surfaces with p_g=q=1, K^2=2, and K ample.

math.AG↗
arXiv2015

The Tate Conjecture for a family of surfaces of general type with p_g=q=1 and K^2=3

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K^2=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of Q, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle \ell-adic cohomology of the surface.

math.AG↗
arXiv2008

A rank inequality for the Tate Conjecture over global function fields

Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be written down.

math.NT↗