Reducible Fibers of Polynomial Maps

晶贝林形变环的不可约分支：极分歧情形

Irreducible components of crystabelline deformation rings: the wildly ramified case

本文针对p≥11的素数上可约非常一般伽罗瓦表示$\bar\rho$，利用ghost series研究其Kisin潜在晶形变环的不可约分支，确定分支总数及各分支特殊纤维的Hilbert-Samuel重数。

Trilinear maps for cryptography

We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the Néron-Severi groups. The discrete logarithm problem is reducible to constructing an explicit description of the algebra generated by two non-commuting endomorphisms, where the explicit description consists of a linear basis with the two endomorphisms expressed in the …

Moduli spaces for families of rational maps on P^1

Let phi: P^1 --> P^1 be a rational map defined over a field K. We construct the moduli space M_d(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M_2(N) is geometrically irreducible for N>1. Restricting ourselves to maps phi of arbitrary degree d >= 2 such that the composition h^{-1} phi h = phi for some nontrivial h in PGL_2, we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.

代数点的希尔伯特不可约性

Hilbert irreducibility for algebraic points

研究数域上曲线覆盖中，给定次数d的代数点使得纤维可约的集合有限性条件，并证明在分支点足够多时该集合有限。

Hurwitz Number Fields

The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The scheme-theoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics.