The moduli space of $G$-algebras

Arithmetic geometry of algebraic curves and their moduli space

We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories to the arithmetic of the moduli space of algebraic curves, especially to automorphic forms on this space.

关于具有复合模数的代数扭，二

On algebraic twists with composite moduli, II

研究GL_{3,Q}上自守系数相关和的界，探讨复合模数迹函数的性质，延续与E.Kowalski和W.Sawin之前的成果。

具有几何系数模的超越幂级数的代数值

Algebraic values of transcendental power series with geometric coefficient moduli

研究在\(\lambda>1\)为实代数数时，构造收敛半径为\(1\)的幂级数\(f(z)\)，其系数代数且模为\(\lambda^m\)，\(f^{(s)}\)在代数点取值代数且超越\(\mathbb{C}(z)\)，通过代数多边形消去与稀疏多项式块论证证明相关结论。

Integrality properties in the Moduli Space of Elliptic Curves: CM Case

A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-α$ is an algebraic unit. The result uses Duke's Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $α\in \bar{\mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-α$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.

Moduli spaces and algebraic cycles in real algebraic geometry

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove t…

Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal de…