arXiv subjects

Haruzo Hida

Publications and source records attributed to Haruzo Hida.

3 recordsLinked to original sources

arXiv2024

On the Frobenius fields of abelian varieties over number fields

Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$ has connected monodromy groups, we show that the set of places $v$ such that $F(A,v)$ is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places.

math.NT↗

arXiv2018

Class groups and local indecomposability for non-CM forms

In the late 1990's, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those $p$-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at $p$. It is expected that such $p$-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for $p$-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the $p$-indivisibility of a certain class group.

math.NT↗

arXiv2017

Horizontal variation of Tate--Shafarevich groups

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $ι: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $χ$ over $K$, let $\mathbb{Q}(χ)$ be the field generated by the image of $χ$ and $\mathfrak{p}_χ$ the prime of $\mathbb{Q}(χ)$ above $p$ determined via $ι_p$. Under mild hypotheses, we show that the number of class group characters $χ$ such that the $χ$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\mathfrak{p}_χ$-part increases with the absolute value of the discriminant of $K$.

math.NT↗