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Chia-Fu Yu

Publications and source records attributed to Chia-Fu Yu.

At least 55 records · Page 3Linked to original sources
arXiv2016

Abelian varieties over finite fields as basic abelian varieties

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

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arXiv2016

On superspecial abelian varieties over finite fields

In this paper we establish a new lattice description for superspecial abelian varieties over a finite field $\mathbb {F}_q$ of $q=p^a$ elements. Our description depends on the parity of the exponent $a$ of $q$. When $q$ is an odd power of the prime $p$, we give an explicit formula for the number of superspecial abelian surfaces over $\mathbb F_q$.

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arXiv2015

Monomial, Gorenstein and Bass Orders

In this article we study a class of orders called {\it monomial orders} in a central simple algebra over a non-Archimedean local field. Monomial orders are easily represented and they may be also viewed as a direct generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two.

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arXiv2015

Notes on locally free class groups

A theorem of Swan states that the locally free class group of a maximal order in a central simple algebra is isomorphic to a restricted ideal class group of the center. In this article we discuss this theorem and its generalization to separable algebras for which it is more applicable to integral representations of finite groups. This is an expository article with aim to introduce the topic for non-specialists.

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arXiv2015

Supersingular abelian surfaces and Eichler class number formula

Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all primes $p$.

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arXiv2014

On finiteness of endomorphism rings of abelian varieties

The endomorphism ring End(A) of an abelian variety A is an order in a semi-simple algebra over Q. The co-index of End(A) is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any algebraically closed field of characteristic p>0, the p-exponents of the co-indices of their endomorphism rings are bounded. We also give a few applications to this finiteness result.

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arXiv2014

Abelian varieties without a prescribed Newton Polygon reduction

In this article we construct for each integer $n\ge 2$ an abelian variety $A$ of dimension $n$ defined over a number field for which there exists a symmetric slope sequence of length $2n$ that does not appear as the slope sequence of $\widetilde A$ for all good reductions $\widetilde A$ of $A$.

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arXiv2013

Fixed points and homology of superelliptic Jacobians

Let $η: C_{f,N}\to \mathbb{P}^1$ be a cyclic cover of $\mathbb{P}^1$ of degree $N$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group $\mathbf{mu}_N\cong \mathbb{Z}/N\mathbb{Z}$ acting on the Jacobian $J_N:=\Jac(C_{f,N})$. For each $\ell$ distinct from the characteristic of the base field, the Tate module $T_\ell J_N$ is shown to be a free module over the ring $\mathbb{Z}_\ell[T]/(\sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian.

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arXiv2013

Endomorphism algebras of factors of certain hypergeometric Jacobians

We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field $E=\mathbb{Q}(ζ_q)$, or a quadratic extension of $E$, or $E\oplus E$. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over $\mathbb{C}$.

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arXiv2013

Embeddings of fields in simple algebras over global fields

Let $F$ be a global field, $A$ a central simple algebra over $F$ and $K$ a finite (separable or not) field extension of $F$ with degree $[K:F]$ dividing the degree of $A$ over $F$. An embedding of $K$ in $A$ over $F$ exists implies an embedding exists locally everywhere. In this paper we give detailed discussions when the converse (i.e. the local-global principle in question) may hold.

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arXiv2012

Class numbers of central simple algebras over global function fields

Let $K$ be a global function field together with a place $\infty$, and $A$ the subring of functions regular outside $\infty$. In this paper we present an effective method to evaluate the (locally free) class number of an arbitrary hereditary $A$-order in an arbitrary definite central simple $K$-algebra. We also show that the class number of any non-principal genus for a hereditary order in $D$ can be reduced to that of the principal genus for another hereditary order in $D$.

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arXiv2012

Embeddings of fields into simple algebras: generalizations and applications

For two semi-simple algebras $A$ and $B$ over an arbitrary ground field $F$, we give a numerical criterion when $\Hom_F(A,B)$, the set of $F$-algebra homomorphisms between them, is non-empty. We also determine when the orbit set $B^\times \backslash \Hom_\falg(A,B)$ is finite and give an explicit formula for its cardinality. A few applications of main results are given.

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arXiv2012

Characteristic polynomials of central simple algebras

We characterize characteristic polynomials of elements in a central simple algebra. We also give an account for the theory of rational canonical forms for separable linear transformations over a central division algebra, and a description of separable conjugacy classes of the multiplicative group.

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arXiv2011

Mass formula of division algebras over global function fields

In this paper we give two proofs of the mass formula for definite central division algebras over global function fields, due to Denert and Van Geel. The first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which we establish the relationship directly between the mass and the value of the associated zeta function at zero.

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arXiv2011

On the existence of maximal orders

We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chapter 5 and 6 of Reiner's book concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.

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