arXiv subjects

Keisuke Arai

Publications and source records attributed to Keisuke Arai.

11 recordsLinked to original sources

arXiv2024

$\mathscr{D}$-elliptic sheaves and the Hasse principle

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathscr{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathscr{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle.

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arXiv2023

An equivalent condition for abelian varieties over finite fields to have QM

In this paper, we give an equivalent condition for an abelian variety over a finite field to have multiplication by a quaternion algebra over a number field. We prove the result by combining Tate's classification of the endomorphism algebras of abelian varieties over finite fields with Yu's criterion of the existence of homomorphisms between semi-simple algebras.

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arXiv2022

Points on Shimura curves rational over imaginary quadratic fields in the non-split case

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational points. In other words, the main result asserts that there is a finite set $P(k)$ of prime numbers depending on $k$ such that: if there is a prime divisor of the discriminant of $B$ which is not in $P(k)$, then $M^B$ has no $k$-rational points. Moreover, we can take $P(k)$ to satisfy the following: There is an effectively computable constant $C(k)$ depending on $k$ such that $p\in P(k)$ implies $p<C(k)$ with at most one possible exception. The case where $k$ splits $B$ was done by Jordan. In the non-split case, the proof is done by studying a canonical isogeny character and its composition with the transfer map.

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arXiv2019

Drinfeld-Stuhler modules and the Hasse principle

We develop a theory of canonical isogeny characters of Drinfeld-Stuhler modules similar to the theory of canonical isogeny characters of abelian surfaces with quaternionic multiplication. We then apply this theory to give explicit criteria for the non-existence of rational points on Drinfeld-Stuhler modular varieties over the finite extensions of $\mathbb{F}_q(T)$. This allows us to produce explicit examples of Drinfeld-Stuhler curves violating the Hasse principle.

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arXiv2014

Non-existence of points rational over number fields on Shimura curves

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we expand their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.

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arXiv2013

Algebraic points on Shimura curves of $Γ_0(p)$-type (III)

In previous articles, we classified the characters associated to algebraic points on Shimura curves of $Γ_0(p)$-type, and over number fields in a certain large class we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. In this article, we prove the non-existence of elliptic points on Shimura curves of $Γ_0(p)$-type under a mild assumption. We also give an explicit example.

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arXiv2012

An effective bound of $p$ for algebraic points on Shimura curves of $Γ_0(p)$-type

In previous articles, we classified the characters associated to algebraic points on Shimura curves of $Γ_0(p)$-type, and over number fields in a certain large class we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. In this article, we obtain an effective bound of $p$ concerning algebraic points on Shimura curves of $Γ_0(p)$-type.

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arXiv2012

On the Rasmussen-Tamagawa conjecture for QM-abelian surfaces

In the previous article, we showed the Rasmussen-Tamagawa conjecture for QM-abelian surfaces over imaginary quadratic fields. In this article, we generalize the previous work to QM-abelian surfaces over number fields of higher degree. We also give several explicit examples.

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arXiv2012

Algebraic points on Shimura curves of $Γ_0(p)$-type (II)

In the previous article, we classified the characters associated to algebraic points on Shimura curves of $Γ_0(p)$-type, and over a quadratic field we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. In this article, we get a similar result for points over number fields of higher degree on Shimura curves of $Γ_0(p)$-type.

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arXiv2012

Algebraic points on Shimura curves of $Γ_0(p)$-type

In this article, we classify the characters associated to algebraic points on Shimura curves of $Γ_0(p)$-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. This is an analogue of the study of rational points or points over a quadratic field on the modular curve $X_0(p)$ by Mazur and one of the author (Momose). We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen-Tamagawa.

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arXiv2007

On uniform lower bound of the Galois images associated to elliptic curves

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the image of rho_{E,p} is open if E has no complex multiplication. For an elliptic curve E over K whose j-invariant does not appear in an exceptional finite set, we give an explicit uniform lower bound of the size of the image of rho_{E,p}.

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