Examples of abelian surfaces with everywhere good reduction
نویسندگان
چکیده
We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler–Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures. Mathematics Subject Classification 11F41 · 11F46 · 11F67 · 11G10
منابع مشابه
Abelian varieties over cyclotomic fields with good reduction everywhere
For every conductor f ∈ {1, 3, 4, 5, 7, 8, 9, 11, 12, 15} there exist non-zero abelian varieties over the cyclotomic field Q(ζf ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X1(f ) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over ...
متن کاملOn an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields
We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields. We prove some conditional results for the p′-part on it, and prove the p′-part of the conjecture for constant or isoconstant Abelian schemes, in particular the p′-part for (1) relative elliptic curves with good reduction or ...
متن کاملAbelian varieties over the field of the 20th roots of unity that have good reduction everywhere
The elliptic curve E given by Y 2 + (i+1)XY + iY = X + iX acquires good reduction everywhere over the cyclotomic field Q(ζ20). We show, under assumption of GRH, that every abelian variety over Q(ζ20) with good reduction everywhere is isogenous to E for some g ≥ 0.
متن کامل$PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings
A module is said to be $PI$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this paper, we focus on direct summands and indecomposable decompositions of $PI$-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper ...
متن کاملSplit Reductions of Simple Abelian Varieties
Consider an absolutely simple abelian variety X over a number field K. We show that if the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then Xp is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is noncommutative, then Xp is reducible for p in a set of positive density. An absolutely simple abelian variety ...
متن کامل