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New examples of modular rigid Calabi-Yau threefolds.

Matthias Schütt (2004)

Collectanea Mathematica

The aim of this article is to present five new examples of modular rigid Calabi-Yau threefolds by giving explicit correspondences to newforms of weight 4 and levels 10, 17, 21 and 73.

Non-abelian congruences between L -values of elliptic curves

Daniel Delbourgo, Tom Ward (2008)

Annales de l’institut Fourier

Let E be a semistable elliptic curve over . We prove weak forms of Kato’s K 1 -congruences for the special values L 1 , E / ( μ p n , Δ p n ) . More precisely, we show that they are true modulo p n + 1 , rather than modulo p 2 n . Whilst not quite enough to establish that there is a non-abelian L -function living in K 1 p [ [ Gal ( ( μ p , Δ p ) / ) ] ] , they do provide strong evidence towards the existence of such an analytic object. For example, if n = 1 these verify the numerical congruences found by Tim and Vladimir Dokchitser.

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

Keisuke Arai (2016)

Acta Arithmetica

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 extend their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.

Non-solvable base change for Hilbert modular representations and zeta functions of twisted quaternionic Shimura varieties

Cristian Virdol (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we prove some non-solvable base change for Hilbert modular representations, and we use this result to show the meromorphic continuation to the entire complex plane of the zeta functions of some twisted quaternionic Shimura varieties. The zeta functions of the twisted quaternionic Shimura varieties are computed at all places.

Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2

Anna Arnth-Jensen, E. Victor Flynn (2009)

Journal de Théorie des Nombres de Bordeaux

We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.

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