Ordinary primes for some varieties with extra endomorphisms

Let A be an abelian variety defined over a number field and of dimension g. When g<3, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for A. In higher dimension very little is known. We show that if g=3 and A has multiplication by an imaginary quadratic field E, then there exists a nonzero density set of ordinary primes for A. We reach the same conclusion if g=4 and the pair (A,E) has signature (2,2). We also obtain partial results when g=3 and A has multiplication by a totally real cubic field. We show that our methods also apply to certain abelian varieties of Albert type IV of higher dimension. These results are derived from an extended version of the l-adic methods of Katz, Ogus, and Serre in the presence of extra endomorphisms.Comment: 13 page

Una predel·la per a la devoció a l'antic retaule major de la Seu Vella de Lleida

Aquest treball se centra en l'estudi de la segona predel·la que fou afegida pels escultors Rotllí Gauler i Jordi Safont, entorn al 1440-1441, a l'antic retaule major de la Seu Vella de Lleida, que havia estat executat per Bartomeu de Robio i taller vers el 1360-1362. L'anàlisi s'aborda des del punt de vista de les noves aportacions del segon gòtic internacional, lligades a la Devotio Moderna, per la qual cosa s'ha mirat d'aprofundir en el programa iconogràfic sobre el cicle de la Passió que s'hi desenvolupà.This work focuses on the study of the second predella that was added by the sculptors Rotllí Gaulter and Jordi Safont, around 1440-1441, to the old altarpiece of the Seu Vella of Lleida, that was executed by Bartomeu de Robio and his workshop towards 1360-1362. The analysis is done from the point of view of the second International Gothic new contributions, related to Devotio Moderna, so that we tried to deep into the iconographic program on the cycle of the Passion that are developed

Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.Comment: minor edits, 42 page

El santuari de Santa Maria del Merli de la vila d'Alguaire: anàlisi artístic del temple del segle XIII

En este breve estudio, se ofrecen los pocos datos históricos sobre el santuario del Merli (Alguaire, Lleida) y la análisis estilístico del edificio, datable en el último cuarto del siglo XIII, así como las transformaciones y ampliación que sufrió en época moderna, dentro del contexto de la arquitectura religiosa del área de Lleida

On the Hilbert eigenvariety at exotic and CM classical weight 1 points

Let $F$ be a totally real number field and let $f$ be a classical cuspidal $p$-regular Hilbert modular eigenform over $F$ of parallel weight $1$. Let $x$ be the point on the $p$-adic Hilbert eigenvariety $\mathcal E$ corresponding to an ordinary $p$-stabilization of $f$. We show that if the $p$-adic Schanuel Conjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If we additionally assume that $F/\mathbb Q$ is Galois, we show that the weight map is \'etale at $x$ if $f$ has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of parallel weight $1$). We prove these results by showing that the completed local ring of the eigenvariety at $x$ is isomorphic to a universal nearly ordinary Galois deformation ring.Comment: The material in the introduction and the final sections was reorganized. The sections on background material were substantially shortene

On a local-global principle for quadratic twists of abelian varieties

Let $A$ and $A^{\prime}$ be abelian varieties defined over a number field $k$ of dimension $g \geq 1$. For $g \leq 3$, we show that the following local-global principle holds: $A$ and $A^{\prime}$ are quadratic twists of each other if and only if, for almost all primes $\mathfrak{p}$ of $k$ of good reduction for $A$ and $A^{\prime}$, the reductions $A_{\mathfrak{p}}$ and $A_{\mathfrak{p}}^{\prime}$ are quadratic twists of each other. This result is known when $g=1$, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $g=4$

Un primer apropament al santoral de la consueta RC. 0031 (Olim Roda 13) de l'Arxiu Capitular de Lleida del segle XV

Tal com s’indica al títol, aquest estudi vol ser un primer apropament al santoral de la consueta RC.0031 conservada a l’Arxiu Capitular de Lleida (ACL), procedent de Roda d’Isàvena. En ell, primerament, fixem la cronologia definitiva del manuscrit entorn de les primeres dècades del segle xv. Tot seguit, ens fixem en les seves característiques, i aprofitem per oferir la transcripció completa de les regulae. Passem després a analitzar el santoral, que comparem amb els d’altres manuscrits de l’antiga catedral, en concret, amb el del calendari del breviari ilerdense del 1451 (ACL, LC.0016) i amb el del leccionari del segle xiv (ACL, RC.0026). Finalment, dediquem un apartat al culte marià, especialment a la litúrgia estacional, i acabem donant notícia sobre els drames litúrgics que se celebraven a la catedral lleidatan