Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for GL(2)/K, proving an etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a regular $p$-stabilised newform of weight $k$ at least 2 without CM by $K$. (1) We construct a two-variable $p$-adic $L$-function attached to the base-change of $f$ to $K$ under assumptions on $f$ that we conjecture always hold, in particular making no assumption on the slope of $f$. (2) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic $L$-functions of classical base-change Bianchi cusp forms in families. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space.Comment: 26 pages, with a 3 page appendix by Carl Wang-Erickson. Comments welcome! Changes for v5: added contents, minor changes to exposition. v4: corrected funding acknowledgements. v3: This version has a new introduction, has been reorganised and greatly shortened, and incorporates minor corrections. v2: minor correction

Exceptional zeros and L-invariants of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an L-invariant Lp, depending only on p and f, such that when the p-adic L-function of f has an exceptional zero at p, its derivative can be related to the classical L-value multiplied by Lp. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate Lp to the L-invariant of F, resolving a conjecture of Trifkovi\'{c} in this case

Families of bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $GL(2)/K$, proving an etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this. (1) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the (two-variable) $p$-adic $L$-functions of classical Bianchi cusp forms in families. (2) Let $f$ be a $p$-stabilised newform of weight $k$ at least 2 without CM by $K$. We construct a two-variable $p$-adic $L$-function attached to the base-change of $f$ to $K$ under assumptions on $f$ that we conjecture always hold, in particular making no assumption on the slope of $f$. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space

On pp-adic LL-functions for GL2nGL_{2n} in finite slope Shalika families

Let $p$ be a prime number and $\mathrm{Gal}_p$ be the Galois group of the maximal abelian extension unramified outside $p\infty$ of a totally real number field $F$ of degree $d$. Using automorphic cycles, we construct evaluation maps on the parahoric overconvergent cohomology for $\mathrm{GL}_{2n}/F$ in degree $d (n^2 + n -1)$, which allow us to attach a distribution on $\mathrm{Gal}_p$ of controlled growth to any finite slope overconvergent $U_p$-eigenclass. When the eigenclass comes from a non-critical refinement $\tilde\pi$ of a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model, we prove this distribution interpolates all Deligne-critical $L$-values of $\pi$, giving the first construction of $p$-adic $L$-functions in this generality beyond the $p$-ordinary setting. Further, under some mild assumptions we use our evaluation maps to show that $\tilde\pi$ belongs to a unique $(d+1)$-dimensional Shalika family in a parabolic eigenvariety for $\mathrm{GL}_{2n}/F$, and that this family is \'etale over the weight space. Finally, under a hypothesis on the local Shalika models at bad places which is empty for $\pi$ of level 1, we construct a $p$-adic $L$-function for the family.Comment: 64 pages (inc. glossary of notation), comments welcome! Changes for v2,v3: minor corrections and expositional improvement

Análisis de desempeño del Sistema Privado de Pensiones : un acercamiento desde la teoría de agencia

La presente investigación tiene por objetivo analizar el desempeño del Sistema Privado de Pensiones desde su estructura organizacional. El referido sistema puede ser analizado de distintas maneras, el presente trabajo lo evaluará desde dos perspectivas: la Teoría de Portafolio de Markowitz y la Teoría de Agencia. En el caso de la Teoría de Portafolio de Markowitz, ésta sirvió para analizar la eficiencia de las decisiones de inversión tomadas por las AFP bajo determinados supuestos (sin restricciones, con restricciones de liquidez y con restricciones legales). De esta forma, el modelo de Markowitz fue utilizado para determinar portafolios óptimos bajo cada supuesto, para cada uno de los años y cada uno de los tipos de fondos de pensiones (Fondo 1, 2 y 3). Con ello, se estimó la ineficiencia o pérdida de rentabilidad causada por la iliquidez del mercado peruano y el marco regulatorio del Sistema Privado de Pensiones (SPP). Luego, se comparó con la rentabilidad ajustada al riesgo de las AFP y en donde se observó que ésta última rentabilidad se encuentra siempre por debajo de las restricciones analizadas. Se concluyó, por lo tanto, que existen otros factores no considerados que causan ineficiencia dentro del SPP. En el caso de la Teoría de Agencia, ésta se utilizó para evaluar las decisiones tomadas por los agentes del SPP (AFP y Estado) en favor de sus principales (los afiliados al SPP) desde la perspectiva del problema de agencia. Es necesario indicar que este análisis se realiza de manera exploratoria con la finalidad de poder encontrar algunas posibles explicaciones a las ineficiencias encontradas con la primera teoría. De esta manera, se determinan los incentivos que tienen los agentes del SPP, los posibles conflictos de interés, los costos de agencia, la posible pérdida acumulada de riqueza para el afiliado al SPP, y finalmente, se realizó un análisis a profundidad de un agente del SPP en un determinado accionar. Se concluyó que algunos agentes del SPP tienen incentivos para afectar a los afiliados, tomando ventaja de su posición debido a la actual estructura organizacional del SPP, y ello puede explicar las ineficiencias halladas en la aplicación del modelo de Markowitz.Tesi

Families of Bianchi modular symbols : critical base-change p-adic L-functions and p-adic Artin formalism

Let K be an imaginary quadratic field. In this article, we study the eigenvariety for GL2/K, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let f be a p-stabilised newform of weight k≥2 without CM by K. Suppose f has finite slope at p and its base-change f/K to K is p-regular. Then: (1) We construct a two-variable p-adic L-function attached to f/K under assumptions on f that conjecturally always hold, in particular with no non-critical assumption on f/K. (2) We construct three-variable p-adic L-functions over the eigenvariety interpolating the p-adic L-functions of classical base-change Bianchi cusp forms. (3) We prove that these base-change p-adic L-functions satisfy a p-adic Artin formalism result, that is, they factorise in the same way as the classical L-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space

Exceptional zeros and L-invariants of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL2 over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an Linvariant Lp, depending only on p and f, such that when the p-adic L-function of f has an exceptional zero at p, its derivative can be related to the classical L-value multiplied by Lp. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL2/Q. When p is not split and f is the base-change of a classical modular form f ˜, we relate Lp to the L-invariant of f ˜, resolving a conjecture of Trifkovi´c in this case.Postprint (author's final draft

Parabolic eigenvarieties via overconvergent cohomology

Let G be a connected reductive group over Q such that G = G/Qp is quasi-split, and let Q ⊂ G be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to Q, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When Q is a Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct Q-parabolic eigenvarieties, which parametrise p-adic families of systems of Hecke eigenvalues that are finite slope at Q, but that allow infinite slope away from Q