arXiv subjects

Santiago Molina Blanco

Publications and source records attributed to Santiago Molina Blanco.

7 recordsLinked to original sources

arXiv2023

Squares of toric period integrals in higher cohomology

Thanks to the Harder-Eichler-Shimura isomorphism we can realize a quaternionic automorphic representation of a fixed weight in the cohomology space of certain arithmetic groups. For many interesting applications, it is convenient to consider the cap-product of a cohomology class in these spaces with a fundamental class associated to a maximal torus. In a recent paper, the author computes the absolute value of such a cap-product, and he relates it to special values of Rankin-Selberg L-functions. This provides a formula analogous to that of Waldspurger in higher cohomology. In this paper we compute the square of the cap-product instead of its absolute value.

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arXiv2023

Exceptional zero formulas for anticyclotomic p-adic L-functions

In this note we define anticyclotomic p-adic measures attached to a finite set of places S above p, a modular elliptic curve E over a general number field F and a quadratic extension K/F. We study the exceptional zero phenomenon that arises when E has multiplicative reduction at some place in S. In this direction, we obtain p-adic Gross-Zagier formulas relating derivatives of the corresponding p-adic L-functions to the extended Mordell-Weil group of E. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result. We obtain a formula that computes the r-th derivative of the p-adic L-function, where r is the number of places in S where E has multiplicative reduction, in terms of plectic points and Tate periods of E.

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arXiv2020

Finite slope Triple Product p-adic L-functions over totally real fields

We construct p-adic L-functions associated with triples of finite slope p-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting.

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arXiv2020

Extremal p-adic L-functions

In this note we propose a new construction of cyclotomic p-adic L-functions attached to classical modular cuspidal eigenforms. This allows us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL2 over Q this allows us to construct the p-adic L-function in the so far uncovered extremal case which arises under the unlikely hypothesis that p-th Hecke polynomial has a double root. Although Tate's conjecture implies that this case should never take place for GL2/Q, the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree and this article proposes a method which should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functions, and relate them to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family.

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arXiv2019

Triple product p-adic L-functions for Shimura curves over totally real number fields

Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p-adic families of finite slope quaternionic automorphic forms over F. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p-adic L-functions to triples of ordinary p-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu-Zhang-Zhang to our families.

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arXiv2018

Anticyclotomic p-adic l-functions and the exceptional zero phenomenon

Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p-adic L-function attached to (A,E), in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation.

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arXiv2017

Eichler-Shimura isomorphism and group cohomology on arithmetic groups

The Eichler-Shimura isomorphism realizes the automorphic representation generated by an automorphic newform in certain cohomology of an arithmetic group. In this short note, we give a cohomological interpretation of the Eichler-Shimura isomorphism as a connection morphism of certain exact sequence of G-modules, for the multiplicative group G of any quaternion algebra over a totally real field.

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