# Special Values of Automorphic L-functions and Associated p-adic L-Functions (18w5053)

Arriving in Oaxaca, Mexico Sunday, September 30 and departing Friday October 5, 2018

## Organizers

Tadashi Ochiai (Osaka University)

Fabian Januszewski (Karlsruher Institut für Technologie (KIT) Institut für Algebra und Geometrie)

Khoai Hà Huy (Thang Long University)

Alexei Panchichkine (Université de Grenoble I)

Vinayak Vatsal (Mathematics Department, University of British Columbia)

## Objectives

The role of p-adic families and p-adic deformation theory has become increasingly important in much recent progress in number theory, from Wiles' work on Fermat's last theorem, to the work of Bhargava on the Birch-Swinnerton-Dyer conjecture, and to the recent work of Scholze. One thread that runs consistently through the subject is that of the p-adic L-function, which is a p-adic counterpart of the usual complex L-functions of Dirichlet, Riemann, Hasse, and Weil.

The first example of p-adic L-function was constructed by Kubota and Leopoldt in the early 1960’s. Their work produced a p-adic analogue of the Riemann zeta-function, as well as of the Dirichlet L-functions. Starting in the 1970s, other constructions of Kubota-Leopoldt p-adic L-functions, and p-adic L-functions for elliptic curves, were found by Mazur, Iwasawa, Hida, Coleman, and others. These works brought deeper understanding of the Kubota-Leopoldt p-adic L-function, and led to much progress on Iwasawa Main Conjecture, which relates the analytic L-function to the structure of certain algebraic objects known as Selmer groups.

At this time, many different kinds of p-adic L-functions have been constructed. The Kubota-Leopoldt p-adic L-functions were generalized to totally real fields and CM fields by Barsky, Cassou-Nouges, Deligne-Ribet and Katz. The case of elliptic curves, and elliptic modular forms, was treated by Manin, Amice-Vélu, Vishik, Mazur and others during early 1970’s. Subsequently, the study of multivariable, and higher dimensional and higher rank L-functions was initiated by Hida, Greenberg, and Coates-Schmidt, while others began to consider the anticyclotomic theory. Supersingular variants were also studied, first by Perrin-Riou. Subsequently some key tools were developed by Coleman.

A key insight in all of the above is that L-functions come in families, and the p-adic L-functions correspond to p-adic families, such as the cyclotomic family, anticyclotomic family, Hida family, and so-on. The behavior of the p-adic L-function in these families is then deeply connected with the algebraic properties of the families in question. For example, Bhargava’s spectacular work on the Birch-Swinnerton-Dyer conjcture relies crucially on the results of Skinner-Urban on p-adic L-functions for elliptic curves, and their proof of the main conjecture of Iwasawa theory for elliptic curves.

It is clear, therefore, that one should attempt to develop the theory of p-adic L-functions in as general and flexible a frame-work as possible, preferably in the context of automorphic forms and representations. This theory is still quite fragmentary, and the only known high rank cases are as follows:

(1) symmetric product of elliptic cusp forms (Coates, Hida, Schmidt, et al),

(2) Hilbert modular cusp forms and GL(2) over general number fields (Manin, Haran, et al),

(3) GL(n) and GL(n) x GL(n-1) under some conditions (Ash, Ginzburg, Schmidt, Januszewski, et al)

(4) standard representation of GSp(4) (Panchishkin et al)

(5) Generalization of Katz method in unitary case (Eischen, Harris, Skinner)

The first objective of this workshop is to develop the general theory of one-variable cyclotomic p-adic L-functions, and to put the known results in the common frame of automorphic representations. The proposed topics include

(i) Conjectural framework of the expected p-adic L-functions after Coates-Perrin-Riou, Panchichkine

(ii) The essential ingredients needed to construct p-adic L-functions (distribution properties, boundedness of denominators, congruences between special values, non-canonical choices)

(iii) Some known examples of automorphic L-functions for which the (partial) construction of the p-adic L-function has been carried out (iv) review of the known cases (1)-(5) above.

During the last decades, there has also been much progress on automorphic representations and automorphic L-functions. It will therefore be relevant to consider the following topics, in order to add them to the p-adic toolkit:

(i) Current state for calculation of local integrals associated to automorphic representations (Oda, Sun et al).

(ii) Present state of understanding for the periods for critical values of automorphic L-functions (Shimura, Deligne, Harris, et al).

(iii) Automorphic frame work of (relative) trace formula to understand special values of automorphic L-functions.

(iv) Normalization of optimal complex periods (Stevens, Vatsal et al)

Thanks to these recent advances on autormorphic side, it is quite timely to put together both experienced specialists on p-adic L-functions and specialists on automorphic L-functions. Combining the knowledge and expertise of these two different backgrounds, we expect to pave the way for new developments on p-adic L-functions and automorphic L-functions. For those who work on the construction and the study of p-adic L-functions, this workshop will provide a technical and conceptual framework for generalizing the subject. For those who are not necessarily specialists of p-adic L-functions but who devote themselves to automorphic representations, this workshop hopefully provides a strong new motivation to generalize some of their very technical technical works, and to make them more accessible.

The first example of p-adic L-function was constructed by Kubota and Leopoldt in the early 1960’s. Their work produced a p-adic analogue of the Riemann zeta-function, as well as of the Dirichlet L-functions. Starting in the 1970s, other constructions of Kubota-Leopoldt p-adic L-functions, and p-adic L-functions for elliptic curves, were found by Mazur, Iwasawa, Hida, Coleman, and others. These works brought deeper understanding of the Kubota-Leopoldt p-adic L-function, and led to much progress on Iwasawa Main Conjecture, which relates the analytic L-function to the structure of certain algebraic objects known as Selmer groups.

At this time, many different kinds of p-adic L-functions have been constructed. The Kubota-Leopoldt p-adic L-functions were generalized to totally real fields and CM fields by Barsky, Cassou-Nouges, Deligne-Ribet and Katz. The case of elliptic curves, and elliptic modular forms, was treated by Manin, Amice-Vélu, Vishik, Mazur and others during early 1970’s. Subsequently, the study of multivariable, and higher dimensional and higher rank L-functions was initiated by Hida, Greenberg, and Coates-Schmidt, while others began to consider the anticyclotomic theory. Supersingular variants were also studied, first by Perrin-Riou. Subsequently some key tools were developed by Coleman.

A key insight in all of the above is that L-functions come in families, and the p-adic L-functions correspond to p-adic families, such as the cyclotomic family, anticyclotomic family, Hida family, and so-on. The behavior of the p-adic L-function in these families is then deeply connected with the algebraic properties of the families in question. For example, Bhargava’s spectacular work on the Birch-Swinnerton-Dyer conjcture relies crucially on the results of Skinner-Urban on p-adic L-functions for elliptic curves, and their proof of the main conjecture of Iwasawa theory for elliptic curves.

It is clear, therefore, that one should attempt to develop the theory of p-adic L-functions in as general and flexible a frame-work as possible, preferably in the context of automorphic forms and representations. This theory is still quite fragmentary, and the only known high rank cases are as follows:

(1) symmetric product of elliptic cusp forms (Coates, Hida, Schmidt, et al),

(2) Hilbert modular cusp forms and GL(2) over general number fields (Manin, Haran, et al),

(3) GL(n) and GL(n) x GL(n-1) under some conditions (Ash, Ginzburg, Schmidt, Januszewski, et al)

(4) standard representation of GSp(4) (Panchishkin et al)

(5) Generalization of Katz method in unitary case (Eischen, Harris, Skinner)

The first objective of this workshop is to develop the general theory of one-variable cyclotomic p-adic L-functions, and to put the known results in the common frame of automorphic representations. The proposed topics include

(i) Conjectural framework of the expected p-adic L-functions after Coates-Perrin-Riou, Panchichkine

(ii) The essential ingredients needed to construct p-adic L-functions (distribution properties, boundedness of denominators, congruences between special values, non-canonical choices)

(iii) Some known examples of automorphic L-functions for which the (partial) construction of the p-adic L-function has been carried out (iv) review of the known cases (1)-(5) above.

During the last decades, there has also been much progress on automorphic representations and automorphic L-functions. It will therefore be relevant to consider the following topics, in order to add them to the p-adic toolkit:

(i) Current state for calculation of local integrals associated to automorphic representations (Oda, Sun et al).

(ii) Present state of understanding for the periods for critical values of automorphic L-functions (Shimura, Deligne, Harris, et al).

(iii) Automorphic frame work of (relative) trace formula to understand special values of automorphic L-functions.

(iv) Normalization of optimal complex periods (Stevens, Vatsal et al)

Thanks to these recent advances on autormorphic side, it is quite timely to put together both experienced specialists on p-adic L-functions and specialists on automorphic L-functions. Combining the knowledge and expertise of these two different backgrounds, we expect to pave the way for new developments on p-adic L-functions and automorphic L-functions. For those who work on the construction and the study of p-adic L-functions, this workshop will provide a technical and conceptual framework for generalizing the subject. For those who are not necessarily specialists of p-adic L-functions but who devote themselves to automorphic representations, this workshop hopefully provides a strong new motivation to generalize some of their very technical technical works, and to make them more accessible.