# Noncommutative Iwasawa theory and p-adic automorphic forms.

Lead Research Organisation:
King's College London

Department Name: Mathematics

### Abstract

The last fifteen years have seen two fairly disjoint developments in Iwasawa theory, and its relationship with come of the basic problems in arithmetic geometry. On the one hand, the precise formulation of the main conjectures in noncommutative Iwasawa theory reached a certain maturity in the work of Fukaya-Kato. An important case of the noncommutative main conjecture was proven by the PI (and independently by Burns-Rtter-Weiss). On the other, the theory of automorphic forms (p-adic and lambda-adic) was systematically developed and applied to prove main conjectures in commutative Iwasawa theory beyond the classical main conjectures by several authors including Hida, Tilouine, Urban and Skinner. It is therefore an appropriate time to combine these two developments to prove new results in both directions. We propose to tackle three inter-related problems in this general area. Firstly, we want to extend our methods used to prove the noncommutative main conjectures for Tate motives to prove new results on noncommutative main conjectures for motives other than Tate motives. To this end we propose to systematically study p-adic and lambda-adic automorphic forms over various totally real fields and relations between these automorphic forms as the fields vary. Secondly, implicit in the conjectures of Fukaya-Kato are certain factorisations of p-adic L-functions. These factorisations, known only in a couple of cases, have deep arithmetic implications such as towards Greenberg's L-invariant conjectures. We propose a new strategy to attack these factorisation problems using the tools developed to tackle our first question. Lastly, we propose to study main conjectures over function fields. The algebraic techniques we have developed have already proven very fruitful in Iwasawa theory over function fields in the work of Burns. There is, however, another family of main conjectures over function fields (e.g. in the work of Trihan and his collaborators). Our third project is to use our algebraic results and techniques used by Burns to attack these main conjectures.

### Planned Impact

Aside from the academic impacts mentioned previously our proposal may be beneficial in many other ways. Number theory has always been an area of strength for the UK. Progress in noncommutative Iwasawa theory has been led by UK mathematician and the UK also has a very strong research group working on p-adic automorphic forms. However, there has been very little research on applying the latter to (noncommutative) Iwasawa theory even though this has generated immense interest in the US and in Europe in recent years. Our proposed research, which compliments the existing expertise in the UK very well, therefore will ensure that the UK maintains and reinforces its leadership in noncommutative Iwasawa theory by exploiting recent developments in other parts of number theory and arithmetic geometry. This may also lead to more international recognition, for example, through increased share of ERC grants.

Number theory is an area of mathematics considered to be most accessible to general public. Therefore it is an appropriate means to introduce a wider audience to the beauty of mathematics especially to encourage enthusiastic young student to study mathematics in their pursuit of higher education. We will take opportunities provided by King's College's various outreach programs to explain number theory related to our proposed research to general public.

We believe that in the later stages of the proposed research computational number theory packages (such as PARI, FLINT) will be useful. These packages are not used just by mathematicians but also by other scientists and engineers. Numerically verifying the predictions we make may lead to a further development of these packages and can have economic benefit for the UK.

Number theory is an area of mathematics considered to be most accessible to general public. Therefore it is an appropriate means to introduce a wider audience to the beauty of mathematics especially to encourage enthusiastic young student to study mathematics in their pursuit of higher education. We will take opportunities provided by King's College's various outreach programs to explain number theory related to our proposed research to general public.

We believe that in the later stages of the proposed research computational number theory packages (such as PARI, FLINT) will be useful. These packages are not used just by mathematicians but also by other scientists and engineers. Numerically verifying the predictions we make may lead to a further development of these packages and can have economic benefit for the UK.

## People |
## ORCID iD |

Mahesh Ramesh Kakde (Principal Investigator) |

### Publications

Bleher F
(2016)

*Higher Chern Classes in Iwasawa theory*
Dasgupta S

*On the Gross-Stark conjecture.*
Kakde M
(2016)

*Elliptic Curves, Modular Forms and Iwasawa Theory*
Kakde M
(2016)

*p -Adic Aspects of Modular Forms*Description | I will leave the response from last year unchanged below this one. Since last year we have made a lot of progress on all our questions. 1. With Samit Dasgupta we proposed to prove a factorisation theorem. A concrete application of this theorem is towards very general conjectures of Greenberg on L-invariants. The Gross-Stark conjecture belongs to this family of conjectures and was made by Gross in 1980. During Dasgupta's visit last year we started discussing this conjecture and managed to prove "rank 2" case of this conjecture (previously the rank 1 case was proven by Dasgupta, Darmon, Pollack under some hypothesis, later removed by Ventullo, and published in the Annals of Mathematics). We have now proven the Gross-Stark conjecture in full generality (i.e. arbitrary rank) with Dasgupta and Ventullo. This is the first result on special L-value formula in higher rank case and is a major progress in the field. We have also made a significant progress on the factorisation problem. We have identified Eisenstein series that we can use to prove the result. However, statements we need about these Eisenstein series do not appear in the literature in the generality we need. Therefore we are making several technical and tedious computations first to prove these generalisations before applying them to our factorisation problem. 2. Congruences in non-commutative Iwasawa theory - we have constructed very interesting q-expansions over non-commutative Iwasawa algebras whose constant terms are non-commutative p-adic zeta functions that we constructed in our earlier work. These q-expansions can be thought of as "non-commutative Hida families" of Eisenstein series. A much more interesting and also a much harder question is the construction of such families for cusp forms. We have made a significant progress on this question jointly with Ashay Burungale. Apart from its intrinsic interest, we realised that these families are needed in our proposed project with Bouganis. We also gave an alternate description of K_1 group of Z_p[GL_2(Z/pZ)]. This predicts new congruences for L-values of some non-CM elliptic curves. These calculations can be extended to find K_1 of the Iwasawa algebra of GL_2(Z_p). 3. We have also started a collaboration with a group of mathematicians in the US on approaches to Iwasawa theory. Traditionally Iwasawa theory ignores pseudo-null submodules. However, we develop a theory of "characteristic symbols" which are refinements of "characteristic elements" and encode information about "small" Iwasawa modules. In some special, but extremely interesting, instances we can formulate and prove a "main conjecture" for pseudo-null submodules. We are working on generalising this "main conjecture" to other situations and have already identified many interesting settings in which they can be formulated. [LAST YEAR]The grant for proposal "Non-commutative Iwasawa Theory and p-adic automorphic forms" began on April 15th this year. The goal of the proposed research is 3 fold. Firstly, to prove congruences between special values of L-functions associated to automorphic forms on unitary groups, to prove factorisation theorems for various p-adic L-functions and to prove main conjectures in function field Iwasawa theory. We have made significant progress on the last goal of the proposed research. One of the main steps towards proving main conjectures in non-commutative Iwasawa theory is understanding K_1 groups of Iwasawa algebras and related rings. With my collaborator, David Burns, I have obatined very general results on K_1 of Iwasawa algebras and related rings. A paper including these results is in preparation. We are also writing a paper which proves main conjecture of non-commutative Iwasawa theory for isotrivial abelian varieties over function fields. These algebraic results provided new insights into K_1 of power series rings over Iwasawa algebra and lead to the paper "Non-commutative q-expansions" where I construct q-expansions over non-commutative Iwasawa algebras which interpolate suitable modificaitons of Iwasawa theoretic Eisenstein series of Deligne and Ribet. We have submitted the paper for publication to a conference proceedings. The first two goals of the proposed research are more ambitious and I have had visits from my collaborators Samit Dasgupta and Thanasis Bouganis. These visits did lead us to very plausible approaches to the problems and we are highly optimistic that they will eventually lead to major results. However, there is nothing concrete to report now. |

Exploitation Route | We have made significant breakthroughs especially on Gross-Stark conjecture, factorisation theorem and theory of characteristic symbols. These techniques will be useful to solve many other similar problems in the area. Our discovery of "non-commutative Hida families" may also turn out to be very useful. Since the proposed research is in its intial stage it is hard to say beyond what has already been said in the proposal. However, I believe that the techniques I use with my collaborators will surely be useful to solve several other problems in number theory. |

Sectors | Other |

Description | Congruences in Non-commutative Iwasawa Theory |

Organisation | Durham University |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | We are working on proving congruences for special values of L-functions attached to automorphic forms on unitary groups. I have proven the algebraic results which give rise to these congruences and proven the congruences for Artin L-functions. I have simplified and generalised the algebraic results so that the congruences in general can be proven. The project is still in its early stages. |

Collaborator Contribution | My collaborator has proven these congruences for L-functions of certain automorphic forms. We are bring together our expertise to prove the result in general. |

Impact | None yet. |

Start Year | 2014 |

Description | Factorisations of p-adic L-functions |

Organisation | University of California, Santa Cruz (UCSC) |

Country | United States |

Sector | Academic/University |

PI Contribution | I provided a novel new approach to proving factorisation theorems for p-adic L-functions. The project is still in early stages and though we are optimistic it is not know yet if the approach yields results. The only proven instances of factorisation theorems use special elements such as elliptic units and Beilinson elements. My approach is automorphic and if it works will give many generalisations. |

Collaborator Contribution | My collaborator has proven one of the only two know instances of factorisation theorems for p-adic L-functions. He is an expert in the theory and his insights is guiding the new approach that I alluded to above. |

Impact | None. |

Start Year | 2013 |

Description | Function field Iwasawa theory |

Organisation | King's College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | I will leave the previous response below - this project has turned into a much bigger one. We are working towards a proof of equivariant BSD (assuming finiteness to sha) for ordinary abelian varieties and semistable abelian varieties. This has also led to many new questions which we hope to tackle in future. We are working on proving main conjectures over function fields over finite fields. These require proving generalisations of my results about K_1 of some Iwasawa algebras and related rings. We prove these results in a paper in preparation. |

Collaborator Contribution | My collaborator is an expert in Iwasawa theory and has proven some versions of main conjecture over function fields. Our approach is to generalise his strategy as much as possible. |

Impact | Two papers are in preparation. |

Start Year | 2013 |