Schedule for: 18w5053 - Special Values of Automorphic L-functions and Associated p-adic L-Functions

In this talk, we will formulate the conjectural framework of the conjecture on the existence of the cyclotomic p-adic L-function. First, we will introduce Hasse-Weil L-function for motives. Then we discuss the critical condition of motives and the algebraicity of special values as well as some fundamental conditions which should be essential to discuss p-adic properties of special values. After, we introduce some basic languages of Iwasawa algebra and measures, we will state the conjecture on the existence of of the cyclotomic p-adic L-function which was formulated thanks to Coates–Perrin-RIou and some others.

Related references:
[De72] P. Deligne, Valeurs de fonctions L et p ́eriodes d’int ́egrales, Automorphic forms, repre- sentations and L-functions, Proc. Sympos. Pure Math., XXXIII Part 2, Amer. Math. Soc., Providence, R.I., pp. 247–289, 1979.

[CP89] J. Coates, B. Perrin-Riou, On p-adic L-functions attached to motives over Q, Algebraic number theory, pp. 23–54, Adv. Stud. Pure Math., 17, Academic Press, 1989.

We will describe the now-classical method of constructing p-adic L-functions for GL(2, Q) based on modular symbols. The ideas of this method go back to Manin, Mazur, and Swinnerton-Dyer, as well as many others.

In a joint work with E. Urban, we prove under mild assumptions Integral Period Re- lations for the quadratic base change of a modular form and we compute the relative congruence number in terms of the value at s = 1 of a quadratic twist of the adjoint L-function. This proves a conjecture of Hida (1999). It also implies, under standard Taylor-Wiles type assumptions, in the real quadratic case, resp. in the imaginary quadratic case, the Bloch-Kato formula, resp. an analogue of it. This analogue would be the exact conjectural Bloch-Kato formula if a Bianchi period defined by E. Urban in his thesis could be related to a Bloch-Kato-Beilinson regulator.

Related references:
S. Bloch, K. Kato: Grothendieck Festschrift vol I, Birkhauser pp 333-400, 1990

H. Hida: Non-critical values of adjoint L-functions for SL(2), Proc. Symp. Pure Math. 66 (1999) Part I, 123-175

I explain the classical construction of the cyclotomic p-adic L-function of elliptic mod- ular forms by the Rankin-Selberg method.

Related references:
1. G. Shimura, The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804.

2. H. Hida, Elementary theory of L-functions and Eisenstein series. London Mathematical So- ciety Student Texts, 26. Cambridge University Press, Cambridge, 1993. xii+386 pp.

3. A. Panchishkin, Two variable p-adic L functions attached to eigenfamilies of positive slope. Invent. Math. 154 (2003), no. 3, 551–615.

4. S. Kobayashi, The p-adic Gross-Zagier formula for elliptic curves at supersingular primes. Invent. Math. 191 (2013), no. 3, 527–629.

This is a joint work with Kazuki Morimoto. In this talk, first we plan to give an outline of the proof of the refined global Gross-Parasd conjecture for special Bessel periods on SO(2n+1). Then we discuss about its consequence to Boecherer ’s conjecture concerning the Fourier coefficients of the Siegel cusp forms of degree two which are Hecke eigenforms.

Related references:
The followings are links to the papers related to this talk.

https://arxiv.org/abs/1611.05567

https://doi.org/10.1007/s00208-016-1440-z

A construction of p-adic L-functions for GL(2) via the modular symbol method is reviewed in this talk. I will summarize some technical points of the construction comparing with the works of F. Januszewski on p-adic L-functions for GL(n + 1) × GL(n). In particular, the behavior under the Tate twists is emphasized in the talk, since it is the most important new ingredient in Januszewski’s recent preprint.

Related references (for talks I to IV):
(Main) F.Januszewski. Non-abelian p-adic Rankin-Selberg L-functions and non-vanishing of central L- values, arXiv:1708.02616, 2017.

K. Namikawa. On p-adic L-functions associated with cusp forms on GL2. manuscr. math. 153, pages 563–622, 2017. (Sub)

B.J. Birch. Elliptic curves over Q, a progress report. 1969 Number Theory Institute. AMS Proc. Symp. Pure Math. XX, 396–400, 1971.

M. Dimitrov. Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties. Amer. J. Math. 135, 1117–1155, 2013.

F. Januszewski. Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields, J. Reine Angew. Math. 653, 1–45, 2011.

F. Januszewski. On p-adic L-functions for GL(n)×GL(n−1) over totally real fields, Int. Math. Res. Not., Vol. 2015, No. 17, 7884–7949.

F. Januszewski. p-adic L-functions for Rankin-Selberg convolutions over number fields, Ann. Math. Quebec 40, special issue in Honor of Glenn Stevens ’60th birthday, 453–489, 2016.

F. Januszewski. On period relations for automorphic L-functions I. To appear in Trans. Amer. Math. Soc., arXiv:1504.06973

H. Kasten and C.-G. Schmidt. On critical values of Rankin-Selberg convolutions. Int. J. Number Theory 9, pages 205–256, 2013. D. Kazhdan, B. Mazur, and C.-G. Schmidt. Relative modular symbols and Rankin-Selberg convolutions, J. Reine Angew. Math. 512, 97–141, 2000.

K. Kitagawa. On standard p-adic L-functions of families of elliptic cusp forms, p-adic mon- odromy and the Birch and Swinnerton-Dyer conjecture (B. Mazur and G. Stevens, eds.), Con- temp. Math. 165, AMS, 81–110, 1994.

J.I. Manin. Non-archimedean integration and p-adic Hecke-Langlands L-series. Russian Math. Surveys 31, 1, 1976.

B. Mazur, and P. Swinnerton-Dyer. Arithmetic of Weil Curves, Invent. Math. 25, 1–62, 1974. B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84, 1–48, 1986.

C.-G. Schmidt. Relative modular symbols and p-adic Rankin-Selberg convolutions, Invent. Math. 112, 31–76, 1993.

C.-G. Schmidt. Period relations and p-adic measures, manuscr. math. 106, 177–201, 2001. B. Sun. The non-vanishing hypothesis at infinity for Rankin-Selberg convolutions. J. Amer. Math. Soc. 30, pages 1–25, 2017.

A. Raghuram. On the Special Values of certain Rankin-Selberg L-functions and Applications to odd symmetric power L-functions of modular forms. Int. Math. Res. Not. 2010, 334–372, 2010.

A. Raghuram. Critical values for Rankin-Selberg L-functions for GL(n) × GL(n − 1) and the symmetric cube L-functions for GL(2). Forum Math. 28, 457–489, 2016.

In this talk I will give a quick overview of the analytic theory of Rankin-Selberg L-functions for GL(n+1) x GL(n) due to Jacquet, Piatetski-Shapiro and Shalika, which are the L-functions of interest in this lecture series. Subsequently I will introduce the modular symbol of Schmidt, Kazhdan-Mazur-Schmidt, Kasten-Schmidt, Raghuram and myself and explain its relation to special values of Rankin-Selberg L-functions and the archimedean non-vanishing hy- pothesis established by Sun.

Consider the universal minimal p-ordinary deformation rT into GL(2, T ) (for a prime p ≥ 5) of a modulo p induced representation from a quadratic field F. For almost all primes p split in F, we describe how to determine T as an algebra over the weight Iwasawa algebra Λ as an extension of degree 1,2,3. This implies that the Pontryagin dual of the adjoint Selmer group of rT is isomorphic to Λ/(Lp) as Lambda-modules for an explicit power series Lp.

For a prime p and a positive integer n, the standard zeta function LF (s) is consid- ered, attached to an Hermitian modular form F = ∑ A(H)qH on the Hermitian upper half H plane Hm of degree n, where H runs through semi-integral positive definite Hermitian matrices of degree n, i.e. H ∈ Λm(O) over the integers O of an imaginary quadratic field K, where qH = exp(2πiTr(HZ)). Analytic p-adic continuation of their zeta functions constructed by A.Bouganis in the ordinary case, is extended to the admissible case via growing p-adic measures. Previously this problem was solved for the Siegel modular forms. Main result is stated in terms of the Hodge polygon PH(t) : [0,d] → R and the Newton polygon PN(t) = PN,p(t) : [0,d] → R of the zeta function LF (s) of degree d = 4n. Main theorem gives a p-adic analytic interpolation of the L values in the form of certain integrals with respect to Mazur-type measures.

Related references:
[BS00] B ̈ocherer, S., and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, N. 5, 1375–1443 (2000).

[Bou16] Bouganis T., p-adic Measures for Hermitian Modular Forms and the Rankin–Selberg Method. in Elliptic Curves, Modular Forms and Iwasawa Theory – Conference in honour of the 70th birthday of John Coates, pp 33–86

[CourPa] Courtieu M, Panchishkin A. A, Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

We give a survey on our work with C.Schmidt on p-adic interpolation for standard L-functions for Siegel modular forms. We emphasize that there is a strong analogy between exterior twists of Eisenstein series and appliction of certain differential operators. The case of triple product L-functions for GL(2) can be handled in a similar way.

Related references:
C.G.Schmidt, S.B ̈ocherer: p-adic measures attached to Siegel modular forms. Ann.Inst.Fourier 50, 1375-1443(2000)

A.Panchishkin, S.B ̈ocherer: p-adic interpolation for triple L-functions: analytic aspects. In: Automorphic Forms and L-functions II: Contemporary Math.489 (2009)

Daniel Barrera Salazar:
Title: Triple product p-adic L-functions over totally real number fields

Abstract: This poster presents some ideas of a work in progress joint with S. Molina, which is about the construction of p-adic L-functions attached to a triple of Hida families over totally real number fields and employs methods developed by Andreatta and Iovita. This work is part of a project joint with S. Molina and V. Rotger, whose main objective is to obtain new advances in the BSD conjecture for elliptic curves over totally real number fields. In this poster we also try to put our work in the framework of such project.



Daniel Disegni:
Title: Local Langlands, local factors, and Zeta integrals in analytic families

Abstract: Let X be a Noetherian scheme over Q (e.g. the spectrum of an affinoid in an eigenvariety) and let \Pi be a nice X-family of automorphic representations. In order to construct a p-adic L-function L_p(\Pi) along X, one starts from the expression of L(\Pi) as a ratio of global and local Zeta integrals. A natural, calculation-free and flexible approach is then to interpolate \emph{both} the global \emph(and} the local (at \ell\neq p) Zeta integrals. We provide the local results in the Rankin-Selberg case (GL_n x GL_m), after studying the interpolation of the Local Langlands correspondence for GL_n along X. The results are largely analogous to those of Emerton, Helm, and Moss, who studied bases which are local rings of mixed characteristic.

This lecture will cover the doubling method for unitary groups and the resulting in- tegral representation of the standard L-functions for cuspidal representations of unitary groups.

This lecture will focus on the case of the unitary group U(1) (so U(1)xU(1) inside U(1,1)) and will explain how Katz’s p-adic L-function for a CM fields is a special case of the doubling method constructions.

In the third talk of this lecture series I will define p-adic L-functions attached to nearly ordinary cohomology classes for GL(n+1) x GL(n) and sketch a proof of the Manin congruences and the functional equation.

In the last talk of this lecture series we will examine the relation between the p-adic L-functions constructed in the previous lecture and the complex analytic p-adic L-functions of Jacquet, Piatetski-Shapiro and Shalika. This will involve p-stabilization in principal series representations, a careful study of local zeta integrals and non-vanishing properties of certain Whittaker vectors.

An earlier paper by Harris and Venkatesh that just appeared in Exp. Math. is good background reading (but not obligatory of course; I will strive to make the lecture reasonably self-contained.)

Related references:
The link is https://www.tandfonline.com/doi/abs/10.1080/10586458.2017.1409144?journalCode=uexm20

P -adic L-functions for cohomological cuspidal automorphic representations of GL(2n) were first constructed by Ash and Ginzburg in the case of trivial coefficients. We will discuss a new construction, which works for arbitrary coefficient systems. The construction relies on the representation theory of p-adic groups as well as properties of the cohomology of p-arithmetic groups. This is a generalization of Spiess’ work on the GL(2)-case.

Related references:
L. Gehrmann, On Shalika models and p-adic L-functions, Israel Journal of Mathematics 226 Issue 1, (June 2018), 237–294

A. Ash and D. Ginzburg, P -adic L-functions for GL(2n), Inventiones mathematicae 116 (1994), 27–73.

M. Spiess, On special zeros of p-adic L-functions of Hilbert modular forms, Inventiones mathe- maticae 196 (2014), 69–138

I will give an overview of in-part conjectural relationships between modular symbols in Eisenstein quotients of homology groups of locally symmetric spaces and cohomological oper- ations on special units such as cyclotomic units. Some focus will be given to aspects related to p-adic L-functions and special values. The talk will be based in part on joint work with Takako Fukaya and Kazuya Kato.

Related references:
Takako Fukaya, Kazuya Kato, and Romyar Sharifi, Modular symbols in Iwasawa theory, in Iwa- sawa Theory 2012 - State of the Art and Recent Advances, Contrib. Math. Comput. Sci. 7, Springer, 2014, 177-219.

Takako Fukaya and Kazuya Kato, On conjectures of Sharifi, preprint, 2012, http://math.ucla.edu/ shar- ifi/sharificonj.pdf

This lecture will review some general facts about p-adic L-functions and the Eisen- stein measure and explain the strategy for constructing one from the other.

This lecture will cover some of the details of the constructions and especially the choice of data and calculations at the p-adic places.