It is an open problem to describe the shape of the reductions of local Galois representations attached to cusp forms at primes away from the level, or more generally, the shape of the reductions of two-dimensional crystalline representations. Partial results go back to Deligne, Fontaine and Edixhoven. One folklore conjecture (attributed to Breuil, Buzzard and Emerton) is that if the weight is even and the slope is fractional, then the reduction is always irreducible.

In this talk we shall state and prove our zig-zag conjecture which deals with large exceptional weights and half-integral slopes. These weights fall squarely outside the scope of the above conjecture. The conjecture states that the reduction in these cases is given by an alternating sequence of irreducible and reducible representations depending on the size of two auxiliary parameters. Special cases of zig-zag have been proved over the years by various authors using Langlands correspondences.

The present general proof uses the reverse of a recent limiting argument due to Chitrao-Ghate-Yasuda in the Colmez-Chenevier rigid analytic blow up space of trianguline representations to reduce the study of the reduction of crystalline representations to results on the reductions of semi-stable representations due to Breuil-Mezard, Guerberoff-Park and most recently Chitrao-Ghate.

Given two distinct cuspidal automorphic `$L$`

-functions (of `$\mathrm{GL}_n$`

and `$\mathrm{GL}_m$`

over `$\mathbb{Q}$`

) one expects that their quotients will have infinitely poles, but this is surprisingly hard to prove. In this talk, I will discuss my recent work on the case `$m=n-2$`

and the primitivity of the `$L$`

-functions of cuspidal automorphic `$L$`

-functions of `$\mathrm{GL}_3$`

. These methods also work for Artin `$L$`

-functions and, more generally, for the `$L$`

-functions of Galois representations under further hypotheses.

One of the central themes in number theory is the study of special values of `$L$`

-functions, in particular, the investigation of their transcendental nature and algebraic relations among them. A special function governing linear relations among the values `$L(1,\chi)$`

as `$\chi$`

varies over Dirichlet characters modulo `$q$`

, is the digamma function, which is the logarithmic derivative of the gamma function. In this talk, we discuss the arithmetic nature and related properties of values of the digamma function at rational arguments, and emphasize their connection with a seemingly unrelated conjecture of Erdos, which is still open.

Families of `$p$`

-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing the correct setup for the study of `$p$`

-adic deformations of automorphic forms. In order to obtain arithmetic applications, such as constructing `$p$`

-adic `$L$`

-functions or proving explicit reciprocity laws for Euler systems, one needs to perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the `$p$`

-stabilization of the automorphic form we are interested in.

While the geometry of an eigenvariety at points of cohomological weight is well understood thanks to classicality results, the study at classical points which are limit of discrete series (such as weight `$1$`

Hilbert modular forms or weight `$(2,2)$`

Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch–Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjecture.

Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint `$p$`

-adic `$L$`

-functions.

In this talk we will illustrate this philosophy based on ideas of Joël Bellaïche.

For all positive powers of primes `$p \geq 5$`

, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight `$1/2$`

weakly holomorphic modular form or a suitable vector-valued harmonic Maassform. To this end, we work with the logarithmic derivatives of these twisted Borcherds products, and offer various numerical examples of non-trivial linear congruences between them modulo `$p=11$`

. In the case of positive powers of primes `$p = 2, 3$`

, we obtain similar results by multiplying the logarithmic derivative with a Hilbert class polynomial as well as a power of the modular discriminant function. Both results confirm a speculation by Ono. (joint work with Andreas Mono).

Since the work of Kubota in the late 1960s, it has been known that certain Gauss sum twisted (multiple) Dirichlet series are closely connected to a theory of automorphic functions on metaplectic covering groups. The representation theory of such covering groups was then initiated by Kazhdan and Patterson in the 1980s, who emphasized the role of a certain non-uniqueness of Whitattaker functionals.

Motivated on the one hand by the recent theory of Weyl group multiple Dirichlet series, and on the other by the so-called “quantum” geometric Langlands correspondence, we explain how to connect the representation theory of metaplectic covers of $p$-adic groups to an object of rather disparate origin, namely a quantum group at a root of unity. This gives us a new point of view on the non-uniqueness of Whittaker functionals and leads, among other things, to a Casselman–Shalika type formula expressed in terms of (Gauss sum) twists of “$q$”-Littlewood–Richardson coefficients, objects of some combinatorial interest.

Joint work with Valentin Buciumas.

Baladi and Vallée shows the limit Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers with bounded denominators based on ergodic methods. We give an analogue of the result for complex continued fractions over imaginary quadratic number fields and discuss applications in value distribution of `$L$`

-functions of `$\mathrm{GL}_2$`

(joint with Dohyeong Kim and Seonhee Lim).

In his seminal paper in 2001, Henri Darmon proposed a systematic construction of `$p$`

-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of local (`$p$`

-adic) cohomology classes/cycles in the `$p$`

-adic Galois representation attached to a cuspidal cohomological automorphic representation of `$\mathrm{PGL}_2$`

over any number field, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato conjecture. This work generalises previous constructions of Rotger-Seveso for elliptic cusp forms and earlier joint work with Williams for Bianchi cusp forms. Time permitting, I will also talk about the plectic analogues of these objects.

In a joint work in progress with K. Prasanna, we
study period relations for the base change to `$\mathrm{GL}_4$`

of a
cohomological cuspidal representation on `$\mathrm{GSp}_4$`

.
An unexpected period occurs in the period relations.

Let `$N$`

be a prime number `$>3$`

. Mazur has defined, from the theory of modular forms, a unit `$u$`

in `$\mathbb{Z}/N$`

. This unit turned out to be, up to a `$6$`

-th root of unity,`$\prod_{k=1}^{(N-1)/2}k^k$`

.
In this talk we will describe how the unit is connected to various objects in number theory. For instance:
–The unit `$u$`

can be understood as a derivative of the zeta function at `$-1$`

, (despite living in a finite field).
– Lecouturier has shown that this unit is the discriminant of the Hasse polynomial: `$\sum_{i=0}^{(N-1)/2}a_i X^i$`

modulo `$N$`

, where `$a_i$`

is the square of the `$i$`

-th binomial coefficient in degree `$N$`

.
– Calegari and Emerton have related `$u$`

to the class group of the quadratic field `$\mathbb{Q}(\sqrt{-N})$`

.
For every prime number `$p$`

dividing `$N-1$`

, It is important to determine when `$u$`

is a `$p$`

-th power in `$(\mathbb{Z}/N)^*$`

.
If time allows, I will describe the connections to modular forms and Galois representations, and the general theory that Lecouturier has developed from this unit. For instance,when `$u$`

is not a `$p$`

-th power, a certain Hecke algebra acting on modular forms is of rank `$1$`

over the ring of `$p$`

-adic integers `$\mathbb{Z}_p$`

(the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.

Let `$p$`

be an odd prime, `$f$`

be a `$p$`

-ordinary newform of weight `$k$`

and `$h$`

be a normalized cuspidal `$p$`

-ordinary Hecke eigenform of weight `$\ell < k$`

. Let `$p$`

be an Eisenstein prime for `$h$`

i.e. the residual Galois representation of `$h$`

at `$p$`

is reducible. In this talk, we show that the `$p$`

-adic `$L$`

-function and the characteristic ideal of the `$p^{\infty}$`

-Selmer group of the Rankin-Selberg convolution of `$f$`

, `$h$`

generate the same ideal modulo `$p$`

in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for `$f \otimes h$`

holds modulo `$p$`

. This is a joint work with Somnath Jha and Sudhanshu Shekhar.

The (tame) class field theory for a smooth variety `$X$`

is the
study of describing the abelianized (tame) {'e}tale fundamental group of
`$X$`

in terms of some groups which are defined using algebraic cycles of `$X$`

.
In this talk, we study the tame class field theory for smooth varieties
over local fields. We will begin with defining few notions and recalling
various results from the past to overview the historical background of the
subject. We will then study abelianized tame fundamental group denoted as
`$\pi^{ab,t}_{1}(X)$`

, with the help of reciprocity map ```
$\rho^{t}_{X} :
C^{t}(X) \rightarrow \pi^{ab,t}_{1}(X)$
```

and will describe the kernel and
topological cokernel of this map. This talk is based on a joint work with
Prof. Amalendu Krishna and Dr. Rahul Gupta.

Let `$K$`

be an imaginary quadratic field of class number `$1$`

such that both `$p$`

and `$q$`

split in `$K$`

. We show that under appropriate hypotheses, the `$p$`

-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic `$\mathbb{Z}_q$`

-extension of `$K$`

. This is joint work with Antonio Lei.

We report on new ideas of Ki-Seng Tan and myself towards the construction of a `$p$`

-adic `$L$`

-function associated to an automorphic overconvergent `$F$`

-isocrystal over a curve over a finite field. This function should be of interest in the Iwasawa theory for such coefficients.

Let `$F$`

be a totally real field. Let `$\pi$`

be a cuspidal cohomological automorphic representation for `$\mathrm{GL}_2/F$`

. Let `$L(s, \mathrm{Ad}^0, \pi)$`

denote the adjoint `$L$`

-function associated to `$\pi$`

. The special values of this `$L$`

-function and its relation to congruence primes have been studied by Hida, Ghate and Dimitrov. Let `$p$`

be an integer prime. In this talk, I will discuss the construction of a `$p$`

-adic adjoint `$L$`

-function in neighbourhoods of very decent points of the Hilbert eigenvariety. As a consequence, we relate the ramification locus of this eigenvariety to the zero set of the `$p$`

-adic `$L$`

-functions. This was first established by Kim when `$F=\mathbb{Q}$`

. We follow Bellaiche’s description of Kim’s method, generalizing it to arbitrary totally real number fields. This is joint work with John Bergdall and Matteo Longo.

The Siegel-Veech transform is a basic tool in homogeneous as well as Teichmuller dynamics. I will introduce the transform and explain how it can be used in counting problems.

In the late 1950s, an important problem in number theory was to extend the notion of `$L$`

-functions attached to cuspforms on the upper-half
plane to automorphic forms on reductive groups. Langlands’s work on non-abelian Harmonic analysis, namely the problem of the spectral decomposition of automorphic forms, led him to a general notion of `$L$`

-functions
attached to cuspforms. We give an introduction to the spectral decomposition of automorphic forms and discuss some contemporary problems.

Over an unramified extension `$F/\mathbb{Q}_p$`

, by the works of Fontaine, Wach, Colmez and Berger, it is well-known that a crystalline representation of the absolute Galois group of `$F$`

is of finite height. Moreover, in this case, to a crystalline representation one can functorially attach a lattice inside the associated etale `$(\varphi, \Gamma)$`

-module called the Wach module. Berger showed that the aforementioned functor induces an equivalence between the category of crystalline representations and Wach modules. Furthermore, this categorical equivalence admits an integral refinement. In this talk, our goal is to generalize the notion of Wach modules to relative `$p$`

-adic Hodge theory. For a “small” unramified base (in the sense of Faltings) and its etale fundamental group, we will generalize the result of Berger to an equivalence between crystalline representations and relative Wach modules as well as establish its integral refinement.

Andreatta, Iovita, and Pilloni have proven the existence of an adic eigencurve, which includes characteristic `$p$`

points at the boundary. In joint work with Ruochuan Liu, using the theory of Crystalline periods, we show that the Galois representations associated to these points satisfy an appropriate trianguline property.

Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the `$\ell$`

-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the `$\ell$`

-primary part of principally polarized Abelian `$3$`

-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over `$\mathbb{Q}$`

defined over at least `$10$`

variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).

I will report on recent work with Lichtenbaum and Suzuki on a new proof of the relation between the arithmetic of an elliptic curve over function fields and surfaces over finite fields.

Let `$F$`

be a totally real field and `$p$`

be an odd prime unramified in `$F$`

. We will give an overview of the problem of determining the explicit mod `$p$`

structure of a modular `$p$`

-adic Galois representation and determining the associated local Serre weights. The Galois representations are attached to Hilbert modular forms over `$F$`

, more precisely to eigenforms on a Shimura curve over `$F$`

. The weight part of the Serre’s modularity conjecture for Hilbert modular forms relates the local Serre weights at a place `$v|p$`

to the structure of the mod `$p$`

Galois representation at the inertia group over `$v$`

. Thus, local Serre weights give good information on the structure of the modular mod `$p$`

Galois representation. The eigenforms considered are of small slope at a fixed place `$\mathbf{p}|p$`

, and with certain constraints on the weight over `$\mathbf{p}$`

. This is based on a joint work with Shalini Bhattacharya.

We introduce a smoothed version of the equivariant `$S$`

-truncated
`$p$`

-adic Artin `$L$`

-function for one-dimensional admissible `$p$`

-adic Lie
extensions of number fields. Integrality of this smoothed `$p$`

-adic
`$L$`

-function, conjectured by Greenberg, has been verified for pro-`$p$`

extensions (assuming the Equivariant Iwasawa Main Conjecture) as well as
`$p$`

-abelian extensions (unconditionally). Integrality in the general case
is also expected to hold, and is the subject of ongoing research.

It is a natural question to count matrices `$A$`

with integer entries in an expanding box of side length `$x$`

with `$\det(A) = r$`

, a fixed integer; or with the characteristic polynomial of `$A = f$`

, a fixed integer polynomial; and there are several results in the literature on these problems. Most of the existing results, which use either Ergodic methods or Harmonic Analysis, give asymptotics for the number of such matrices as `$x$`

goes to infinity and in the only result we have been able to find that gives a bound on the error term, the bound is not very satisfactory. The aim of this talk will be to present an ongoing joint work with Rachita Guria in which, for the easiest case of `$2 \times 2$`

matrices, we have been able to obtain reasonable bounds for the error terms for the above problems by employing elementary Fourier Analysis and results from the theory of Automorphic Forms.

We bound a short second moment average of `$\mathrm{GL}(3)$`

and `$\mathrm{GL}(3) \times \mathrm{GL}(1)$`

`$L$`

-functions. These yield `$t$`

-aspect and depth aspect subconvexity bounds respectively, and improve upon the earlier subconvexity exponents. This moment estimate provides an analogue for cusp forms of Ivic’s bound for the sixth moment of the zeta function, and is the first time a short second moment has been used to obtain a subconvex bound in higher rank. This is a joint work with Ritabrata Munshi and Wing Hong Leung.

I will answer some questions (admissibility, dimensions of invariants by Moy-Prasad groups)
on representations of reductive `$p$`

-adic groups and on Hecke algebras modules raised in my paper for the 2022-I.C.M. Noether lecture.

If `$\theta$`

is an involution on a group `$G$`

with fixed points `$H$`

,
it is a question of considerable interest to classify irreducible representations of `$G$`

which carry an `$H$`

-invariant linear form. We will discuss some cases of this
question paying attention to finite dimensional representation of compact groups
where it is called the Cartan-Helgason theorem.

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions “on average” for moduli `$q$`

in the range `$q \le x^{1/2 -\epsilon }$`

for any `$\epsilon>0$`

. In 1989, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli `$q$`

is extended to `$q \le x/ (\log x)^B$`

for any `$B>1$`

. We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.

Hilbert modular forms are generalization of classical modular forms over totally real number fields. The Fourier coefficients of a modular form are of great importance owing to their rich arithmetic and algebraic properties. In the theory of modular forms one of the classical problems is to determine a modular form by a subset of all Fourier coefficient. In this talk, we discuss about to determination of a Hilbert modular form by the Fourier coefficients indexed by square-free integral ideals. In particular, we talk about the following result.

Given any `$\epsilon>0$`

, a non zero Hilbert cusp form `$\mathbf{f}$`

of weight `$k=(k_1,k_2,\ldots, k_n)\in (\mathbb{Z}^{+})^n$`

and square-free level `$\mathfrak{n}$`

with Fourier coefficients
`$C(\mathbf{f},\mathfrak{m})$`

, then there exists a square-free integral ideal `$\mathfrak{m}$`

with `$N(\mathfrak{m})\ll k_0^{3n+\epsilon} N(\mathfrak{m})^{\frac{6n^2 +1}{2}+\epsilon}$`

such that `$C(\mathbf{f},\mathfrak{m})\neq 0$`

. The implied constant depend on `$\epsilon , F.$`

Let `$F$`

be a global field and `$\Gamma_F$`

its absolute Galois group. Given
a continuous representation `$\bar{\rho}: \Gamma_F \to G(k)$`

, where `$G$`

is a split
reductive group and `$k$`

is a finite field, it is of interest to know when `$\bar{\rho}$`

lifts
to a representation `$\rho: \Gamma_F \to G(O)$`

, where `$O$`

is a complete discrete
valuation ring of characteristic zero with residue field `$k$`

. One would also like to control
the local behaviour of `$\rho$`

at places of `$F$`

, especially at primes dividing `$p = \mathrm{char}(k)$`

(if `$F$`

is a number field). In this talk I will give an overview of a method developed in joint work with
Chandrashekhar Khare and Stefan Patrikis which allows one to construct such lifts in many cases.

The modularity lifting theorem of Boxer-Calegari-Gee-Pilloni established for the first time the existence of infinitely many modular abelian surfaces `$A / \mathbb{Q}$`

upto twist with `$\text{End}_{\mathbb{C}}(A) = \mathbb{Z}$`

. We render this explicit by first finding some abelian surfaces whose associated mod-`$p$`

representation is residually modular and for which the modularity lifting theorem is applicable, and then transferring modularity in a family of abelian surfaces with fixed `$3$`

-torsion representation. Let `$\rho: G_{\mathbb{Q}} \rightarrow GSp(4,\mathbb{F}_3)$`

be a Galois representation with cyclotomic similitude character. Then, the transfer of modularity happens in the moduli space of genus `$2$`

curves `$C$`

such that `$C$`

has a rational Weierstrass point and `$\mathrm{Jac}(C)[3] \simeq \rho$`

. Using invariant theory, we find explicit parametrization of the universal curve over this space. The talk will feature demos of relevant code in Magma.

Let `$k$`

be a nonarchimedian local field, `$\widetilde{G}$`

a connected reductive `$k$`

-group, `$\Gamma$`

a finite group of automorphisms of `$\widetilde{G}$,`

and `$G:= (\widetilde{G}^\Gamma)^\circ$`

the connected part
of the group of `$\Gamma$`

-fixed points of `$\widetilde{G}$`

.
The first half of my talk will concern motivation: a desire for a more explicit understanding of base change and other liftings of representations. Toward this end, we adapt some results of Kaletha-Prasad-Yu. Namely, if one assumes that the residual characteristic of `$k$`

does not divide the order of `$\Gamma$`

, then they show, roughly speaking, that `$G$`

is reductive, the building `$\mathcal{B}(G)$`

of `$G$`

embeds in the set of `$\Gamma$`

-fixed points of `$\mathcal{B}(\widetilde{G})$`

, and similarly for reductive quotients of parahoric subgroups.

We prove similar statements, but under a different hypothesis on `$\Gamma$`

. Our hypothesis does not imply that of K-P-Y, nor vice versa. I will include some comments on how to resolve such a totally unacceptable situation.

(This is joint work with Joshua Lansky and Loren Spice.)

Euler systems are cohomological tools that play a crucial role in the study of special values of `$L$`

-functions; for instance, they have been used to prove cases of the Birch–Swinnerton-Dyer conjecture and have recently been used to prove cases of the more general Bloch–Kato conjecture. A fundamental technique in these recent advances is to show that Euler systems vary in `$p$`

-adic families. In this talk, we will first give a general introduction to the theme of `$p$`

-adic variation in number theory and introduce the necessary background from the theory of Euler systems; we will then explain the idea and importance of `$p$`

-adically varying Euler systems, and finally discuss current work in progress on `$p$`

-adically varying the Asai–Flach Euler system, which is an Euler system arising from quadratic Hilbert modular eigenforms.

I will talk about recent work pertaining to the existence of abelian varieties not isogenous to Jacobians over fields of both characteristic zero and p. This is joint work with Jacob Tsimerman.

(Joint work with Andy O’Desky) There is a very classical formula counting the number of irreducible polynomials in one variable over a finite field. We study the analogous question in many variables and generalize Gauss’ formula. Our techniques can be used to answer many other questions about the space of irreducible polynomials in many variables such as it’s euler characteristic or euler hodge-deligne polynomial. To prove these results, we define a generalization of the classical ring of symmetric functions and use natural basis in it to help us compute the answer to the above questions.

A conjecture of Katz and Sarnak predicts that the distribution of spacings between ``straightened” Hecke angles (corresponding to Fourier coefficients of Hecke newforms) matches that of a uniformly distributed, random sequence in the unit interval. This comparison is made with the help of local spacing statistics, such as the level spacing distribution and various types of correlations of the Hecke angles. In previous joint work with Baskar Balasubramanyam and ongoing joint work with my PhD student Jewel Mahajan, we have provided evidence in favour of this conjecture, by showing that the pair correlation function of the Hecke angles, averaged over families of Hecke newforms, is expected to be Poissonnian, with variance converging to zero as we take larger and larger families. In this talk, we will explore various types of questions arising in the study of the local behaviour of sequences of Hecke angles, and explain the above-mentioned results.

Lambert series lie at the heart of modular forms and the theory of the Riemann zeta function. The early pioneers in the subject were Ramanujan and Wigert. We discuss Ramanujan’s formula for odd zeta values and its generalizations and analogues obtained by the speaker with his co-authors culminating into a recent transformation for `$\sum_{n=1}^{\infty}\sigma_a(n)e^{-ny}$`

for `$a\in\mathbb{C}$`

and Re`$(y)>0$`

. We will discuss several applications of this result. A formula of Wigert and its recent analogue found by Soumyarup Banerjee, Shivajee Gupta and the author will be discussed and its application in the zeta-function theory will be given. This talk is an amalagamation of results of the author on this topic from various papers co-authored with Bibekananda Maji, Rahul Kumar, Rajat Gupta, Soumyarup Banerjee and Shivajee Gupta.

Given a group `$G$`

and two Gelfand subgroups `$H$`

and `$K$`

of `$G$`

, associated to an irreducible representation `$\pi$`

of `$G$`

, there is a notion of `$H$`

and `$K$`

being correlated with respect to `$\pi$`

in `$G$`

. This notion is defined by Benedict Gross in 1991. We discuss this theme and give some details in a specific example (which is joint work with Arindam Jana).

The (`$p^{\infty}$`

) fine Selmer group (also called the `$0$`

-Selmer group) of an elliptic curve is a subgroup of the usual `$p^{\infty}$`

Selmer group of an elliptic curve and is related to the first and the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group over the cyclotomic `$\mathbb{Z}_p$`

-extension of a number field `$K$`

is intricately related to Iwasawa’s `$\mu$`

-invariant vanishing conjecture on the growth of `$p$`

-part of the ideal class group of `$K$`

in the cyclotomic tower. In this talk, we will discuss the structure and properties of the fine Selmer group over certain `$p$`

-adic Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.

In this talk I will explain new research on `$L$`

-invariants of modular forms, including ongoing joint work with Robert Pollack. `$L$`

-invariants, which are `$p$`

-adic invariants of modular forms, were discovered in the 1980’s, by Mazur, Tate, and Teitelbaum. They were formulating a `$p$`

-adic analogue of Birch and Swinnerton-Dyer’s conjecture on elliptic curves. In the decades since, `$L$`

-invariants have shown up in a ton of places: `$p$`

-adic `$L$`

-series for higher weight modular forms or higher rank automorphic forms, the Banach space representation theory of `$\mathrm{GL}(2,\mathbb{Q}_p)$`

, `$p$`

-adic families of modular forms, Coleman integration on the `$p$`

-adic upper half-plane, and Fontaine’s `$p$`

-adic Hodge theory for Galois representations. In this talk I will focus on recent numerical and statistical investigations of these `$L$`

-invariants, which touch on many of the theories just mentioned. I will try to put everything into the context of practical questions in the theory of automorphic forms and Galois representations and explain what the future holds.

We will talk on the an analogue of the Tamagawa Number conjecture, with coefficients over varieties over finite fields. This a joint work with O. Brinon (Bordeaux) and a work in progress.

Euler solved the famous Basel problem and discovered that Riemann zeta functions at positive even integers are rational multiples of powers of `$\pi$`

. Multiple zeta values (MSVs) are a multi-dimensional generalization of the Riemann zeta values, and MZVs which are rational multiples of powers of `$\pi$`

is called Eulerian MZVs. In 1996, Borwein-Bradley-Broadhurst discovered a series of conjecturally Eulerian MZVs which together with the known Eulerian family seems to exhaust all Eulerian MZVs at least numerically. A few years later, Borwein-Bradley-Broadhurst-Lisonek discovered two families of interesting conjectural relations among MZVs generalizing the previous conjecture of Eulerian MZVs, which were later extended further by Charlton in light of alternating block structure. In this talk, I would like to present my recent joint work with Minoru Hirose concerning block shuffle relations that simultaneously resolve and generalize the conjectures of Charlton.

We reprove the main equidistribution instance in the Ferrero–Washington proof of the vanishing of cyclotomic Iwasawa `$\mu$`

-invariant, based on the ergodicity of a certain `$p$`

-adic skew extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).

In his 1976 proof of the converse to Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-`$p$`

extensions of the `$p$`

-th cyclotomic field when `$p$`

is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-`$p$`

extensions of `$\mathbb{Q}(N^{1/p})$`

when `$N$`

is a prime that is congruent to `$-1$`

mod `$p$`

. This answers a question posted on Frank Calegari’s blog.

Following a joint work with Sara Arias-de-Reyna and François Legrand, we present a new kind of families of modular forms. They come from representations of the absolute Galois group of rational function fields over `$\mathbb{Q}$`

. As a motivation and illustration, we discuss in some details one example: an infinite Galois family of Katz modular forms of weight one in characteristic `$7$`

, all members of which are non-liftable. This may be surprising because non-liftability is a feature that one might expect to occur only occasionally.

Let `$R$`

be the Iwasawa algebra over a compact, `$p$`

-adic, pro-`$p$`

group
`$G$`

, where `$G$`

arises as a Galois group of number fields from Galois representations.
Suppose `$M$`

is a finitely generated `$R$`

-module. In the late 1970’s , Harris studied the
asymptotic growth of the ranks of certain coinvariants of `$M$`

arising from the action
of open subgroups of `$G$`

and related them to the codimension of `$M$`

. In this talk, we
explain how Harris’ proofs can be simplified and improved upon, with possible
applications to studying some natural subquotients of the Galois groups of number fields.

This talk will be a report of work in progress with Ming-Lun Hsieh. Just as in classical Iwasawa theory where one studies congruences involving Hecke eigenvalues associated to Eisenstein series, we study congruences involving `$p$`

-adic families of Hecke eigensystems associated to the space of Yoshida lifts of two Hida families. Our goal is to show that under suitable assumptions, the characteristic ideal of a dual Selmer group is contained inside the congruence ideal.

Let `$H$`

be a subgroup of a group `$G$`

. For an irreducible representation `$\sigma$`

of `$H$`

, the triple `$(G,H, \sigma)$`

is called a Gelfand triple if `$\sigma$`

appears at most once in any irreducible representation of `$G$`

. Given a triple, it is usually difficult to determine whether a given triple is a Gelfand triple. One has a sufficient condition which is geometric in nature to determine if a given triple is a Gelfand triple, called Gelfand criterion. We will discuss some examples of the Gelfand triple which give us multiplicity one theorem for non-degenerate Whittaker models of `${\mathrm GL}_n$`

over finite chain rings, such as `$\mathbb{Z}/p^n\mathbb{Z}$`

.
This is a joint work with Pooja Singla.

The theory of `$\delta$`

-geometry was developed by A. Buium based on the analogy with differential algebra where the analogue of a differential operator is played by a `$\pi$`

-derivation `$\delta$`

. A `$\pi$`

-derivation `$\delta$`

arises from the `$\pi$`

-typical Witt vectors and naturally associates with a lift of Frobenius `$\phi$`

. In this talk, we will discuss the theory of `$\delta$`

-geometry for Anderson modules. Anderson modules are higher dimensional generalizations of Drinfeld modules.

As an application of the above, we will construct a canonical `$z$`

-isocrystal `$\mathbb{H}(E)$`

with a Hodge- Pink structure associated to an Anderson module `$E$`

defined over a `$\pi$`

-adically complete ring `$R$`

with a fixed `$\pi$`

-derivation `$\delta$`

on it. Depending on a `$\delta$`

-modular parameter, we show that the `$z$`

-isocrystal `$\mathbb{H}(E)$`

is weakly admissible in the case of Drinfeld modules of rank `$2$`

. Hence, by the analogue of Fontaine’s mysterious functor in the positive characteristic case (as constructed by Hartl), one associates a Galois representation to such an `$\mathbb{H}(E)$`

. The relation of our construction with the usual Galois representation arising from the Tate module of `$E$`

is currently not clear. This is a joint work with Sudip Pandit.

In the 1980’s, Greene defined hypergeometric functions over finite fields using
Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric
series studied by Gauss, Kummer and others. These functions have played important roles in the study of Apery-style supercongruences, the Eichler-Selberg trace
formula, Galois representations, and zeta-functions of arithmetic varieties. In this
talk we discuss the distributions (over large finite fields) of natural families of these
functions. For the `$_2F_1$`

functions, the limiting distribution is semicircular, whereas
the distribution for the `$_3F_2$`

functions is Batman distribution.

The delta symbol is the key in solving many different problems in the analytic theory of numbers. In recent years this has been used to solve various sub-convexity problems for higher rank `$L$`

-functions. This talk will be a brief report on some new progresses. In particular, I will mention the results obtained in recent joint works with Roman Holowinsky & Zhi Qi and Sumit Kumar & Saurabh Singh.

We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed with respect to the Haar measure. However, for representations of `$PGL_2(\mathbb{F}_q)$`

, they are clustered around `$1$`

and `$-1$`

for odd `$p$`

and around `$1$`

for `$p=2$`

. This is joint work with Sameer Kulkarni.

Let `$K$`

be a finite extension of `$\mathbb{Q}_p$`

. The theory of `$(\varphi, \Gamma)$`

-modules constructed by Fontaine provides a good category to study `$p$`

-adic representations of the absolute Galois group `$Gal(\bar{K}/K)$`

. This theory arises from a ‘‘devissage’’ of the extension `$\bar{K}/K$`

through an intermediate extension `$K_{\infty}/K$`

which is the cyclotomic extension of `$K$`

. The notion of `$(\varphi, \tau)$`

-modules generalizes Fontaine’s constructions by using Kummer extensions other than the cyclotomic one. It encapsulates the important notion of Breuil-Kisin modules among others. It is thus desirable to establish properties of `$(\varphi, \tau)$`

-modules parallel to the cyclotomic case. In this talk, we explain the construction of a functor that associates to a family of `$p$`

-adic Galois representations a family of `$(\varphi, \tau)$`

-modules. The analogous functor in the `$(\varphi, \Gamma)$`

-modules case was constructed by Berger and Colmez . This is joint work with Leo Poyeton.

In this talk, we will discuss genericity of cuspidal representations of `$p$`

-adic unitary groups. Generic representations play a central role in the local Langlands correspondences and explicit knowledge of such representations will be useful in understanding the local Langlands correspondence in a more explicit way. After a brief review of `$p$`

-adic unitary groups, their unipotent subgroups, Whittaker functionals and genericity of cuspidal representations in this context, we will discuss the arithmetic nature of the problem.

Let `$F$`

be a non-archimedean local field of residue characteristic `$p$`

. The classical local Langlands correspondence is a 1-1 correspondence between 2-dimensional irreducible complex representations of the Weil group of `$F$`

and certain smooth irreducible complex representations of `$GL_2(F)$`

. The number-theoretic applications made it necessary to seek such correspondence of representations on vector spaces over a field of characteristic `$p$`

. In this talk, however, I will show that for `$F$`

of residue degree `$> 1$`

, unfortunately, there is no such 1-1 mod `$p$`

correspondence. This result is an elaboration of the arguments of Breuil and Paskunas to an arbitrary local field of residue degree `$> 1$`

.

Last updated: 20 Apr 2024