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Monday, 09 September 2019
Time Speaker Title Resources
09:00 to 10:00 James Borger Introduction to Witt vectors, delta-rings, and prisms (Lecture 1)

We will first give a development of the theory of Witt vectors from the point of view of delta-rings, following Joyal. This is not the traditional approach, but it has the advantages of being more direct and not needing any mysterious formulas or arguments with p-adic congruences. In the later lectures, we will talk about the theory of prisms, as recently introduced by Bhatt-Scholze, which are certain kinds of delta-rings.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Kiran Kedlaya Perfectoid spaces (Lecture 1)

In this course, we will introduce perfectoid spaces, various examples, and the tilting equivalence. Additional topics will be treated as time permits.

11:45 to 12:45 Denis Benois Introduction to p-adic Hodge theory (Lecture 1)

In the first part of this course we review the classification of p-adic representations in terms of different rings of p-adic periods. In the second part, we define the curve of Fontaine-Fargues $X^{FF}$ and overview the theory of vector bundles on $X^{FF}$ with applications to p-adic representations.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Ehud De Shalit Moduli of p-divisible groups (Lecture 1)

p-divisible groups and their moduli spaces play a prominent role in p-adic Hodge theory and in the study of Shimura varieties. Recent work of Scholze, Weinstein, Fargues and Fontaine has changed our perspective on them and yielded new results as well as new proofs of old theorems. This mini-course will be an introduction to moduli of p-divisible groups, starting with basics of Dieudonne' theory and ending with a sketch of Faltings' theorem on the isomorphism between the Lubin-Tate tower and the Drinfel'd tower at the infinite level.

15:15 to 16:15 Utsav Choudhury Introduction to the category of Adic spaces (Lecture 1)

In these talks we will introduce the category of adic spaces. We will show that the category of adic spaces is a natural generalization of formal schemes and rigid analytic varieties. In fact we will construct fully fauthful functors from the category of  formal schemes ( resp. from the category of rigid analytic varieties) to the category of adic spaces.

16:15 to 16:30 -- Tea Break
Tuesday, 10 September 2019
Time Speaker Title Resources
09:00 to 10:00 James Borger Introduction to Witt vectors, delta-rings, and prisms (Lecture 2)

We will first give a development of the theory of Witt vectors from the point of view of delta-rings, following Joyal. This is not the traditional approach, but it has the advantages of being more direct and not needing any mysterious formulas or arguments with p-adic congruences. In the later lectures, we will talk about the theory of prisms, as recently introduced by Bhatt-Scholze, which are certain kinds of delta-rings.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Kiran Kedlaya Perfectoid spaces (Lecture 2)

In this course, we will introduce perfectoid spaces, various examples, and the tilting equivalence. Additional topics will be treated as time permits.

11:45 to 12:45 Denis Benois Introduction to p-adic Hodge theory (Lecture 2)

In the first part of this course we review the classification of p-adic representations in terms of different rings of p-adic periods. In the second part, we define the curve of Fontaine-Fargues $X^{FF}$ and overview the theory of vector bundles on $X^{FF}$ with applications to p-adic representations.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Ehud De Shalit Moduli of p-divisible groups (Lecture 2)

p-divisible groups and their moduli spaces play a prominent role in p-adic Hodge theory and in the study of Shimura varieties. Recent work of Scholze, Weinstein, Fargues and Fontaine has changed our perspective on them and yielded new results as well as new proofs of old theorems. This mini-course will be an introduction to moduli of p-divisible groups, starting with basics of Dieudonne' theory and ending with a sketch of Faltings' theorem on the isomorphism between the Lubin-Tate tower and the Drinfel'd tower at the infinite level.

15:15 to 16:15 Utsav Choudhury Introduction to the category of Adic spaces (Lecture 2)

In these talks we will introduce the category of adic spaces. We will show that the category of adic spaces is a natural generalization of formal schemes and rigid analytic varieties. In fact we will construct fully fauthful functors from the category of  formal schemes ( resp. from the category of rigid analytic varieties) to the category of adic spaces.

16:15 to 16:30 -- Tea Break
16:30 to 17:30 Aditya Karnataki p-adic automorphic forms in the sense of Scholze (Lecture 1)

In this course, we will introduce p-adic automorphic forms in the sense of Scholze. Completed cohomology, canonical and anti-canonical subgroups will be introduced. We will study how to consider anti-canonical subgroups in a tower. If time permits, we will study Galois representations using above.

Wednesday, 11 September 2019
Time Speaker Title Resources
09:00 to 10:00 James Borger Introduction to Witt vectors, delta-rings, and prisms (Lecture 3)

We will first give a development of the theory of Witt vectors from the point of view of delta-rings, following Joyal. This is not the traditional approach, but it has the advantages of being more direct and not needing any mysterious formulas or arguments with p-adic congruences. In the later lectures, we will talk about the theory of prisms, as recently introduced by Bhatt-Scholze, which are certain kinds of delta-rings.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Kiran Kedlaya Perfectoid spaces (Lecture 3)

In this course, we will introduce perfectoid spaces, various examples, and the tilting equivalence. Additional topics will be treated as time permits.

11:45 to 12:45 Denis Benois Introduction to p-adic Hodge theory (Lecture 3)

In the first part of this course we review the classification of p-adic representations in terms of different rings of p-adic periods. In the second part, we define the curve of Fontaine-Fargues $X^{FF}$ and overview the theory of vector bundles on $X^{FF}$ with applications to p-adic representations.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Ehud De Shalit Moduli of p-divisible groups (Lecture 3)

p-divisible groups and their moduli spaces play a prominent role in p-adic Hodge theory and in the study of Shimura varieties. Recent work of Scholze, Weinstein, Fargues and Fontaine has changed our perspective on them and yielded new results as well as new proofs of old theorems. This mini-course will be an introduction to moduli of p-divisible groups, starting with basics of Dieudonne' theory and ending with a sketch of Faltings' theorem on the isomorphism between the Lubin-Tate tower and the Drinfel'd tower at the infinite level.

15:15 to 16:15 Chitrabhanu Chaudhuri Introduction to the category of Adic spaces

In these talks we will introduce the category of adic spaces. We will show that the category of adic spaces is a natural generalization of formal schemes and rigid analytic varieties. In fact we will construct fully fauthful functors from the category of  formal schemes ( resp. from the category of rigid analytic varieties) to the category of adic spaces.

16:15 to 16:30 -- Tea Break
16:30 to 17:30 Debargha Banerjee p-adic automorphic forms in the sense of Scholze (Lecture 1)

In this course, we will introduce p-adic automorphic forms in the sense of Scholze. Completed cohomology, canonical and anti-canonical subgroups will be introduced. We will study how to consider anti-canonical subgroups in a tower. If time permits, we will study Galois representations using above.

Thursday, 12 September 2019
Time Speaker Title Resources
09:00 to 10:00 Ehud De Shalit Moduli of p-divisible groups (Lecture 4)

p-divisible groups and their moduli spaces play a prominent role in p-adic Hodge theory and in the study of Shimura varieties. Recent work of Scholze, Weinstein, Fargues and Fontaine has changed our perspective on them and yielded new results as well as new proofs of old theorems. This mini-course will be an introduction to moduli of p-divisible groups, starting with basics of Dieudonne' theory and ending with a sketch of Faltings' theorem on the isomorphism between the Lubin-Tate tower and the Drinfel'd tower at the infinite level.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Kiran Kedlaya Perfectoid spaces (Lecture 4)

In this course, we will introduce perfectoid spaces, various examples, and the tilting equivalence. Additional topics will be treated as time permits.

11:45 to 12:45 Denis Benois Introduction to p-adic Hodge theory (Lecture 4)

In the first part of this course we review the classification of p-adic representations in terms of different rings of p-adic periods. In the second part, we define the curve of Fontaine-Fargues $X^{FF}$ and overview the theory of vector bundles on $X^{FF}$ with applications to p-adic representations.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Arnab Saha Introduction to Witt vectors, delta-rings, and prisms (Lecture 1)

We will first give a development of the theory of Witt vectors from the point of view of delta-rings, following Joyal. This is not the traditional approach, but it has the advantages of being more direct and not needing any mysterious formulas or arguments with p-adic congruences. In the later lectures, we will talk about the theory of prisms, as recently introduced by Bhatt-Scholze, which are certain kinds of delta-rings.

15:15 to 16:15 Rajneesh Singh Local Shtukas and Divisible Local Anderson Modules

I will talk about the development of  the analogue of crystalline Dieudonne theory for p-divisible groups in the arithmetic of function fields. In our theory, p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes.

16:15 to 16:30 -- Tea Break
16:30 to 17:30 Debargha Banerjee p-adic automorphic forms in the sense of Scholze (Lecture 2)

In this course, we will introduce p-adic automorphic forms in the sense of Scholze. Completed cohomology, canonical and anti-canonical subgroups will be introduced. We will study how to consider anti-canonical subgroups in a tower. If time permits, we will study Galois representations using above.

Friday, 13 September 2019
Time Speaker Title Resources
09:00 to 10:00 Kiran Kedlaya Perfectoid spaces (Lecture 5)

In this course, we will introduce perfectoid spaces, various examples, and the tilting equivalence. Additional topics will be treated as time permits.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Denis Benois Introduction to p-adic Hodge theory (Lecture 5)

In the first part of this course we review the classification of p-adic representations in terms of different rings of p-adic periods. In the second part, we define the curve of Fontaine-Fargues $X^{FF}$ and overview the theory of vector bundles on $X^{FF}$ with applications to p-adic representations.

11:45 to 12:45 Ehud De Shalit Moduli of p-divisible groups (Lecture 5)

p-divisible groups and their moduli spaces play a prominent role in p-adic Hodge theory and in the study of Shimura varieties. Recent work of Scholze, Weinstein, Fargues and Fontaine has changed our perspective on them and yielded new results as well as new proofs of old theorems. This mini-course will be an introduction to moduli of p-divisible groups, starting with basics of Dieudonne' theory and ending with a sketch of Faltings' theorem on the isomorphism between the Lubin-Tate tower and the Drinfel'd tower at the infinite level.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Arnab Saha Introduction to Witt vectors, delta-rings, and prisms (Lecture 5)

We will first give a development of the theory of Witt vectors from the point of view of delta-rings, following Joyal. This is not the traditional approach, but it has the advantages of being more direct and not needing any mysterious formulas or arguments with p-adic congruences. In the later lectures, we will talk about the theory of prisms, as recently introduced by Bhatt-Scholze, which are certain kinds of delta-rings.

15:15 to 16:15 Aditya Karnataki p-adic automorphic forms in the sense of Scholze (Lecture 2)

In this course, we will introduce p-adic automorphic forms in the sense of Scholze. Completed cohomology, canonical and anti-canonical subgroups will be introduced. We will study how to consider anti-canonical subgroups in a tower. If time permits, we will study Galois representations using above.

16:15 to 16:30 -- Tea Break
Monday, 16 September 2019
Time Speaker Title Resources
09:00 to 10:00 Kiran Kedlaya Coherent (phi, Gamma)-modules and cohomology of local systems

One of the motivating points for the theory of perfectoid spaces is the geometric form of Fontaine's theory of (phi, Gamma)-modules, in which one interprets etale Q_p-local systems on rigid analytic spaces in termrs of locally free sheaves for the pro-etale topology over a certain base ring with a Frobenius action (the extended Robba ring). We describe an enlargement of this category to an abelian category of coherent (phi, Gamma)-modules which can be used to control the cohomology of Q_p-local systems. Joint work with Ruochuan Liu.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Shalini Bhattacharya Reduction of crystalline representations and local constancy in the weight space
11:45 to 12:45 Denis Benois On extra zeros of Rankin-Selberg L-functions

In this talk, we discuss a Mazur-Tate-Teitelbaum formula for the Rankin-Selberg product of two modular forms  of the same weight K>=2 (joint work with S. Horte).

13:00 to 14:00 -- Lunch
14:00 to 15:00 Mahesh Kakde Explicit formulae for Gross-Stark units and Hilbert’s 12th problem

In this talk I will report on my joint work in progress with Samit Dasgupta on the tower of fields conjecture first formulated by Gross. This proves a conjecture of Dasgupta on explicit p-adic analytic formulae for Gross-Stark units. These units, when considered for all primes of a totally real number field F, generate the maximal abelian CM extension of F and therefore our work can be considered as giving a p-adic analytic solution to Hilbert’s 12th problem. Further, the tower of fields conjecture also proves a conjecture of Dasgupta and Spiess which gives a p-adic analytic formulae, in terms of Eisenstein cocycles, for the characteristic polynomial of the Gross regulator matrix.

15:00 to 15:30 -- Tea Break
15:30 to 16:30 Alberto Vezzani Cohomologies for rigid analytic varieties via motivic homotopy theory

We introduce Ayoub's categories of derived mixed motives in the context of rigid geometry and outline their most important features. As an application, we give streamlined methods to define and study classical p-adic cohomology theories, via a new realization functor associating to a rigid variety over a perfectoid field in positive characteristic a vector bundle on the associated Fargues-Fontaine curve (developed with A.-C. Le Bras).

Tuesday, 17 September 2019
Time Speaker Title Resources
09:00 to 10:00 James Borger Canonical lifts in families

I'll present recent work with Lance Gurney extending the theory of canonical lifts of abelian varieties to arbitrary families. Every family of ordinary abelian varieties over a base S on which p is nilpotent lifts to a unique family over W(S) admitting a delta-structure, in the sense of Joyal, Buium, and Bousfield.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Ehud De Shalit Foliations on Unitary Shimura Varieties in Characteristic p
11:45 to 12:45 Eknath Ghate Zig-zag and the theta operator

For some years now, my collaborators and I have been computing the mod p reductions of two-dimensional local p-adic modular Galois representations at good primes p, using the p-adic and mod p Local Langlands Correspondences. A complete description of the reduction is now available for all weights and primes p at least 5, for slopes at most 2.

Two recent general patterns have emerged from these computations: (1) a zig-zag conjecture that describes the reductions for small half-integral slopes in the tricky case of exceptional weights, and, (2) a pattern showing that some of the reductions in slope v + 1 are related to the reductions in slope v by twisting.

In this talk, we recall some evidence towards (1). We also describe a recent breakthrough proving pattern (2) under an assumption on the weight. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms (joint work with Arvind Kumar). Finally, we show that the seemingly unrelated patterns (1) and (2) are compatible.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Elmar Grosse-Klönne Hecke modules and Galois representations into the dual group

Let $F/{\mathbb Q}_p$ be a finite extension. Let $G$ be a simply connected semisimple split algebraic group over ${\mathcal O}_F$, let $G^{\vee}$ be its dual. Let $H$ be the pro-$p$ Iwahori Hecke algebra of $G(F)$, with coefficients in a finite extension $k$ of the residue field of ${\mathcal O}_F$. Motivated by the search for mod-$p$ local Langlands correspondences we suggest to assign to irreducible supersingular $H$-modules certain homomorphisms of the absolute Galois group of $F$ into $G^{\vee}(k)$. We then ask if such an assignment can be upgraded into an exact functor between suitable abelian categories.

15:00 to 15:30 -- Tea Break
15:30 to 16:30 Chandrakant Aribam Galois cohomology for Lubin-Tate $(\varphi,\Gamma_{LT})$ modules

We extend the work of Herr by introducing a complex which allows us to compute cohomology groups over Lubin-Tate extensions. We also compare it with Galois cohomology groups.

16:30 to 17:00 Shaunak Deo Hilbert modular eigenvariety at exotic and CM classical points of parallel weight one

We sketch our recent results about the geometry of the p-adic eigenvariety constructed by Andreatta-Iovita-Pilloni, which interpolates Hilbert modular eigenforms over a totally real field F, at classical, regular points of parallel weight one which either are CM or have exotic projective image. To prove these results, we assume the p-adic Schanuel conjecture in most of the cases. The key ingredient in our proof is the calculation of the dimension of the tangent spaces of some Galois deformation problems. This talk is based on joint work with A. Betina and F. Fite.

Wednesday, 18 September 2019
Time Speaker Title Resources
09:00 to 10:00 Somnath Jha p r -Selmer companion modular forms
10:00 to 10:30 -- Tea Break
10:30 to 11:30 Narasimha Kumar Cheraku On sign changes of the Fourier coefficient of modular forms over number fields

In this talk, we shall discuss some results concerning the simultaneous non-vanishing, sign changes  of Fourier coefficients of (two) Hilbert  cusp forms. This is in a joint work with Tarun Dalal. We shall also discuss a variant of multiplicity one theorems for half-integral weight cuspidal eigenforms in terms of signs of the Fourier coefficients.

11:45 to 12:45 A. Raghuram Arithmetic of L-functions for orthogonal groups

This talk will be a report on my joint work with Chandrasheel Bhagwat on the special values of L-functions for orthogonal groups. For an even positive integer n, we prove rationality results for the ratios of critical values of L-functions for SO(n,n) x GL(1). The proofs involve (1) Langlands-Shahidi method with the ambient reductive group being O(n+1,n+1) and the maximal parabolic subgroup having Levi quotient O(n,n) x GL(1), and (2) Eisenstein cohomology for even orthogonal groups.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Mladen Dimitrov Geometry of the eigencurve at CM points and trivial zeros of Katz p-adic L-functions

This talk is based on a joint work with Adel Betina studying the geometry of the p-adic eigencurve at a weight one theta series f irregular at p. We show that f belongs to exactly four Hida families and study their mutual congruences. In particular, we show that the congruence ideal of one of the CM families has a simple zero at f if, and only if, a certain L-invariant $L(\varphi)$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst $L(\varphi)$ and $L(\varphi^{−1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz p-adic L-function of $\varphi$ has a simple (trivial) zero at s=0, if $L(\varphi)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz p-adic L-function of φ at s=0 thus extending a conjecture of Gross.

15:00 to 15:30 -- Tea Break
15:30 to 16:30 Fabrizio Andreatta Classicality of overconvergent modular symbols
16:30 to 17:00 Aranya Lahiri Rigid Analytic Vector in Locally Analytic Representations: Exactness and Applications

Rigid analytic vectors of a locally analytic representation has played a significant part in Emerton's point of view of Locally Analytic Representations. In this talk we will discuss the exactness properties of the functor "taking rigid analytic vectors of an wide open subgroup" for an admissible locally analytic representation V. If time permits we will compute some examples of rigid analytic vectors and indicate some applications of the result.

Thursday, 19 September 2019
Time Speaker Title Resources
09:00 to 10:00 Sujatha Ramdorai Iwasawa theory of the fine Selmer groups of Galois representations

In this talk, we give an overview of some recent results of the structure and Galois cohomology of the fine Selmer groups and Selmer groups of  the Galois representations attached to elliptic curves and modular forms.These objects are considered as modules over associated Iwasawa algebras and hence the behaviour of the representation at a prime p intervenes.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Jacques Tilouine cohomology of GL(N), adjoint Selmer groups and simplicial deformation rings

In a joint work with E. Urban, we relate the cohomology of GL(N) to the adjoint Selmer group of a Galois representation (constructed by Scholze using perfectoids) by using Galatius-Venkatesh simplicial deformation theory.

11:45 to 12:45 Marie France Vigneras Smooth representations of reductive p-adic groups over arbitrary fields

General results on scalar extension allow to show that  classification theorems  in  the  theory of admissible  irreducible  representations of a reductive $p$-adic group proved over an algebraically closed field remain valid over an arbitrary field of the same characteristic.

13:00 to 14:00 -- Lunch
14:00 to 15:00 Sudhanshu Shekhar Non-commutative Twisted Euler characteristic in Iwasawa theory
15:00 to 15:30 -- Tea Break
15:30 to 16:30 Kestutis Cesnavicius Purity for flat cohomology

The absolute cohomological purity for étale cohomology of Gabber--Thomasson implies that an étale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

16:30 to 17:00 Tathagata Mandal On quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

We study the variance of local epsilon factor for a modular form with arbitrary nebentypus with respect to twisting by a certain quadratic character. As an application, we will determine the type of the supercuspidal representation from that if the attached representation at a prime p is supercuspidal. For modular forms with trivial nebentypus, similar results are proved by Pacetti. In the ramified principal series (with p || N and p odd) and unramified supercuspidal representation of level zero case, we relate the variance with Morita's p-adic Gamma function. This is joint work with Debargha Banerjee.

Friday, 20 September 2019
Time Speaker Title Resources
09:00 to 10:00 Arnab Saha Isocrystals associated to arithmetic jet spaces

We will talk about the construction of a canonical isocrystal associated to an abelian scheme, arising from the theory of arithmetic jet spaces. This isocrystal admits a 'Hodge-type' filtration and maps to the crystalline cohomology preserving the filtration.  However the semilinear structure of this object is distinct from that of the crystalline cohomology and the exact connection between these two objects is not yet clear.  In the case of elliptic curves, depending on a modular parameter, this isocrystal is also weakly admissible which leads to a new crystalline Galois representation attached to the elliptic curve via the Fontaine functor. This is joint work with J. Borger.

10:00 to 10:30 -- Tea Break
10:30 to 11:30 Guhan Venkat Stark-Heegner cycles for Bianchi modular forms

In his seminal paper in 2001, Henri Darmon came up with a systematic construction of p-adic points, viz. Stark-Heegner points, on elliptic curves over the rationals. In this talk, I will report on the construction of local (p-adic) cohomology classes in the Harris-Taylor-Soudry representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger-Seveso. These local cohomology classes are conjecturally the restriction of global cohomology classes in an appropriate Bloch-Kato Selmer group and have consequences towards the Bloch-Kato-Beilinson conjecture as well as Gross-Zagier type results. This is based on a joint work with Chris Williams (Imperial College London).

11:45 to 12:45 Ildar Gaisin Fargues' conjecture in the GL_2-case

Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.