Abstracts

Johannes Anschütz: Introduction to prismatization

Prismatization is a powerful way of describing p-adic cohomology theories via quasi-coherent cohomology of associated stacks. This talk gives an introduction to the prismatization of p-adic formal schemes after Drinfeld/Bhatt-Lurie, and if time permits it will describe the hypothetical analytic prismatization of Q_p.

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Bijay Raj Bhatta: Zilber Pink Conjecture for simple PEL type Shimura Varieties

In this talk, we will discuss about the Zilber-Pink conjecture on unlikely intersections in moduli space A_g of principally polarised abelian varieties of dimension g. In particular, the case (Parameter Height Bound in Pila Zannier strategy) when the associated endomorphism algebra is of Type III or IV (Albert types). This extends the work of Daw and Orr who proved Type I and II cases assuming Large Galois Orbits(LGO), solving cases conditionally on LGO for all simple PEL type Shimura Varieties! If time permits, we will talk about the known cases for LGO due to Daw-Orr (multiplicative degeneration case).

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Ben Forrás: A Kida formula and an integrality statement for signed Selmer groups in ramified extensions

We consider an elliptic curve E over a number field with good reduction above a prime $p>3$ such that there is at least one $p$-adic place with good supersingular reduction, and we study signed Selmer groups associated with E. As opposed to existing works on the subject, we do not assume that supersingular places are unramified over $Q_p$, instead replacing this by a weaker assumption. By investigating projectivity properties of the Pontryagin dual of the signed Selmer group, we obtain a Kida formula as well as an integrality property of the corresponding characteristic element, the latter being conjecturally related to a $p$-adic $L$-function.

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Alexandros Groutides: Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_{2} \times \mathrm{GL}_{2}$

In recent years, local representation theory of $p$-adic groups and zeta integrals, have been linked to Euler system norm relations. In this talk, we will briefly touch upon this idea, initially introduced by Loeffler-Skinner-Zerbes, and discuss recent developments in the integral version of the theory. Using a representation-theoretic framework, we show that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for $\mathrm{GL}_{2} \times \mathrm{GL}_{2} $ satisfy the conjectured optimal integral behaviour; i.e. any construction of this type with any choice of integral input data would give local factors appearing in tame norm relations at $p$ which are integrally divisible by the Rankin-Selberg Euler factor $\mathcal{P}_p^{'}(\mathrm{Frob}_{p}^{-1})$ modulo $p-1$.

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Yukako Kezuka: Non-commutative Iwasawa theory of abelian varieties

Non-commutative Iwasawa theory has emerged as a powerful framework for understanding deep arithmetic properties over number fields contained in a p-adic Lie extension and their precise relationship to special values of complex L-functions. This talk aims to explore non-commutative Iwasawa theory over global function fields. We consider an abelian variety A defined over various base fields F, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we investigate the arithmetic of A over different p-adic Lie extensions without assuming the finiteness of the Selmer group of A over the base field F, as well as its relation to the order of vanishing of the L-function of A/F at s = 1.

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Daniel Kriz: Horizontal p-adic L-functions with applications to L-values and their derivatives

Given a positive integer d, a fundamental question in algebraic and analytic number theory is to determine how many order d character twists of a central L-value or derivative of a modular form are non-vanishing. For d = 2 this question is addressed by Goldfeld's conjecture, where substantial progress has been made in recent years via analytic number-theoretic and Iwasawa-theoretic techniques. For d > 2, a conjecture of David-Fearnley-Kisilevsky predicts 100% non-vanishing of order d twists, but little was previously known toward this conjecture. In this talk I will describe a novel approach to these questions using a new construction called horizontal p-adic L-functions. The non-vanishing of these functions is related to Kolyvagin's conjecture and similar questions in the theory of Euler systems. Using a structure theorem for horizontal Iwasawa algebras we give strong lower bounds on the non-vanishing of order d twists of L-values of newforms and their derivatives, as well as similar lower bounds on simultaneous non-vanishing. For 100% of elliptic curves we improve the previous best-known lower bounds in the d = 2 case due to Ono (r = 0) and Kumar-Mallesham-Sharma-Singh (r = 1), giving the best general results toward Goldfeld's conjecture and the first general results toward David-Fearnley-Kisilevsky's conjecture. This is joint work with Asbjørn Nordentoft.

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Georg Linden: Equivariant Vector Bundles on the Drinfeld Upper Half Space

We consider vector bundles on the Drinfeld upper half space over a non-archimedean local field $K$ which are the restriction of homogeneous vector bundles on projective space. Their global sections are so called non-archimedean holomorphic discrete series representations of $\mathrm{GL}_d(K)$ and can be studied in the framework of locally analytic representations. We describe the structure of these representations via a generalization of the functors $\mathcal{F}^G_P$ due to Orlik and Strauch, thus extending work of Orlik to $K$ of positive characteristic.

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Yucheng Liu: Continuum envelope on Fargues--Fontaine curves and elliptic curves

I will discuss some applications of the theory of Bridgeland stability condition, which is originated from string theoruy, on Fargues--Fontaine cueves. This is joint work with Heng Dui and Qingyuan JIang.

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Andreas Mihatsch: Arithmetic theta series

Starting from a positive definite quadratic lattice, one may consider the generating series with n-th coefficient equal to the number of lattice vectors of length n. This series defines a modular form called the theta series of the lattice. In the context of Shimura varieties, a similar mechanism provides generating series with coefficients in the (arithmetic) Chow group of the variety. A central conjecture states that these arithmetic theta series are modular as well. Moreover, they seem to be closely related to derivatives of L-functions and Eisenstein series, even though there is currently no general formulation of such a relation. In my talk, I will give an introduction to these ideas.

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Julian Quast: On local Galois deformation rings

In joint work with Vytautas Paškūnas, we show that the universal framed deformation ring of an arbitrary mod p representation of the absolute Galois group of a p-adic local field valued in a possibly disconnected reductive group G is flat, local complete intersection and of the expected dimension. In particular, any such mod p representation has a lift to characteristic 0. The work extends results of Böckle, Iyengar and Paškūnas in the case G=GL_n. We give an overview of the proof of this main result.

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Calle Sönne: Associating Galois Representations to mod p Hilbert Modular Forms using the Goren-Oort Stratification

We give an outline of a possible alternative proof of a result due to Diamond-Sasaki which associates Galois representations to mod p geometric Hilbert modular eigenforms (of arbitrary weight). This uses the existence and properties of the Goren-Oort stratification on Hilbert modular varieties.

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Reinier Sorgdrager: Pro-p Iwahori Hecke action and the mod p Langlands program

Let K be a p-adic field. Given a mod p representation of dimension 2 of the absolute Galois group of K, one can consider its isotypic component in the mod p cohomology of a suitably chosen Shimura variety to obtain a mod p representation of GL2(K). When K=Q_p this association is essentially the mod p Langlands correspondence for GL2(Q_p), by local-global compatibility. Beyond this case no such correspondence has been established and a first step in establishing it, could be to prove the resulting representation of GL2(K) is independent of the global choices made in its construction (such as the choice of the Shimura variety). This talk will be about a work in progress (at the time of writing the abstract) where we compute the action of the (classical) pro-p Iwahori Hecke algebra on the (derived) pro-p Iwahori invariants of the GL2(K)-representation. We work in the case where K is unramified over Q_p and under some genericity conditions on the Galois representation, so that we can apply the work of Breuil-Herzig-Hu-Morra-Schraen. It turns out that the Hecke action is indeed independent of global choices.

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Sören Sprehe: Antisymmetry in the theory of rigid meromorphic cocycles

In 2021 Darmon and Vonk initated the theory of p-adic singular moduli for real quadratic fields by defining 'rigid meromorphic cocycles'. These are elements of the first cohomology group of Ihara's group $SL_2(Z[1/p])$ with values in the group of rigid meromorphic functions on Drinfeld's upper half-plane. Using rigid meromorphic cocycles Darmon and Vonk assign to each pair of real quadratic irrationalities a p-adic number. The two irrationalities play a vastly different role in the construction of this assignement. However, it is expected to behave like the difference of two classical singular moduli - in particular, it should be antisymmetric in the argument. We will use the recent work of Darmon, Gehrmann and Lipnowski on rigid meromorphic cocycles for higher dimensional orthogonal groups to give a new, symmetric construction of this function. The main objects in the latter work are divisor valued cocycles which are defined by analogues of Heegner divisors on orthogonal Shimura varieties. This is work in progress.

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Benchao Su: On Breuil's locally $\mathrm{Ext}^1$ conjecture in the $\mathrm{GL}_2(L)$ case

Let L be a finite extension of Qp. In this talk, we will present a new method to compute Ext1 groups for certain locally analytic representations that appear in the completed cohomology of some unitary Shimura curves, and are related to some two dimensional de Rham representations of Gal(barL/L) with irreducible underlying Weil-Deligne representations. If time permits, we will discuss some work on Breuil's Ext1 conjecture in this case.

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James Taylor: Equivariant Vector Bundles with Connection on Drinfeld Symmetric Spaces

Recently Ardakov and Wadsley classified torsion equivariant line bundles with connection on the Drinfeld upper half-plane in terms of characters of O_D^x, the Galois group of the Drinfeld tower. In this talk we explain some recent work which extends this classification to vector bundles of any rank (describing these in terms of representations of O_D^x of any dimension) and to Drinfeld symmetric spaces of any dimension. We will then explain how these results can be used to deduce properties of the representations of GL_n(F) arising from the global sections of the Drinfeld tower, which are known to realise a part of the p-adic Langlands correspondence when F = Qp.

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Yingying Wang: Geometry and cohomology of compactified Deligne--Lusztig varieties

Deligne--Lusztig varieties are fundamental objects in the study of irreducible representations in characteristic 0 of finite reductive groups. I will discuss the cohomology of the structure and canonical sheaf for compactified Deligne--Lusztig varieties, as one of the first steps in understanding the situation in defining characteristic.

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Zhixiang Wu: Bernstein-Zelevinsky duality for locally analytic principal series representations

We calculate the Bernstein-Zelevinsky dual of locally analytic principal series representations of p-adic Lie groups, using the Kohlhaase-Schraen resolutions of these representations. This is joint work with Matthias Strauch.

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Deding Yang: Positivity of automorphic vector bundles on the special fiber of unitary Shimura varieties

Let (G,\mu) be a Shimura datum, and let X be the special fiber of the corresponding Shimura variety with hyperspecial level. We can construct automorphic vector bundles on X, which is a natural generalization of the modular line bundle over modular curves. The coherent cohomology of these bundles play an important role in many arithmetic problems. In this talk, we study the geometry of X when G is a split unitary group and prove some positivity results of these vector bundles.

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Riccardo Zuffetti: The Lefschetz decomposition of the Kudla-Millson theta function

In the 80's Kudla and Millson introduced a theta function in two variables, nowadays known as the Kudla--Millson theta function. This behaves as a modular form with respect to one variable, and as a closed differential form on an orthogonal Shimura variety with respect to the other variable. In this talk I show that the Lefschetz decomposition of (the cohomology class of) this theta function provides simultaneously the modular decomposition in Eisenstein and cuspidal parts. This is joint work with J. Bruinier.

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