Arithmetic aspects of automorphic forms
Org:
Antonio Lei (University of Ottawa) and
Giovanni Rosso (Concordia University)
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 ANTONIO CAUCHI, Concordia University
Towards new Euler systems for automorphic Galois representations [PDF]

The construction of Euler systems for Galois representations associated to automorphic forms often relies on the existence of RankinSelberg integrals which calculate the corresponding Lfunction.
I will discuss a new RankinSelberg integral, which represents a twist of the degree 5 $L$function of cusp forms on $\mathrm{GSp}_4$, and its application to the study of the arithmetic of the standard Galois representation associated to cusp forms on $\mathrm{GSp}_4$. This is joint work with Armando Gutierrez.
 HENRI DARMON, McGill
Generalised Hecke eigenvectors [PDF]

If $M$ is a module over a Hecke algebra and $v\in M$ is a simulatenous eigenvector for the Hecke operators, a generalised eigenvector attached to $v$ is a an element $v'\in M$
satisfying $$T_\ell v' = a_\ell v' + a_\ell' v.$$
The scalars $a_\ell'$ often carry rich arithmetic information. When $M$ is the space of forms of weight one, they are logarithms of algebraic numbers and are a key to explicit class field theory for real quadratic fields. I will discuss the case where $M$ is a space of modular forms of weight two with fourier coefficients in $\mathbb Z/p\mathbb Z$, where these quantites appear to be related to classes in $K$theory considered by Beilinson and Flach.
This is an account of work in progress with Alice Pozzi.
 CÉDRIC DION, Université Laval
Refined conjectures on Fitting ideals of Selmer groups [PDF]

Fix an odd prime number $p$ and let $K_n$ be the nth layer of the cyclotomic $\mathbb{Z}_p$extension of $\mathbb{Q}$. Kim and Kurihara showed that the Pontryagin dual of the Selmer group over $K_n$ attached to an elliptic curve $E$ with good ordinary reduction at p is generated by the MazurTate element at level $K_n$. When the reduction is supersingular at $p$, they show that the same result holds, up to an explicit error term. In this talk, we discuss generalization of these results for the Selmer group of E over the $\mathbb{Z}_p^2$extension of an imaginary quadratic field where p splits.
 RYLAN GAJEKLEONARD, Union College
Iwasawa invariants of nonordinary modular forms [PDF]

This talk will outline a method for computing the (analytic) Iwasawa invariants of cuspidal newforms with $a_p=0$ in terms of the associated sequence of MazurTate elements. The connection between MazurTate elements and $p$adic Lfunctions is wellknown for weight two modular forms, but the relation is less clear at higher weights. In the $a_p=0$ case, we use Pollack's decomposition of the padic Lfunction to construct explicit lifts of MazurTate elements to the full Iwasawa algebra. By studying the behavior of these lifts upon projection to layer n, we relate the Iwasawa invariants of MazurTate elements to those of the corresponding $p$adic Lfunctions. Corollaries include a relation between the Iwasawa invariants attached to certain pcongruent pairs of modular forms and a description of the padic valuation of critical Lvalues for modular forms with $a_p = 0$.
 EYAL GOREN, McGill University
Foliations on Shimura varieties in positive characteristic [PDF]

In joint work with E. De Shalit (Hebrew U) we developed a theory of foliations on Shimura varieties in positive characteristic, offering as "case studies" the examples of Hilbert modular varieties and unitary Shimura varieties. Those reveal intimate connections with modular forms mod p, Shimura varieties with parahoric level structure, inseparable morphisms and deformation theory. I will provide an overview, using two particular examples: HilbertBlumenthal surfaces and Picard modular surfaces.
 JEFFREY HATLEY, Union College
Vanishing anticyclotomic $\mu$invariants for nonordinary modular forms [PDF]

Let $E_{/\mathbf{Q}}$ be an elliptic curve and $p$ a prime such that $E[p]$ is irreducible as a $G_{\mathbf{Q}}$module. A fundamental conjecture (due to Greenberg and PerrinRiou) states that the Iwasawa $\mu$invariant(s) associated to $E$ over the cyclotomic $\mathbf{Z}_p$extension of $\mathbf{Q}$ must vanish. Despite many focused efforts, this conjecture is still wide open. One may extend this conjecture to more general modular forms and more general $\mathbf{Z}_p$extensions of number fields. In this talk, we discuss work (joint with Antonio Lei) which establishes some cases of this conjecture over anticyclotomic $\mathbf{Z}_p$extensions of imaginary quadratic fields.
 DEBANJANA KUNDU, Fields Institute
$\lambda$invariant stability in Families of Modular Galois Representations [PDF]

Consider a family of modular forms, all of whose residual $\pmod{p}$ Galois representations are isomorphic. It is wellknown that their corresponding Iwasawa $\lambda$invariants may vary. We will discuss this variation from a quantitative perspective, providing lower bounds on the frequency with which these $\lambda$invariants grow or remain stable. This is joint work with Jeff Hatley.
 JEF LAGA, Princeton University
Rational torsion on abelian surfaces with quaternionic multiplication [PDF]

Mazur classified all possible rational torsion subgroups of elliptic curves over $\mathbb{Q}$. In joint work with Ciaran Schembri, Ari Shnidman and John Voight, we put strong constraints on the torsion subgroup of a class of abelian surfaces whose geometric endomorphism algebra is large, namely an indefinite quaternion algebra. The proof uses quaternion arithmetic, Neron models, and the modularity of abelian surfaces of $\text{GL}_2$type.
 HEEJONG LEE, University of Toronto
EmertonGee stacks for $\mathrm{GSp}_4$ and Serre weight conjectures [PDF]

In the Langlands program, we want to construct a certain correspondence between automorphic representations and Galois representations. The meaning of this correspondence can be explained in terms of the $L$functions. However, one can also ask how the structure of one side is reflected on the other side. Serre weight conjectures explicitly explain that how (Serre) weights of the automorphic side and the ramification behavior on the Galois side are related.
During the first half of my talk, I will discuss the Serre weight conjectures and their relation to local Galois representations. This will motivate us to understand certain Galois deformation rings. Then I will discuss EmertonGee stacks (which allows a more geometric approach to Galois representations) and local models of LeLe HungLevinMorra (which can describe parts of EmertonGee stacks explicitly), as well as their generalizations to the group $\mathrm{GSp}_4$.
 ADAM LOGAN, Government of Canada
A conjectural uniform construction of many rigid CalabiYau threefolds [PDF]

Given a rational Hecke eigenform $f$ of weight $2$, EichlerShimura theory gives a construction of an elliptic curve over ${\mathbb Q}$ whose associated modular form is $f$. Mazur, van Straten, and others have asked whether there is an analogous construction for Hecke eigenforms $f$ of weight $k>2$ that produces a variety for which the Galois representation on its etale ${\mathrm H}^{k1}$ (modulo classes of cycles if $k$ is odd) is that of $f$. In weight $3$ this is understood by work of Elkies and Schütt, but in higher weight it remains mysterious, despite many examples in weight $4$. In this talk I will present a new construction based on families of K3 surfaces of Picard number $19$ that recovers many existing examples in weight $4$ and produces almost $20$ new ones.
 MOHAMMADREZA MOHAJER, University of Ottawa
Linear relations of padic periods of 1motives [PDF]

1periods are complex numbers arising from degree 1 Bettide Rham comparison isomorphism or from Deligne 1motives. Due to Wüstholz, Huber and other research works, KontsevichZagier period conjecture is known for these periods. In our research, we are aiming to draw padic analogies with the wellestablished results that are known for these periods. Specifically, in this talk, we will explore the padic periods of curve type. Our main goal is to study the transcendence and linear relations of these padic numbers. We will begin by introducing the formalism of padic periods where it provides us a tool to state the padic period conjecture and the padic version of subgroup theorem. We will then move on to the padic periods of 1motives with good reduction which arise from crystallinede Rham realisations and we will compare them with those padic periods coming from padic integration theory.
 KATHARINA MUELLER, Universite Laval
On the Iwasawa invariants of BDPSelmer groups and BDP padic Lfunctions [PDF]

Let $f_1$ and $f_2$ be weight two Hecke cusp forms with isomorphic residual Galois representations. Let $K$ be an imaginary quadratic field and assume that the generalized Heegner hypothesis holds for the pairs $(f_1,K)$ and $(f_2,K)$. Let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$extension. We will analyze the relation between the Iwasawa invariants of the BDPSelmer groups over $K_\infty$ and the BDP $p$adic $L$fucntions for $f_1$ and $f_2$.
This is joint work with Antonio Lei and Jiacheng Xia.
 SIDDARTH SANKARAN, University of Manitoba
Arithmetic SiegelWeil formiulas for zero dimensional varieties. [PDF]

The arithmetic SiegelWeil formula is a conjectural identity, due to Kudla, that predicts relations between certain arithmetic 'special' cycles on Shimura varieties and derivatives of Eisenstein series. In the zerodimensional case, notable examples include work of KudlaRapoportYang, Howard, and AndreattaGorenHowardMadapusi; these results are ultimately proved by an explicit computation.
This talk is part of an ongoing effort to understand these results from a more conceptual point of view; after reviewing the formula in a simple case, we will place these results in a more general context, and explain how the arithmetic SiegelWeil formula can be seen as an application of the usual SiegelWeil formula. The main novelty is the introduction of a padic "Green function" in the zerodimensional setting, mirroring a construction of Kudla at the archimedean place.
 JIACHENG XIA, Université Laval
The convergence problem in Kudla's modularity conjectures [PDF]

Kudla's modularity conjectures generalize the classical GrossKohnenZagier theorem for modular curves and Heegner points to Shimura varieties for orthogonal and unitary groups and special cycles of arbitrary codimensions. These conjectures were confirmed by YuanZhangZhang and Liu assuming convergence of the generating series of special cycles over totally real fields and CM fields. To fill this gap of convergence, in a joint work in progress with Qiao He, we prove an algebraicity result of these generating series which is a crucial step in our approach towards the convergence problem.
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