Time: Tuesdays 2 pm - 3 pm
Location: ENR2 S395
Organizers: Serin Hong and Sandra Nair
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Time: Tuesdays 2 pm - 3 pm Location: ENR2 S395 Organizers: Serin Hong and Sandra Nair |
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September 8 |
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September 15 |
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September 22 |
Ziqi Guo (Peking University)
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September 29 |
JiWoong Park (UT Austin)
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October 6 |
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October 13 |
Nikolas Castro (UCSD)
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October 20 |
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October 27 |
Jacob Swenberg (UCLA)
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November 3 |
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November 10 |
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November 17 |
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November 24 |
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December 1 |
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December 8 |
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September 10 |
Robert Pollack (University of Arizona)
The distribution of invariants of modular forms has been studied in many contexts. The Sato-Tate conjecture makes a precise prediction on the distribution of normalized Hecke-eigenvalues for modular forms. Here one fixes a form and varies the eigenvalue. One could also fix the eigenvalue and vary the form and still this invariant has a beautifully predictable distribution. |
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October 15 |
Linli Shi (University of Connecticut)
The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group \(\mathrm{GU}(2,1)\). |
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October 22 |
Serin Hong, (University of Arizona)
The notion of Newton stratification originates from Grothendieck's work on the moduli space of abelian varieties. Since its inception, the notion has significantly evolved to find many surprising applications in arithmetic geometry. In this talk, we provide a friendly overview of this notion and discuss some recent developments, with a particular focus on the question of determining all nonempty strata. |
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October 29 |
Anna Medvedovsky (University of Arizona)
We tell the story of mod-p modular forms, focusing on the example of \(p\) = 2 and \(p\) = 3 and level 1: the Hecke algebra acting on the space of forms, the Galois representation carried thereby, and explicit matrix realization thereof. An application is studying the *density* of a mod-\(p\) modular form, which captures the distribution of its prime Fourier coefficients. The \(p\) = 2 case relies on published and unpublished work of Bellaïche, Nicolas, and Serre. |
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November 5 |
Ben Savoie (Rice University)
The Emerton-Gee stack for \(\mathrm{GL}_2\) serves as a moduli space for 2-dimensional representations of the absolute Galois group of \(K\), where \(K\) is a finite, unramified extension of \(\mathbb{Q}_p\). This stack is of significant interest because it is expected to play the role of the stack of L-parameters in the conjectural categorical \(p\)-adic Langlands correspondence for \(\mathrm{GL}_2(K)\). In this talk, I will present recent joint work with Kalyani Kansal, where we determine which of the irreducible components of the Emerton-Gee stack are smooth. Among the non-smooth components, we also identify those which are normal or Cohen-Macaulay. This allows us to show that the normalization of every component has fairly mild (resolution-rational) singularities. The talk will begin with a review of Galois representations and modular forms, followed by a discussion of key ideas in the construction of the Emerton-Gee stack. Finally, I will describe how our results update expectations about the categorical \(p\)-adic Langlands conjecture. |
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November 13, 2PM |
Florian Sprung (Arizona State University)
For a modular form* \(f\) of weight two, one can attach a \(p\)-adic L-function, which is `good' if \(p\) is an ordinary prime, i.e. the \(p\)-th Fourier \(a_p\) of \(f\) is a \(p\)-adic unit. `Good' means `Iwasawa function,' or in even simpler terms `coefficients are bounded.' When \(p\) is non-ordinary (i.e. \(a_p\) is not a p-adic unit), the \(p\)-adic L-functions are `bad' -- they have unbounded growth behavior on their coefficients (and note the plural -- there are now two of them). However, one can factor out the badness. This was done in R. Pollack's PhD thesis when \(a_p=0\), which gave rise to two functions \(\log^+\) and \(\log^-\), the `signed logarithms'. The speaker handled the general non-ordinary case via a \(2 \times 2\) matrix \(\mathrm{Log}(a_p)\). This matrix is, when \(a_p=0\), essentially a diagonal matrix in which \(\log^+\) and \(\log^-\) appear. But where does \(\mathrm{Log}(a_p)\) come from? We give a simple description in terms of \(p\)-adic digits. |
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November 19 |
Shaunak Deo (Indian Institute of Science)
Let \(p\) and \(\ell\) be primes such that \(p > 3\) and \(p \mid \ell-1\) and \(k\) be an even integer. We will give a necessary and sufficient condition for the \(\mathbb{Z}_p\)-rank of the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight \(k\) and level \(\Gamma_0(\ell)\) at the maximal Eisenstein ideal containing \(p\) to be greater than \(1\) in terms of vanishing of the cup products of certain global Galois cohomology classes. We will begin with a brief review of modular forms. |
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November 26 |
Chengyang Bao (UCLA)
Crystalline deformation rings play an important role in Kisin's proof of the Fontaine-Mazur conjecture for \(\mathrm{GL}_2\) in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally. |
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December 10 |
Kimball Martin (University of Oklahoma)
Root numbers are signs that determine symmetries of L-functions. While asymptotically root numbers are +1 half the time and -1 half the time, there is in fact a bias towards sign +1. Moreover, an unexpected correlation between root numbers and Fourier coefficients of modular forms, termed murmurations, was recently discovered. I will explain these phenomena, as well as analogues for local root numbers. |
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January 21 |
Deewang Bhamidipat (UC Santa Cruz)
Unitary Shimura varieties are moduli spaces of abelian varieties with a certain extra structure, including a signature condition. An effective way to understand these spaces in positive characteristic is by stratifying them, of which two are of interest: the Ekedah-Oort (EO) stratification, defined using the \(p\)-torsion group scheme structure up to isomorphism, and the Newton stratification, defined using the \(p\)-divisible group structure up to isogeny. We will see in several concrete examples that these two stratifications are very different, reflecting that these two invariants capture very different attributes of the abelian varieties. In joint work with E. Anne, M. Fox, H. Goodson, S. Groen, and S. Nair, we take a specific stratum in the Newton stratification - the supersingular stratum - and study its intersection with the EO stratification in some low signature cases. |
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January 28 |
Kirti Joshi (University of Arizona)
The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields. |
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February 4 |
Kirti Joshi (University of Arizona)
The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields. |
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February 11 |
Kirti Joshi (University of Arizona)
The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields. |
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February 18 |
Jinyue Luo (University of Chicago)
In application to modularity, one often seeks for an R=T theorem, that is, the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudo representations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample. |
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March 4 |
Jaclyn Lang (Temple University)
In his celebrated Eisenstein ideal paper, Mazur studied congruences modulo a prime \(p\) between Eisenstein series and cusp forms in prime level \(N\). If \(p\) is at least \(5\), he showed that such congruences exist if and only if \(N\) is congruent to \(1\) modulo \(p\). I will discuss recent work with Preston Wake in which we investigate Eisenstein-cuspidal congruences when the level is \(N^2\), where \(N\) is a prime congruent to \(-1\) modulo \(p\). We show that such congruences exist in this case, and that they are remarkably uniform compared with Mazur’s setting. Moreover, one can use a mild extension of Ribet’s method to produce from our congruences nontrivial elements in the class group of \(\mathbb{Q}(N^{1/p})\). |
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March 25 |
Gaurish Korpal (University of Arizona)
Chevyrev and Galbraith (2013) and Goren and Love (2023) show that the successive minima of the Gross lattice of a supersingular elliptic curve can be used to characterize the endomorphism ring of that curve. We show that the third successive minimum \(D_3\) of the Gross lattice gives necessary and sufficient conditions for the curve to be defined over the field \(\mathbb{F}_p\) or over the field \(\mathbb{F}_{p^2}\). In the case where the curve \(E\) is defined over \(\mathbb{F}_p\), the value of \(D_3\) can even yield finer information about the endomorphism ring of \(E\). This talk is based on my joint work with Chenfeng He, Ha Tran, and Christelle Vincent. |
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April 1 |
Pan Yan (University of Arizona)
In the usual paradigm of the Rankin-Selberg method, the Eulerian factorization of a global integral relies on the uniqueness of a model such as the Whittaker model, or the uniqueness of an invariant bilinear form between an irreducible representation and its contragredient. Examples of Rankin-Selberg integrals which unfold to non-unique models are very rare because standard tools for local unramified computation such as the Casselman-Shalika formula are not applicable. In this talk we derive new global integrals for \(\mathrm{Sp}(2n) \times \mathrm{GL}(k)\) where \(n\) is even, from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and extending a previous result of Ginzburg and Soudry on the case \(n=k=2\). We show that these new global integrals unfold to non-unique models on \(\mathrm{Sp}(2n)\). Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the L-functions for \(\mathrm{Sp}(2n) \times \mathrm{GL}(k)\), generalizing a previous work of Piatetski-Shapiro and Rallis on \(\mathrm{Sp}(2n) \times \mathrm{GL}(1)\). This is joint work with Yubo Jin. |
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April 8 |
Katharine Woo (Princeton University)
We resolve Manin's conjecture for all Châtelet surfaces over \(\mathbb{Q}\) (surfaces given by equations of the form \(x^2 + ay^2 = f(z)\)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values. |
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April 22 |
Serin Hong (University of Arizona)
In order to study automorphic forms on general groups, one must work with not only scalar-valued forms but also vector-valued forms. One of major challenges for studying vector-valued forms is lack of explicit examples to work with. In this talk, we describe a method for constructing vector-valued automorphic forms on unitary groups from scalar-valued ones, inspired by the work of Clery and van der Geer for Siegel modular forms. This is joint work with T. Browning, P. Coupek, E. Eischen, C. Freschette, S. Y. Lee, and D. Marcil, which began as a project from the 2022 Arizona Winter School. |
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April 22 |
Chapman Howard (University of Arizona)
The philosophy of using L-functions (analytic invariants) to study algebraic objects has produced many important results in the last century. We’ll discuss how this philosophy can be refined to a more interesting invariant (the epsilon factor or local root number), which in our case completely determines the representations we will study. With the importance of the local root number well understood, we’ll move toward the conjecture, explain previously proven cases, what work we have done so far, and how one might prove our case. |
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September 9 |
Doug Ulmer (University of Arizona)
We take a section \(P\) of infinite order on an elliptic surface and consider points where some multiple \(nP\) is tangent to the zero section. (These are "unlikely intersections" and our consideration of them is motivated by a question in geography of surfaces.) In characteristic zero, Urzua and I show finiteness and give a sharp upper bound, relying heavily on a canonical parallel transport in a family of elliptic curves (the "Betti foliation") and a certain real-analytic one-form. Although the finiteness statement looks completely reasonable in characteristic \(p\), it's not clear what would replace the (non-algebraic) 1-form. More recently with Felipe Voloch, we connect tangencies to \(p\)-descent maps and bound them in characteristic \(p\). We also find a new family of unlikely intersections in characteristic zero related to a famous homomorphism of Manin, and we correct inaccuracies in the literature about this homomorphism. |
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September 30 |
Hang Xue (University of Arizona)
This is an introductory talk which aims to explain what the Gan--Gross--Prasad (GGP) conjecture is, where it is coming from, and what we can say about it. |
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October 7 |
Rob Pollack (University of Arizona)
The slope of an eigenform is the \(p\)-adic valuation of its \(\mathrm{U}_p\)-eigenvalue. If we fix a mod \(p\) Galois representation rhobar, a standard construction is to consider all modular forms whose residual representation is rhobar. One could, for instance, ask which slopes occur for such eigenforms. In this talk, we will instead vary rhobar and study how the corresponding slopes change. More precisely, we let rhobar range over the EG-stack and study how slopes change from component to component. |
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October 14 |
Ralph Greenberg (University of Washington)
We will begin with a general discussion of so-called "pseudo-null" modules over the rings of interest in this talk (namely, formal power series rings over the \(p\)-adic integers). The specific modules that we plan to discuss were first studied in Iwasawa's classical theory about the behavior of ideal class groups in certain towers of number fields. We will discuss various questions and conjectures about these modules and recent attempts to say something nontrivial about their structure. In particular, we will describe recent joint work with F. Bleher, T. Chinburg, M. Kakde, R. Sharifi, and M. Taylor. |
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October 21 |
Doug Haessig (University of Arizona)
Dwork introduced unit root L-functions in the 1970s and conjectured they are \(p\)-adic meromorphic when they come from geometry. This was proven by Wan in 2000 in a series of papers. Very little is known about these L-functions aside from meromorphy. In this talk, we will discuss some recent results on the \(p\)-adic unit root L-function of the hyper-Kloosterman family of exponential sums over finite fields. |
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October 28 |
Bryan Hu (UCSD)
We discuss arithmeticity results for L-functions associated to automorphic forms on quaternionic groups. We describe a method, originating from Shimura, to attack Deligne's conjecture for critical values of L-functions. In particular, we investigate quaternionic modular forms (QMFs), which are non-holomorphic automorphic forms associated to the so-called quaternionic discrete series studied by Gross and Wallach. We describe recent work on the arithmeticity of Fourier coefficients for QMFs, as well as a notion of Maass-Shimura operators for QMFs. |
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November 4 |
Serin Hong (University of Arizona)
The Fargues-Fontaine curve is a \(p\)-adic analogue of the complex projective line. Vector bundles on the Fargues-Fontaine curve have been playing a fundamental role in the recent development of arithmetic geometry, highlighted by the seminal work of Fargues-Scholze on the local Langlands correspondence. We present several classification theorems for vector bundles on the Fargues-Fontaine curve and discuss their applications. |
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November 18 |
Kirti Joshi (University of Arizona)
Anabelomorphy is a notion I formulated to formalize Mochizuki's ideas about anabelian way of changing base rings. In the concrete context of \(p\)-adic fields this means understanding arithmetic while keeping the absolute Galois group of a \(p\)-adic field fixed, but not fixing the field. Since the Local Langlands correspondence deals with representations of Galois (or more precisely Weil-Deligne) group, one may ask: To what extent is the \(p\)-adic field \(K\) tied to the representation theory of topological groups appearing in the Local Langlands Correspondence. My results are quite unexpected and my talk will be a panorama of some of the results obtained to date. |
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November 25 |
Poornima B (UCSD)
The theory of \(p\)-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent \(F\)-isocrystals and the subcategory of overconvergent \(F\)-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent \(F\)-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent \(F\)-isocrystal in terms of data associated to the corresponding convergent \(F\)-isocrystal. |
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December 2 |
Junmyeong Jang (University of Ulsan)
A complex K3 surface has a complex multiplication if the Hodge structure of the trascendental part has an abelian Hodge group. A K3 surface over a field of positive characteristic is supersingular if the Picard number 22. Kazuhiro Ito had a criterion to determine at which place, the reduction of a CM K3 surface is supersingular. In this talk, we will see all but finitely many supersingular reductions of a CM K3 surface is of special type. This result can be regarded as a kind of counter part of that every K3 surface of finite height over a finite field has a CM lifting. |
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December 9 |
Antonio Lei (University of Ottawa)
Classical Iwasawa theory explores how arithmetic objects behave in infinite towers of field extensions, typically focusing on the \(p\)-primary components of objects such as ideal class groups and Selmer groups over extensions built using the same prime \(p\), for example, those generated by \(p\)-power roots of unity. But what happens when the primes differ? Can we describe the \(p\)-primary structure over extensions constructed using another prime \(q\)? Washington and Sinnott answered this question for class groups. In the case of elliptic curves, a folklore conjecture predicts the behavior of their L-values in this non-equal characteristic setting. In this talk, I will present recent joint work with Debanjana Kundu that makes partial progress toward resolving this conjecture. |
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January 20 |
Bryden Cais (University of Arizona)
In this talk, I will describe a novel Iwasawa theory for unramified \(\mathbb{Z}_p\)-extensions of global function fields over an algebraically closed field of characteristic \(p\). In this context, the \(p\)-adic slopes of Frobenius acting on the first crystalline cohomology of the associated \(\mathbb{Z}_p\)-tower of algebraic curves provide a new kind of Iwasawa-theoretic object to study, and I will present evidence for a recent conjecture about the limiting behavior of these slopes. |
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January 27 |
Hang Xue (University of Arizona)
I will explain how to prove some cases of the Tate conjectures for Shimura varieties using automorphic methods, starting from the pioneering work of Harder, Langlands and Rapoport on Hilbert modular surfaces, and culminating on some recent work (in progress) on Shimura varieties attached to \(\mathrm{U}(n, 1)\). A large part of this talk is accessible to graduate students. |
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February 10 |
Sandra Nair (Colorado State University)
The Harris-Viehmann conjecture establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris-Viehmann conjecture have been approached with the additional condition of Hodge-Newton reducibility on the underlying local Shimura datum. Building on previous work by Mantovan (EL/PEL case) and Hong (Hodge case), we extend the proof of the conjecture to non-basic local Shimura data of abelian type under the assumption of Hodge-Newton reducibility. We leverage Shen's construction of Rapoport-Zink spaces of abelian type. This is joint work with Xinyu Zhou. |
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February 17 |
Jack Garzella (UCSD)
The zeta function of a variety in characteristic \(p\) captures a lot of arithmetic information about that variety. Calculating this zeta function as fast as possible is a classical problem in computational number theory. We describe a cohomological approach to this problem, which involves a \(p\)-adic formula for the Frobenius action on cohomology. Costa and Harvey came up with a fast algorithm called *controlled reduction* which uses this method and is the state of the art for varieties of dimension greater than one. We describe various ways one can improve the algorithms of Costa and Harvey, including an "abstract controlled reduction problem" which abstracts the algorithm away from the specifics of any particular class of varieties. Using our algorithms, we find many examples of varieties with interesting arithmetic invariants (like Newton polygons and domino numbers). All work is joint with Batubara, Huang, and Mellberg. |
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February 24 |
Rohit Kumar (Duke University)
In a joint work with Victor Rotger, we study the irregular elliptic Stark conjecture of Darmon, Rotger, and Lauder. The fixed data consists of a rational elliptic curve \(E_f\) and \(2\)-dimensional artin representations \(\rho_g, \rho_h : \mathrm{Gal}(H/\mathbb{Q}) \to \mathrm{GL}_2(L)\) such that the selmer group \(\mathrm{Hom}_{G_{\mathbb{Q}}}(E(H), \rho_g \otimes \rho_h)\) is two-dimensional over \(L\). When the eigencurve is etale at \(g\) (regular case), the conjectures construct a certain regulator from ``rational points'' in \(\mathrm{Hom}_{G_{\mathbb{Q}}}(E(H), \rho_g \otimes \rho_h)\) and relate to a certain \(p\)-adic iterated integral \(\pi_g(e_{\mathrm{ord}}(d^{-1} f^{[p]} \times h))\). We investigate the case when the eigencurve is not smooth at \(g\) (irregular case). In this case, first we connect this \(p\)-adic iterated integral to \(p\)-adic triple product L-functions. Then, we use this to prove instances of this conjecture when \(g = h\) are induced from cubic unramified characters of an imaginary quadratic field \(K\). |
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March 18 |
Preston Wake (Michigan State University)
Given an irreducible two-dimensional mod-\(p\) Galois representation, Serre's conjecture tells you whether or not it arises as the mod-\(p\) Galois representation of a modular form. Even better, Serre gave a precise prediction for minimal weight, level, and character of such a modular form. |
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March 24 |
Klaus Lux (University of Arizona)
In this talk we will give an introduction to several types of tables related to a finite group. We will give an overview of the current computational techniques used to compute these tables, such as the ordinary character table and the modular (Brauer) character tables of the group. Some recent results such as the \(5\)-modular character table of the sporadic Lyons group will be highlighted. Time permitting we will also describe an ongoing project dealing with algorithms that enable one to compute the trivial source character table of a group (a table related to the Burnside table of marks). |
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March 31 |
John Bergdall (University of Arkansas)
The goal of this talk is to discuss a local-global phenomenon in the Langlands program on automorphic forms and Galois representations. Spaces of modular forms are parametrized by weights, and those same weights parametrize local deformation rings defined by conditions in \(p\)-adic Hodge theory. In the mid-2000's, M. Kisin's work on modularity showed that modular points can be found on each and every component of one of these local deformation rings. The purpose of this talk is to describe an extension of Kisin's result. We give a specific method for counting how many points lie on a given component in terms of that component's special fiber geometry. This is joint work with Chengyang Bao (Imperial College, London) and Brandon Levin (Rice University, Houston). |
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April 7 |
Swati (University of South Carolina)
Let \(p\) be prime, and let \(p_{[1,p]}(n)\) denote the function whose generating function is \(\prod (1-q^n)^{-1}(1-q^{pn})^{-1}\). This function and its generalizations \(p_{[c^\ell,d^m]}(n)\) are the subject of study in several recent papers. Let \(\ell\geq 5\), let \(j\geq 1\), and let \(p\in\{2,3,5\}\). In this paper, we prove that the generating function for \(p_{[1,p]}(n)\) in the progression \(\beta_{p,\ell ,j}\) modulo \(\ell^j\) with \(24\beta_{p,\ell,j} \equiv p + 1 \,(\mathrm{mod} ~\ell^j)\) lies in a Hecke-invariant subspace of type \(\{\eta(Dz)\eta(Dpz)F(Dz):F(z)\in M_s(\Gamma_0(p), \chi)\}\) for suitable \(D \geq 1\), \(s \geq 0\), and character \(\chi\). When \(p\in\{2,3,5\}\), we use the Hecke-invariance of these subspaces to prove that for distinct primes \(\ell\) and \(m \geq 5\) and \(j \geq 1\), congruences of the form \[ p_{[1,p]}\left(\frac{\ell^j m^k n + 1}{D}\right) \equiv 0 \pmod{\ell^j} \] for all \(n \geq 1\) with \(m \nmid n\), where \(k\) is explicitly computable and depends on the forms in the invariant subspace. This is a joint work with Matthew Boylan. |
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April 21 |
Chi-Yun Hsu (Santa Clara University)
Coleman showed that the \((k-1)\)st power of the theta operator \(q d/dq\) defines a map from overconvergent modular forms of weight \(2-k\) and slope \(0\) to weight \(k\) and slope \(k-1\). Moreover, the critical \(p\)-stabilization of a classical CM form is the image of a \(p\)-adic CM form, strengthening the fact that its Galois representation splits locally at \(p\). In the \(\mathrm{GSp}_4\) setting, the Galois representation of a Yoshida lift splits locally into two 2-by-2 blocks at \(p\). We prove an analogous strengthening in the joint work in progress with Bharathwaj Palvannan. The relevant theta operator arises from the last differential of the dual BGG complex. We computed its explicit effect on \(q\)-expansions. Using the explicit Fourier coefficients of Yoshida lifts by Hsieh--Namikawa, we show that Yoshida lifts lie in the image of this theta operator, with a certain choice of \(p\)-stabilization. |
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April 28 |
Hang Xue (University of Arizona)
I will explain some approaches to the Tate conjectures via automorphic methods, which eventually lead to a proof of the Tate conjecture for Shimura varieties attached to \(\mathrm{U}(n, 1)\) in many cases. I will begin the talk with some general introductions which should be accessible to the graduate students. |
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May 5 |
Anna Medvedovsky (University of Arizona)
The Euler-Kronecker constant of a number field \(K\) captures the constant term of the Laurent expansion of the Dedekind zeta function \(\zeta_K(s)\) near \(s = 1\) --- that is, the second-order term after the residue at \(s = 1\). We consider generalizations to Dirichlet series associated to multiplicative frobenian functions, specifically those associated to sets of the form \(\{n: a_n(f)\text{ doesn't vanish mod }p\}\) for \(f\) a modular eigenform and \(p\) a prime. Joint with Steven Charlton and Pieter Moree. Preprint: https://arxiv.org/abs/2412.01803. Talk should be largely accessible to graduate students with some background in algebraic number theory and representations of finite groups! |