Johns Hopkins Number Theory Seminar

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the $p$th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic $L$-functions when split by root number. In my talk, I will discuss this root number correlation bias when the average is taken over all weight $k$ modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly.

Whittaker models are realizations of representations on a space of functions that transform in a certain way under the action of a unipotent group, and they have been central in both the local representation theory and the global theory of automorphic representations. For groups other than $\mathrm{GL}_n$, however, most classes of representations do not admit Whittaker models. In this talk, we will discuss several variants of Whittaker models, most notably reduced Whittaker models (for lack of a better name), and their applications to automorphic forms.

The Fontaine--Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation $\rho: \mathrm{Gal}_{\mathbb{Q}} \to \mathrm{GL}_2(\overline{\mathbb{Q}}_p)$ is modular if it is unramified outside finitely many places and de Rham at $p$. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations $\rho: \mathrm{Gal}_{\mathbb{Q}} \to \mathrm{GL}_2(L)$ when L is instead a non-archimedean local field of characteristic $p$.

The twisted Gan--Gross--Prasad conjectures consider the restriction of representations from $\mathrm{GL}_n$ to a unitary group over a quadratic extension $E/F$. In this talk, I will explain the relative trace formula approach to the global twisted GGP conjecture. In particular, I will discuss some differences between the twisted case and the Fourier--Jacobi case of the GGP conjecture for unitary groups, and how the fundamental lemma that arises can be reduced to the Jacquet--Rallis fundamental lemma. We obtain the global twisted GGP conjecture under some unramifiedness assumptions and local conditions.

The class number formula describes the behavior of the Dedekind zeta function at $s=0$ and $s=1$. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin $L$-functions and $p$-adic $L$-functions at $s=0$ and $s=1$ in terms of units and class numbers. The Harris--Venkatesh conjecture describes the residue of Stark units modulo $p$, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader Prasanna--Venkatesh conjectures. In this talk, I will introduce this big picture, formulate a unified conjecture combining Harris--Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

We show that V. Lafforgue's global Langlands correspondence is compatible with Fargues--Scholze's semisimplified local Langlands correspondence. As a result, we canonically lift Fargues--Scholze's construction to a non-semisimplified local Langlands correspondence for fields of characteristic $\geq 5$. We also deduce that Fargues--Scholze's construction agrees with that of Genestier--Lafforgue, answering a question of Fargues--Scholze, Hansen, Harris, and Kaletha.

Langlands introduced stable transfer operators as a fundamental part of his proposal of Beyond Endoscopy. They are intended to be used in comparisons of his proposed refinements of stable trace formulas, in an analogous role to that of endoscopic transfer operators in the theory of Endoscopy. The existence of stable transfer operators is readily established, but the problem is to obtain explicit formulas for their distributional kernels, the so-called stable transfer factors or functorial transfer kernels. We will discuss stable transfer between tori, complex groups, and work in progress on stable transfer from real groups to tori, which would include a generalisation of the Gelfand--Graev formula for stable discrete series characters of $\mathrm{SL}(2)$.

In the first part of this talk, we will introduce Langlands' notion of Beyond Endoscopy and discuss the basic concepts which are central to it, as well as the work of Ali Altug, with an emphasis on his first paper ('Beyond Endoscopy via the Trace Formula I'). In the second part, we will talk about the extension of this work to the case of $\mathrm{Sp}_4(\mathbb Q)$. In particular, we will discuss the change from rational conjugacy classes to stable conjugacy classes and how to characterize stable classes of elliptic tori for the symplectic group, focusing on explicit examples (as time permits).

The strong form of Serre's conjecture states that a two-dimensional mod $p$ representation of the absolute Galois group of $\mathbb Q$ arises from a modular form of a specific weight, level and character. Serre restricted to modular forms of weight at least 2, but Edixhoven later refined this conjecture to include weight one modular forms. In this talk we explore analogues of Edixhoven's refinement for Galois representations of totally real fields, extending recent work of Diamond--Sasaki. In particular, we show how modularity of partial weight one Hilbert modular forms can be related to modularity of Hilbert modular forms with regular weights, and vice versa. We will also discuss the applications of this for $p$-adic Hodge theory.

In the recent framework proposed by Ben-Zvi--Sakellaridis--Venkatesh, automorphic periods ought to, very roughly speaking, come in Langlands dual pairs. I will give a short introduction of this prediction and motivate the need to consider certain singular automorphic periods. In particular, I will present an example in joint work with Akshay Venkatesh, where we establish duality using a generalization of $L$-functions.

We will introduce a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is an average of Rankin--Selberg $L$-functions for $\mathrm{GL}(n+1) \times \mathrm{GL}(n)$ over the full spectrum, and the geometric side consists of Rankin--Selberg $L$-functions for $\mathrm{GL}(n) \times \mathrm{GL}(n)$, and certain explicit meromorphic functions. The formula yields new results towards central $L$-values for $\mathrm{GL}(n+1) \times \mathrm{GL}(n)$: the second moment evaluation, and simultaneous nonvnaishing in the level aspect.

Let $E$ be an elliptic curve, $p$ an odd prime and $K$ an imaginary quadratic field where $p$ is unramified. One may use Iwasawa theory to study the growth of Mordell—Weil ranks and the $p$-part of the Shafarevich—Tate groups inside the anticyclotomic $\mathbb{Z}_p$ extension of $K$. In this talk, we will discuss the case where $E$ has good supersingular reduction at $p$. In particular, we discuss generalizations of Kobayashi’s plus and minus Selmer groups and the corresponding Iwasawa main conjectures.

Oort conjectured in 1995 that isogeny classes in the moduli space $A_g$ of principally polarised abelian varieties in characteristic $p$ are Zariski-dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture to the special fibres of Shimura varieties of Hodge type, and in this talk, I will present a proof of this conjecture. I will mostly focus on the case of $A_g$ since many of the new ideas can already be explained in this special case. This is joint work with Marco D'Addezio.

Let $F$ be a non-archimedean local field and $G$ be a connected reductive group over $F$. For a Bernstein block in the category of smooth complex representations of $G(F)$, we have two kinds of progenerators: the compactly induced representation $\mathrm{ind}_K^{G(F)} (\rho)$ of a type $(K, \rho)$, and the parabolically induced representation $I_{P}^{G}(\Pi^{M})$ of a progenerator $\Pi^{M}$ of a Bernstein block for a Levi subgroup $M$ of $G$. In this talk, we construct an explicit isomorphism of these two progenerators. We also explain that the induced isomorphism between the endomorphism algebras is compatible with their descriptions in terms of affine Hecke algebras.

I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a $p$-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using $p$-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.

In the ongoing work with Getz, Gu and Leslie, we constructed a zeta integral that relates to triple product $L$-functions for arbitrary rank. It is essentially the integral of Garrett in the triple $\mathrmn{GL}_2$ case, but with degenerate Eisenstein series replaced with Schwartz functions on the affine closure of certain Braverman--Kazhdan space. This replacement is necessary for our construction of zeta integrals in higher rank. In the talk, we will give a quick introduction to the Poisson summation conjecture and describe a family of BK-spaces where the conjecture is well understood by the joint work with Getz and Leslie. We will then set up the zeta integral and state our expected results.

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an $R=T$ theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

In this talk I will construct an analogue to the Shimura--Shintani correspondence in the setting of the so-called rigid analytic cocycles, which were defined by Darmon and Vonk and can be thought as objects living on the $p$-adic upper half-plane. Building such a correspondence fits into the nascent $p$-adic Kudla program, an emerging $p$-adic version of the classical Kudla program which studies relations between automorphic forms and generating series of $p$-adic objects. If time permits, I will also give a classification of certain rigid meromorphic cocycles of higher weight.

In his 1976 proof of the converse of 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 posed on Frank Calegari’s blog.

We define rigid inner forms for local and global function fields and discuss the role they play in studying the local and global Langlands correspondence. We focus particularly on global rigid inner forms; more specifically, their relationship with transfer factors and how they can beused to give a (conjectural) formula for multiplicity of discrete automorphic representations.

Over the last century, the Hodge and Tate conjectures have inspired much activity in algebraic and arithmetic geometry. These conjectures give predictions for when certain topological objects come from geometry. Simpson and Fontaine--Mazur introduced non-abelian analogs of these conjectures. In joint work with Daniel Litt, we prove these analogs for low rank local systems on generic curves, resolving conjectures of Esnault--Kerz and Budur--Wang as well as answering questions of Kisin and Whang.

In the relative Langlands programme, we are interested in the relation between periods of automorphic representations and special values of $L$-functions. One of the first interesting examples in the programme is Guo--Jacquet's conjecture which aims to generalise Waldspurger’s well-known theorem relating toric periods to central values of automorphic $L$-functions for $\mathrm{GL}(2)$. To attack such a conjecture, a promising tool is the comparison of relative trace formulae. However, we are faced with some analytic difficulties such as divergent integrals when applying this method. In this talk, we shall recall the background of Guo--Jacquet trace formulae and introduce an infinitesimal variant of these formulae, where we deal with the divergence problem on the relevant infinitesimal symmetric spaces via an analogue of Arthur's truncation. In case the audience is interested, we shall talk about the local comparison of most terms in our formulae at the representation theory seminar on Friday.

Gan, Gross and Prasad have conjectured a criterion for the nontriviality of certain algebraic cycles arising from maps of unitary Shimura varieties. (The cycles can be thought of as higher-dimensional analogues of Heegner points.) I will talk about an analogue in which the criterion is the non-vanishing of the derivative of a Rankin--Selberg $p$-adic $L$-function. This supports variants of the Birch and Swinnerton-Dyer conjecture. The proof is based on a comparison of relative-trace formulas in $p$-adic coefficients. Joint work with Wei Zhang.

I will review the formulations of the Langlands correspondence in various settting (obtained by a sequence of analogies) and formulate a new categorified version of the Langlands correspondence applicable in any geometric setting, which interpolates between these. Of particular interest is the setting of $\ell$-adic sheaves on curves in finite characteristic. A key ingredient is a new moduli space of Galois representations, the stack of local systems with restricted variation. I will describe this moduli space and its various remarkable properties. A key result will be a categorical version of the spectral decomposition of automorphic forms. Upon taking the categorical trace of Frobenius, this gives an enhancement of V. Lafforgue's spectral decomposition of the space of automorphic forms. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.

In this talk I will discuss the local and global conjectures for some strongly tempered spherical varieties. Both conjectures are very similar to the Gan--Gross--Prasad models. More specifically, globally the square of the period integrals should be related to the central value of some $L$-functions of symplectic type. Locally each tempered $L$-packet should contain a unique distinguished element with multiplicity one and the unique distinguished element should be determined by certainepsilon factors (i.e. epsilon dichotmy). I will also discuss the proof of the local conjecture in many cases. This is a joint work with Lei Zhang.

The Toda flow is a famous integrable system and appears in many parts of mathematics. It comes in many different (but related) flavours (tropical, discrete, continuous, quantum) and amusingly the discrete Toda flow (which induces all other flows) can be traced back to work of Frobenius. We hope to convince the audience that the Toda flow can be viewed as a candidate for the mysterious and elusive Frobenius flow of the number field $\mathbb Q$, extending the scope of earlier proposals in the literature. The main goal of our talk is to explain 1) the link between the tropical Toda flow and the Riemann hypothesis (due to Tokihiro and Mada). 2) how the tropical Toda flow is conjecturally linked to knot theory in a very natural way. 3) how a beautiful (but mysterious) conjecture of Shintani (about explicit class field theory for number fields) appears naturally in the quantum Toda framework, surprisingly in perfect analogy with Drinfeld’s solution of the explicit class field theory problem for function fields.

Let $N$ and $p \geq 5$ be primes such that $p$ divides $N−1$. In his landmark paper on the Eisenstein ideal, Mazur proved the $p$-part of the BSD conjecture for the $p$-Eisenstein quotient $J(p)$ of $J_0(N)$ over $\mathbb{Q}$. Using recent results and techniques of the work of Venkatesh and Sharifi on the Sharifi conjecture, we prove unconditionally a weak form of the BSD conjecture for $J(p)$ over aquadratic field $K$ (which can be real or imaginary). This includes results in positive analytic rank, as the analytic rank of $J(p)$ over $K$ can be $\geq 2$ for well-chosen $K$. This is joint work with Jun Wang (MCM Beijing).

This is joint work with Joakim Faergeman. We prove that cuspidal $D$-modules on $\mathrm{Bun}_G$ have non-vanishing Whittaker coefficients, generalizing a result of Gaitsgory in the $\mathrm{GL}_n$ case (whose argument imitates the proof of a theorem of Shalika in the function theoretic setting). More generally, we prove that tempered $D$-modules have non-vanishing Whittaker coefficients, yielding an optimal result. The main idea is to relate Whittaker coefficients with singular support properties of automorphic $D$-modules. Specifically, we reduce to the case of nilpotent singular support using AGKRRV theory. Here, we show that Whittaker coefficients behave like a microstalk, which has natural implications for conservativeness properties.

Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of automorphic representations, these correspond to objects on the metaplectic doublecover of $\mathrm{SL}(2)$. In this talk, I will outline a theory of modular forms of half-integral weight on double covers of exceptional groups, generalizing the integral weight theory developed by Gross--Wallach, Gan--Gross--Savin, and Pollack. Furthermore, I discuss a particular example of a weight $1/2$ modular form on $G_2$ whose Fourier coefficients encode the $2$-torsion in the narrow class groups of totally real cubic fields. This is built by studying a remarkable automorphic representation of the double cover of the exceptional group $F_4$. This is joint work with Aaron Pollack.

Work in progress joint with Samuele Anni and Alexandru Ghitza. For N prime to $p$, we count the number of classical modular forms of level $Np$ and weight $k$ with fixed mod-$p$ Galois representation and Atkin-Lehner-at-$p$ sign, generalizing both recent results of Martin generalizing work of Wakatsuki and Yamauchi (no residual representation constraint) and the $\rhobar$-dimension-counting formulas of Bergdall--Pollack and Jochnowitz. Working with the Atkin-Lehner involution typically requires inverting $p$, which naturally complicates investigations modulo $p$. To resolve this tension, we use the trace formula to establish up-to-semisimplification isomorphisms between certain mod-$p$ Hecke modules (namely, refinements of the weight-filtration graded pieces $W_k$) by exhibiting ever-deeper congruences between traces of prime-power Hecke operators acting on characteristic-zero Hecke modules. This last technique, relying on our refinement of the Brauer--Nesbitt theorem for a single operator, is new, purely algebraic/combinatorial, and may well be of independent interest. We will begin with this algebra theorem, then discuss the classical Atkin-Lehner dimension split, and only then move on to our refined dimension-counting results. The first two-thirds of the talk should be accessible to a wide algebra/number-theory audience.

The quantum modular invariant $j^{qt}$ is a multi-valued and discontinuous generalization of the modular invariant, defined on the moduli space of quantum tori, which may also be defined for function fields over a finite field $\mathbb{F}_{q}$. For $f\in K$, a real quadratic extension $\mathbb{F}_{q}[T]$, $j^{qt}(f)$ has finitely many values, each of which is the modular invariant of a rank 1 Drinfeld module. When $f$ is a fundamental unit, we show that the product of the multivalues of $j^{qt}(f)$ generates the Hilbert class field of $K$. In addition, there is an allied notion of quantum Drinfeld module, and ray class fields of $K$ may be generated by the sums of the multivalues of its quantum torsion points. Together with the previous result, this gives a real quadratic analog of the Theorem of Weber--Fueter, the latter being the main inspiration for Hilbert's 12th problem. In the case of a real quadratic number field $K/\mathbb{Q}$, if $\theta\in K$ is a fundamental unit, $j^{qt}(\theta)$ is a Cantor set, and it is conjectured that the Hilbert class field $H_{K}$ is generated over $K$ by a multiplicative average over $j^{qt}(\theta)$. In this case, the multivalues of $j^{qt}(\theta)$ are the modular invariants of 1 dimensional solenoids associated to quasicrystals in $K$, which are introduced here as the zero characteristic analogs of Drinfeld modules. This talk is based on work with Carlos Castaño Bernard, Luca Demangos, Eric Leichtnam and Pierre Lochak.

Modular forms are global sections of certain line bundles on the modular curve, while $p$-adic overconvergent modular forms are defined only over a strict neighborhood of the ordinary locus. The philosophy of classicality theorems is that when the $p$-adic valuation of $U_p$-eigenvalue is small compared to the weight (called a small slope condition), an overconvergent $U_p$ eigenform is automatically classical, namely it can be extended to the whole modular curve. In the case of Hilbert modular forms, there are the partially classical forms which are defined over a strict neighborhood of a “partially ordinary locus”. Modifying Kassaei’s method of analytic continuation, we show that under a weaker small slope condition, an overconvergent form is automatically partially classical.

In the 1970s Andrew Odlyzko proved good lower bounds for the discriminant of a number field. He also showed that his results could be sharpened by assuming the Generalized Riemann Hypothesis. Someyears later Odlyzko suggested that it might be possible to do without GRH. I shall explain Odlyzko's ideas and sketch how for number fields of reasonably small degree (say up to degree 11 or 12) one can indeed improve the lower known bounds by exploiting hypothetical violations of GRH. This is joint work with Karim Belabas, Francisco Diaz y Diaz and Salvador Reyes, extending unpublished results of Matias Atria.

I will explain a joint work with Juan Esteban Rodríguez Camargo where we reformulate the theory of locally analytic representations of a $p$-adic Lie group as developed by Schneider and Teitelbaum from the perspective of Clausen and Scholze's Condensed Mathematics. As an application, we will show various comparison results between continuous, locally analytic, and Lie algebra cohomology, generalizing classical results of Lazard.

We give a conjectural description of Zeta-values of arithmetic schemes at $s=n$ for any integer $n\in\mathbb{Z}$, in terms of two perfect complexes of abelian groups. The first complex is called Weil-étale motivic cohomology with compact support.The second complex can be thought of as derived de Rham cohomology modulo the Hodge filtration relatively to the sphere spectrum, and is defined using topological Hochschild homology. The functional equation of Zeta functions together with our description of Zeta-values implies a formula relating these complexes, special values of the archimedean Euler factors and Bloch's conductor. We will state this formula, which can actually be proven. This is joint work with Matthias Flach.

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a hyperbolic surface grows more slowly than any positive power of the eigenvalue. In dimensions three and higher, this was shown to be false by Iwaniec--Sarnak and Donnelly. I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to $\mathrm{SO}(p,q)$.

In joint work with Denis Benois, we give an étale construction of Bellaïche's $p$-adic $L$-functions about $\theta$-critical points on the Coleman--Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of $p$-adic regulators on what we call the thick Selmer groups, which come attached to the infinitesimal deformation at the said $\theta$-critical point along the eigencurve, and an exotic ($\Lambda$-adic) $\mathcal{L}$-invariant. Besides our interpolation of the Beilinson--Kato elements about this point, the key input to prove the interpolative properties of this $p$-adic $L$-function is a new $p$-adic Hodge-theoretic eigenspace-transition via differentiation principle.

Condensed objects are a recent invention by Peter Scholze and Dustin Clausen that improve the interactions between algebra and topology in many settings. This leads to an abelian category of topological groups, a solution to the Whitehead problem, a sensible notion of a completed tensor product and many other results that are still being studied. We will go through some of the problems that make condensed mathematics necessary, as well as an introduction to condensed objects and some of its applications.

The Langlands program is a far-reaching collection of conjectures thatrelate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of $p$-adic groups. I will provide an overview of our understanding of the representations of $p$-adic groups, with an emphasis on recent progress including joint work with Kaletha and Spice that introduces a twist to the story, and outline some applications.

It is well known that Siegel modular forms of degree 2 are related to automorphic representations of $\mathrm{GSp}(4)$. This talk will explore this connection to count a specific set of cuspidal automorphic representations of $\mathrm{GSp}(4)$ using known dimension formulas of Siegel cusp forms of degree 2 and prime level. Dimension and codimension formulas for the spaces of Siegel cusp forms of degree 2 are known for the prime levels and some squarefree levels. However, the dimensions of the spaces of Siegel cusp forms of non-squarefree levels are mostly not available in the literature. This talk will present new dimension formulas of Siegel cusp forms of degree 2, weight $k$, and level 4 for two congruence subgroups. Our method relies on counting another set of cuspidal automorphic representations of $\mathrm{GSp}(4)$. This work is joint with Ralf Schmidt and Shaoyun Yi.

We will first talk about the Ceresa class, which is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. In joint work with Dean Bisogno, Wanlin Li and Daniel Litt, we construct a non-hyperelliptic genus 3 quotient of the Fricke--Macbeath curve with vanishing Ceresa class, using the character theory of the automorphism group of the curve, namely, $\mathrm{PSL}_2(\mathbb{F}_8)$. This will also include the tale of another genus 3 curve by Schoen that was lost and then found again! Time permitting, we will also talk about some Galois cohomology classes that obstruct the existence of rational points on curves, by obstructing splittings to natural exact sequences coming from the fundamental group of a curve. In joint work with Wanlin Li, Daniel Litt and Nick Salter, we use these obstruction classes to give a new proof of Grothendieck’s section conjecture for the generic curve of genus $g > 2$. An analysis of the degeneration of these classes at the boundary of the moduli space of curves, combined with a specialization argument lets us prove the existence of infinitely many curves of each genus over $p$-adic fields and number fields that satisfy the section conjecture.

This talk will report on recent work proving new cases of the Heegner Point Main Conjecture of Perrin-Riou. I'll explain the statement of the conjecture and the method of patched bipartite Euler systems used in the proof. This method reduces the HPMC to a 'main conjecture of Bertolini and Darmon at infinite level', which can be resolved using the work of Skinner--Urban along with a deformation-theoretic input following methods of Fakhruddin--Khare--Patrikis. One consequence of the results isan improved $p$-converse theorem to the work of Gross--Zagier and Kolyvagin: $p$-Selmer rank one implies analytic rank one.

We will formulate a list of properties that uniquely characterize the local Langlands correspondence for discrete Langlands parameters with trivial monodromy. Suitably interpreted, this characterization holds for any local field, but requires an assumption on p in the non-archimedean case. We will then discuss an explicit construction of this correspondence, as a realization of functorial transfer from elliptic maximal tori.

I will report on joint work with Vytautas Paškūnas and Gebhard Böckle concerning deformation rings for mod $p$ Galois representations of $p$-adic local fields. After giving a short introduction to the subject, I will explain our main result which says that framed local deformation rings are complete intersections of the 'expected dimension', and which gives a classification ofthe irreducible components in terms of the determinant map. I will explainsome of the ingredients that go into our proof, which involves studying pseudo-characters and moduli spaces of representations with fixed pseudo-character. If time permits I will discuss an application to density of crystalline points in deformation spaces.

An arithmetic locally symmetric space is the quotient of a symmetric space by an arithmetic lattice in its isometry group. Margulis conjectured that once we fix the symmetric space, all its compact arithmetic quotients havea uniform lower bound on the systole (the length of theshortest non contractible loop). This conjecture would follow from the positive answer to the celebrated Lehmer conjecture, but so far it remains wide open. One the corollaries of the Margulis conjectureis the homotopy typeconjecture proposed by Gelander in 2004. It asserts that any arithmetic locally symmetric space M can be triangulated using at most constant times the volume of M simplices. In particular, many natural topological invariants like the Bettinumbers or logarithm of the torsion in homology should be bounded linearly in volume. In this talk I will describe the work in progress with Jean Raimbault and Sebastian Hurtado Salazar in which we prove the homotopy type conjecture. Our approach combinesa new “diophantine version of Margulis Lemma with a careful comparison between the orbital integrals of the characteristic functions of balls of different radii.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm{GL}_2$. Just like holomorphic modular forms, they are defined by having their realcomponent be one of a particularly nice class (in this case, called quaternionic discrete series). We try to count quaternionic automorphic representations on the exceptional group $G_2$ by developing a $G_2$ version of the classical Eichler--Selberg trace formulafor holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in $L$-packets with non-quaternionic members. Standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same $L$-packet. We therefore need to use the more sophisticated stable trace formula. Second, quaternionic discrete series do not satisfy a technical condition of being 'regular', so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity. Using some computations of Mundy, we show that this miraculously does not happen for the specific case of quaternionic representations on $G_2$. Finally, because we are only studying level-1 forms, we can apply some tricks of Chenevier and Taïbi to reduce the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This way, we avoid involved computationson the geometric side of the trace formula.