Étant donné un groupe réductif $G$ sur une extension de degré fini de $\mathbb {Q}_p$ on classifie les $G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge $p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble $B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.

It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

Let $\mathcal {N}(b)$ be the set of real numbers that are normal to base b. A well-known result of Ki and Linton [19] is that $\mathcal {N}(b)$ is $\boldsymbol {\Pi }^0_3$-complete. We show that the set ${\mathcal {N}}^\perp (b)$ of reals, which preserve $\mathcal {N}(b)$ under addition, is also $\boldsymbol {\Pi }^0_3$-complete. We use the characterization of ${\mathcal {N}}^\perp (b),$ given by Rauzy, in terms of an entropy-like quantity called the noise. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of ${\mathcal {N}}^\perp (b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol {\Pi }^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [10] and Schmeling in [36] in the context of $\beta $-transformations.

The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group$\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$, let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$. We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline {n})$ for all $\overline {n}\in \mathbb {Z}_m$, where m is an even integer.

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

The notion of $\theta $-congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $-congruent over certain real number fields.

In this paper we sharpen Hildebrand’s earlier result on a conjecture of Erdős on limit points of the sequence ${\{d(n)/d(n+1)\}}$.

In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:

$$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$

Let $n$ be either $2$ or an odd integer greater than $1$, and fix a prime $p>2(n+1)$. Under standard ‘adequate image’ assumptions, we show that the set of components of $n$-dimensional $p$-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on $n$) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2) 179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$-valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$.

We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

In an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

For positive integers n and d > 0, let $G(\mathbb {Q}^n,\; d)$ denote the graph whose vertices are the set of rational points $\mathbb {Q}^n$, with $u,v \in \mathbb {Q}^n$ being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $\mathbb {Q}^n$. In this paper, we show that a space $\mathbb {Q}^n$ has the property that all pairs of non-trivial distance graphs $G(\mathbb {Q}^n,\; d_1)$ and $G(\mathbb {Q}^n,\; d_2)$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $G(\mathbb {Q}^n,\; d)$.

Let A be the product of an abelian variety and a torus over a number field K, and let $$m \ge 2$$ be a square-free integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of K such that the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order coprime to m. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$ -adic integrals, where $\ell$ varies in the set of prime divisors of m. We deduce that the density is a rational number, whose denominator is bounded (up to powers of m) in a very strong sense. This extends the results of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author, where the case m prime is established.