Let $F$ be a CM field and let ${\left({\bar{r}}_{\pi ,\lambda }\right)}_{\lambda }$ be the compatible system of residual ${\mathcal{𝒢}}_n$-valued representations of ${Gal}_F$ attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation $\pi$ of ${GL}_n\left(\mathbb{𝔸}\right)$, as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to ${\bar{r}}_{\pi ,\lambda }$ are unobstructed for all places $\lambda$ in a subset of Dirichlet density $1$, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framework for proving unobstructedness (with future applications in mind) and an $R=T$-theorem, relating the universal crystalline deformation ring of ${\bar{r}}_{\pi ,\lambda }$ and a certain unitary fixed-type Hecke algebra.

Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel–Jacobi map. We compute the scheme structure of the Hilbert scheme component of ${Hilb}_J$ containing the Abel–Jacobi embedding as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism ${\mathcal{ℳ}}_g\to {\mathcal{𝒜}}_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.

We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular we find that this set has cardinality 92. The essential part of the classification consists in determining the set of quadratic imaginary fields $M$ with class group $C_2\times C_2$ for which there exists an abelian surface $A$ defined over $\mathbb{Q}$ which is geometrically isogenous to the square of an elliptic curve with CM by $M$. We first study the interplay between the field of definition of the geometric endomorphisms of $A$ and the field $M$. This reduces the problem to the situation in which $E$ is a $\mathbb{Q}$-curve in the sense of Gross. We can then conclude our analysis by employing Nakamura’s method to compute the endomorphism algebra of the restriction of scalars of a Gross $\mathbb{Q}$-curve.

We prove a uniform version of non-Archimedean Yomdin–Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of ${\mathbb{𝔽}}_q\left[t\right]$-points of bounded degrees of algebraic varieties, uniformly in the cardinality $q$ of the finite field ${\mathbb{𝔽}}_q$ and the degree, generalizing work by Sedunova for fixed $q$. We also deduce a uniform non-Archimedean Pila–Wilkie theorem, generalizing work by Cluckers–Comte–Loeser.

A central tool in the study of systems of linear equations with integer coefficients is the generalised von Neumann theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\parallel f{\parallel }_{U^{s+1}\left[N\right]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the Möbius function over certain sequences.

The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues.

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov’s formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young’s hybrid subconvexity estimates for twisted modular $L$-functions.

This paper concerns a faithful representation $V$ of a simple linear algebraic group $G$. Under mild assumptions, we show that if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$. That is, the stabilizer in $\text{ Lie}\left(G\right)$ of a generic vector in $V$ is zero. The bound on $\text{ dim }V$ grows like ${\left(\text{ rank }G\right)}^2$ and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position.

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our main theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellekens list holomorphic VOA. These include the affine algebra $E_{8,2}$ and Höhn’s baby monster VOA ${VB}_{\left(0\right)}^{♮}$ but the other $13$ seem to be new. The idea in the proof of our main theorem is to exploit properties of a family of vector-valued modular forms with rational functions as Fourier coefficients, which solves a family of modular linear differential equations in terms of generalized hypergeometric series.