We first show that every algebraic torus over any field, not necessarily split, can be realized as the special fiber of a semi-abelian scheme whose generic fiber is an absolutely simple abelian variety. Then we investigate which algebraic tori can be thus obtained, when we require the generic fiber of the semi-abelian scheme to carry non-trivial endomorphism structures.

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

Let f be a modular form of weight k≥2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function Lp (f,K,s) , as done by Bertolini–Darmon–Iovita–Spieß in [Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411–449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,…,k−1 , and one may be interested in the values of its derivative in this range. We construct, for k≥4 , a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel–Jacobi map. Our main result generalizes the result obtained by Iovita and Spieß in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), 333–384], which gives a similar formula for the central value s=k/2 . Even in this case our construction is different from the one found by Iovita and Spieß.

The $\ell$-adic parabolic cohomology groups attached to noncongruence subgroups of $\mathop{\rm SL_2(\mathbb{Z})$ are finite-dimensional $\ell$-adic representations of $\mathop{\rm Gal}(\overline{/mathbb{K}})$ for some number field $K$. We exhibit examples (with $K=\overline{\mathbb{Q}}$) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of $\mathfrak{sl}_n$ containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski.

It is shown that, given continuous, absolutely irreducible representations of ${\rm Gal}(\mathbb{Q}^{\rm ac}/\mathbb{Q})$ with values in ${\rm GL}_2(\mathbb{F}_2)$ and ${\rm GL}_2(\mathbb{F}_3)$, and having cyclotomic determinant, there is a weight 2 newform of some level whose mod $2$ and mod $3$ representations are equivalent to those given.