Davide Lombardo

Generalised Kummer theory. Instead of considering $\ell^n$-torsion points of abelian varieties (or more generally commutative algebraic groups), one can consider $\ell^n$-division points of a fixed rational point $\alpha \in A(K)$. Many natural problems can be cast in this setting, and while the situation is well understood when $\dim A=1$ and the ground field is $\mathbb{Q}$, other cases seem to be much more mysterious.

Integral Galois theory. Let $L/K$ be a Galois extension of local or global fields with group $G$. What can be said about the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$-module?

Groups of components of Sato-Tate groups. The Sato-Tate conjecture is in a sense a refinement of the Mumford-Tate conjecture; Serre has shown how to attach a certain algebraic group $\operatorname{ST}(A/K)$ to an Abelian variety $A$ over a number field $K$, and the group of components $\operatorname{ST}(A/K) / \operatorname{ST}(A/K)^0$ is thought to have significant arithmetical interest. There are many questions that can be asked about this object, for example: there are some 'obvious' constraints on which finite groups can be component groups of Sato-Tate groups over a given field $K$. Are these conditions sufficient?

Geometric realisation of endomorphisms of Jacobians. Every endomorphism of $\operatorname{Jac}(C)$, where $C$ is a smooth projective curve, can be represented by a divisor on $C \times C$. In some situations, these divisors have 'conceptual' interpretations, and I would like to (identify and) explore more of these cases.

Local-global principles for isogenies of abelian varieties.

Integral Lie theory. A wonderful result by Richard Pink gives a very handy description of the pro-$p$ subgroups of $\operatorname{SL}_2(\mathbb{Z}_p)$ for $p>2$. I have managed to partially extend this to also cover the case $p=2$, but no similar result is available for algebraic groups other than $\operatorname{SL}_2$. Lazard's theory of Lie algebras should be relevant here, but work remains to be done.

The inverse problem for automorphism groups. Let $G$ be an algebraic group. Is there a projective variety $X$ with $\operatorname{Aut}(X) \cong G?$ We now know the answer when $G$ is an abelian variety, and I would love to understand whether certain linear algebraic groups are automorphism groups of smooth varieties or not.

Rational points of moduli stacks of curves. It is well known that elliptic curves over an algebraically closed field are classified by their $j$-invariant. Elliptic curves $E$ over a non-algebraically closed field $K$ of characteristic $\neq 2,3$ are classified by $j(E)$ together with the datum of a class in $K^\times/K^{\times d}$, where $d = \# \operatorname{Aut}(E_{\overline{K}})$. Can a similar description be found for genus 2 curves -- or, more interestingly, can one show that no such description exists (under suitable 'functoriality' assumptions)?