Related papers: Bogomolov conjecture for curves of genus 2 over fu…

Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to…

A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…

We study an analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture…

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our…

In this short note we prove that the Bloch's conjecture holds for a surface of general type of numerical Godeaux type or some class of numerical Campedelli type, with geometric genus zero equipped with an involution, when the quotient of…

We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups.

A quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms is proved. We also give an application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux.

We give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for separated schemes of finite type over a finite field, as stated and proved by Burns and the author. We will also comment briefly on…

We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2.…

We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-\'etale) covers of infinite degree with geometrically connected fibers. The…

We prove the Burghelea Conjecture for groups satisfying some additional cohomological property.

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our…

We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.

We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves…

Let $\Cal C,\Cal C'$ be curves over a base scheme $S$ with $g(\Cal C)\ge 2$. Then the functor $T\mapsto\{$generically smooth $T$-morphisms $T\times_S\Cal C'\to T\times_S\Cal C\}$ from $((S$-schemes)) to ((sets)) is represented by a…

It is shown that a valuation of residue characteristic different from $2$ and $3$ on a field $E$ has at most one extension to the function field of an elliptic curve over $E$, for which the residue field extension is transcendental but not…

We compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to $n$, (in some sense like the Christoffel Darboux theorem for non-mixed correlation functions). We also derive new…