Enumerative Galois theory for cubics and quartics

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden

Linearizing torsion classes in the Picard group of algebraic curves over finite fields

We address the problem of computing in the group of $\ell^k$-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.Comment: To appear in Journal of Algebr

Zero divisors of support size 33 in group algebras and trinomials divided by irreducible polynomials over GF(2)GF(2)

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length $2$ cannot be happen. The first unsettled case is the existence of zero divisors of length $3$. Here we study possible length $3$ zero divisors in rational group algebras and in the group algebras over the field with $p$ elements for some prime $p$

Motivic height zeta functions

Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic fiber $X_F$ is a smooth equivariant compactification of $G$ such that $D=X_F\setminus G_F$ is a divisor with strict normal crossings, let $U$ be a surjective and flat model of $G$ over $C_0$. We consider a motivic height zeta function, a formal power series with coefficients in a suitable Grothendieck ring of varieties, which takes into account the spaces of sections $s$ of $X\to C$ of given degree with respect to (a model of) the log-anticanonical divisor $-K_{X_F}(D)$ such that $s(C_0)$ is contained in $U$. We prove that this power series is rational, that its "largest pole" is at $\mathbf L^{-1}$, the inverse of the class of the affine line in the Grothendieck ring, and compute the "order" of this pole as a sum of dimensions of various Clemens complexes at places of $C\setminus C_0$. This is a geometric analogue of a result over number fields by the first author and Yuri Tschinkel (Duke Math. J., 2012). The proof relies on the Poisson summation formula in motivic integration, established by Ehud Hrushovski and David Kazhdan (Moscow Math. J, 2009).Comment: 54 pages; revise