Prime Hasse principles via Diophantine second moments

We show that almost all primes $p\not\equiv \pm 4 \bmod{9}$ are sums of three cubes, assuming a conjecture due to Hooley, Manin, et al. on cubic fourfolds. This conjecture could be proven under standard number-theoretic hypotheses, following the author's thesis work.Comment: 19 page

The critical polynomial of a graph

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where $M_G={\rm Diag}(x_1,\dots,x_n)-A_G$. We investigate in this article the set $V_{d_G}(r)$ of non-negative values taken by this polynomial when $x_1, \dots, x_n \geq r \geq 1$. We show that $V_{d_G}(1) = {\mathbb Z}_{\geq 0}$. We show that for a large class of graphs one also has $V_{d_G}(2) = {\mathbb Z}_{\geq 0}$. When $V_{d_G}(2) \neq {\mathbb Z}_{\geq 0}$, we show that for many graphs $V_{d_G}(2)$ is dense in ${\mathbb Z}_{\geq 0}$. We give numerical evidence that in many cases, the complement of $V_{d_G}(2)$ in ${\mathbb Z}_{\geq 0}$ might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial

Sums of cubes and the Ratios Conjectures

Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of diameter $X\ge 1$, the bound takes the form $N\le C(\varepsilon) X^{3+\varepsilon}$ ($\varepsilon>0$). We attribute the $\varepsilon$ to several subtly interacting proof factors; we then remove the $\varepsilon$ assuming some standard number-theoretic hypotheses, mainly featuring the Ratios and Square-free Sieve Conjectures. In fact, our softest hypotheses imply conjectures of Hooley and Manin on $N$, and show that almost all integers $a\not\equiv \pm 4 \bmod{9}$ are sums of three cubes. Our fullest hypotheses are capable of proving power-saving asymptotics for $N$, and producing almost all primes $p\not\equiv \pm 4 \bmod{9}$.Comment: 61 pages; updated references; changed title; conceptual improvements; moved some material elsewher

On the density of sums of three cubes.

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