Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{deg}\, q$ is even and coprime with 7.
Let $q=3q_0^2$, where $q_0=3^s$ and $s\ge3$. We prove that a smooth, projective, geometrically irreducible curve over $\F_q$ with $q^3+1$ rational points and genus \[ \frac32q_0(q-1)(q+q_0+1) \] is $\F_q$-isomorphic to the Ree curve.
Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ i...
Let $q \equiv 1 \bmod 8$ be a prime power. We prove that $L\left(\tfrac{1}{2}, \chi\right) \neq 0$ for 100% of the imaginary quadratic Dirichlet characters of $\mathbb{F}_q(T)$.
Peter Koymans, Carlo Pagano, M. Shusterman· 0 citations
We show that if $X$ is a projective variety of dimension $\geq 2$ that is not a rational surface, then the blow up of $X$ at sufficiently many very general points has no nontrivial automorphism. Similar results were known before for $\mathbb{P}^2$ and $\mathbb{P}^3.$
Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.
We classify the modular curves $X_0(N)$ that admit a smooth embedding in $\mathbb P^1 \times \mathbb P^1$ and obtain a partial classification for the Shimura curves $X_0^D(N)$. We also prove that, for $D>1$, a Shimura curve $X_0^D(N)$ admits a smooth plane model if and only if its genus is at most one.
Oana Padurariu· 0 citations
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