Related papers: Kato-Kuzumaki's properties for function fields ove…

Let k be a global field and L be a finite dimensional \'etale algebra over k. In this paper, we assume that L is a product of cyclic extensions of k. Let T_{L/k} be the multinorm-one torus defined by the multinorm equation: N_{L/k} (t) = 1.…

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper…

Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$…

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(\mu_{p^\infty})$ to $p$-adic Lie…

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. Andr\'{e} proved this conjecture for…

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them…

We prove a generalization of Shafarevich's Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the…

We study the boundedness of the Mordell-Weil rank and the growth of the $v$-primary part of the Tate-Shafarevich group of $p$-supersingular abelian varieties of ${\rm GL}_2$-type with real multiplication over $\mathbb{Z}_p$-extensions of…

The classical McShane-Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally…

In this paper we investigate the Tate--Shafarevich group Sha^1(k, T) of a multinorm-one torus $T$ over a global field $k$. We establish a few functorial maps among cohomology groups and explore their relations. Using these properties and…

Let $(S,\mathfrak n)$ be a regular local ring and $f$ a non-zero element of $\mathfrak n^2$. A theorem due to Kn\"orrer states that there are finitely many isomorphism classes of maximal Cohen-Macaulay $R=S/(f)$-modules if and only if the…

We show that if $K$ is an arbitrary field and $G$ is a finite group then there exists a curve over $K$ with automorphism group $G$. We also give a positive solution to the weak inverse Galois problem for function fields over an arbitrary…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

We give a function field specific, algebraic proof of the main results of class field theory for abelian extensions of degree coprime to the characteristic. By adapting some methods known for number fields and combining them in a new way,…

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach…

In their previous works arXiv:2105.11026, arXiv:2206.10749, Cristofaro-Gardiner, Humili\`ere, Mak, Seyfaddini and Smith defined links spectral invariants on connected compact surfaces and used them to show various results on the algebraic…

Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let…

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite. We investigate…

Suppose that (K, $\nu$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $\omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K…