Narrow Results

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient $\mathbb{Z}/n\mathbb{Z}$$(n\ge 2)$ associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient $\mathbb{Z}/2\mathbb{Z}$ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient $\mathbb{Z}/n\mathbb{Z}$ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient $\mathbb{Z}/n\mathbb{Z}$) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].

The object of study is the group of units 𝒪*(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite cyclic cover of a rational variety. On a cyclic cover X → 𝔸^m of affine m-space over k such that the ramification divisor is irreducible and the degree is prime, it is shown that 𝒪*(X) is equal to k*, the non-zero scalars. The same conclusion holds, if X is a sufficiently general affine hyperelliptic curve. If X has a projective completion such that the divisor at infinity has r components, then sufficient conditions are given for 𝒪*(X)/k* to be isomorphic to ℤ^(r-1).

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin–Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree $4$, we show that $\mathrm{\text{ed}}({\mathbf{\text{PGL}}}_4)\ge 4$ over a field of characteristic $2$.

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$. Let $D$ be a central division $F$-algebra of $p$-power degree. We prove that the subgroup of $F^{\ast }$ consisting of reduced norms of $D$ is exactly the kernel of the cup product map $\lambda \in F^{\ast }\mapsto (D)\cup (\lambda )\in H^3(F,{\mathbb{Q}}_p/{\mathbb{Z}}_p(2))$, if either $D$ is tamely ramified or of period $p$. This gives a $p$-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-$p$ degree.

We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the $u$-invariant, the reduced stability index and the symbol lengths for Galois cohomology groups with coefficients in ${\mu }_2=\{+1,-1\}$, as well as a new invariant called the splitting height.

We describe the group of continous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify 𝔽-forms of these superalgebras over any field 𝔽 of characteristic zero.

Let F be a number field, abelian over ℚ, and fix a prime p ≠ 2. Consider the cyclotomic ℤ_p-extension F_∞/F and denote F_n the nth finite subfield and C_n its group of circular units as defined by Sinnott. Then the Galois groups G_m,n = Gal(F_m/F_n) act naturally on the C_m's (for any m ≥ n ≫ 0). We compute the Tate cohomology groups for i = -1,0 without assuming anything else neither on F nor on p.

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for , where K = ℚ(ζ_n), G = Gal(K/ℚ), 𝔽₂ = ℤ/2ℤ and U_K is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζ_n) such that . Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687–2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζ_n) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζ_n)_v with v | p but not a square in ℚ(ζ_n), if n has more than three distinct prime factors.

Let $E$ be a $\mathbb{Q}$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb{Q}$-curve. We show that the question has a positive answer if and only if $E$ has a model that is completely defined over an abelian number field. Next, if $E$ is completely defined over a quadratic or biquadratic number field $L$, we classify all strongly modular twists of $E$ over $L$ in terms of the arithmetic of $L$. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of $L$.

Let $N/K$ be a Galois extension of $p$-adic number fields and let $V$ be a de Rham representation of the absolute Galois group $G_K$ of $K$. In the case $V={\mathbb{Q}}_p(1)$, the equivariant local $𝜖$-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin $L$-functions and it can be formulated as the vanishing of a certain element $R_{N/K}$ in $K_0({\mathbb{Z}}_p[G],{\mathbb{Q}}_p^c[G])$; a similar approach can be followed also in the case of unramified twists ${\mathbb{Q}}_p(1)({\rho }^{\mathrm{\text{nr}}})$ of ${\mathbb{Q}}_p(1)$ (see [W. Bley and A. Cobbe, The equivariant local $𝜀$-constant conjecture for unramified twists of ${\mathbb{Z}}_p(1)$, Acta Arith.178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx.28(2) (2016) 485–521]). One of the main technical difficulties in the computation of $R_{N/K}$ arises from the so-called cohomological term $C_{N/K}$, which requires the construction of a bounded complex $C_{N,T}^{•}$ of cohomologically trivial modules which represents $R\mathrm{\Gamma }(N,T)$ for a full $G_K$-stable ${\mathbb{Z}}_p$-sublattice $T$ of $V$. In this paper, we generalize the construction of $C_{N,T}^{•}$ in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local $𝜀$-constant conjecture for unramified twists of ${\mathbb{Z}}_p(1)$, Acta Arith.178(4) (2017) 313–383] to the case of a higher dimensional $T$.

We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to ${𝔽}_p^{\oplus r}$ ($r\le 2$), we show that the product of the restriction maps for places in a set of places $S$ is injective if the Dirichlet density of $S$ is strictly larger than $1-p^{-r}$. We give applications to the local–global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.

In their 2019 paper, Lee and Park presented a formula for the arithmetic Chern–Simons invariant. This formula gives a relation between this invariant and the local Hasse invariants at certain finite primes. Given a number field having real embeddings, we present alternative formulas to give relations between the arithmetic Chern–Simons invariant and the local Hasse invariants at certain primes including all real ones. As an application, we propose a new problem which concerns the existence of a certain $3$-cocycle. If the answer to this problem is positive, the obtained statement is an analogue of the Albert–Brauer–Hasse–Noether theorem.

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.

Let F be a number field and $p\ge 7$ a rational prime. We obtain a simple descent criterion characterizing those projective Galois representations $\overline{\rho }:G_F\to {\mathrm{\text{PGL}}}_2({𝔽}_p)$ for which the corresponding twist $X_{\overline{\rho }}(p)$ of the principal modular curve of level p is defined over $\mathbb{Q}$. We also give a more concrete version of this result for representations which arise from elliptic curves over cyclic number fields.

In 1962 Serre posed a conjecture, now referred to as Conjecture II, which states that principal homogeneous spaces under semisimple simply connected linear algebraic groups over perfect fields of cohomological dimension two have rational points. In this talk, after summarising the status of Conjecture II, we shall discuss progress concerning the study of principal homogeneous spaces under linear algebraic groups over function fields of two-dimensional schemes: surfaces over algebraically closed fields, strict Henselian two dimensional local domains and arithmetic surfaces that are relative curves over p-adic integers.

Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions [BuR97]. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry.

The goal of this paper is to survey some of this research. I have tried to explain the underlying ideas informally through motivational remarks, examples and proof outlines (often in special cases, where the argument is more transparent), referring an interested reader to the literature for a more detailed treatment. The sections are arranged in rough chronological order, from the definition of essential dimension to open problems.

Let k be a p-adic field and K a function field of a curve over k. It was proved in ([PS3]) that if p ≠ 2, then the u-invariant of K is 8. Let l be a prime number not equal to p. Suppose that K contains a primitive l^th root of unity. It was also proved that every element in H³(K,ℤ/lℤ) is a symbol ([PS3]) and that every element in H²(K,ℤ/lℤ) is a sum of two symbols ([Su]). In this article we discuss these results and explain how the Galois cohomology methods used in the proof lead to consequences beyond the u-invariant computation.