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Academy of Mathematics and Systems Science, CAS Colloquia & Seminars：Positivity and model complete fields
正性 模型完备域 有理数域
2023/11/13
In recent work with Will Johnson, we have shown that given a curve C over the rational number field Q with genus at least 2 and C(Q) is empty, the class of fields K of characteristic 0 such that C(K) ...
Random Galois extensions of Hilbertian fields
Random Galois extensions Hilbertian fields Number Theory
2012/6/21
Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.
Mersenne Primes in Real Quadratic Fields
Mersenne Primes Real Quadratic Fields Number Theory
2012/5/24
The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenn...
Counting rational points over number fields on a singular cubic surface
rational points over number fields singular cubic surface Number Theory
2012/4/18
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successful...
On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields
Surjectivity Galois Representations Elliptic Curves over Number Fields Number Theory
2012/4/17
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states t...
Given a finite set $S$ of places of a number field, we prove that the field of totally $S$-adic algebraic numbers is not Hilbertian.
Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields
Non-existence of elliptic curves quadratic fields Number Theory
2011/9/19
Abstract: We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields.
Graphs associated with the map $x \mapsto x+x^{-1}$ in finite fields of characteristic two
Graphs finite fields of characteristic two Number Theory
2011/9/19
Abstract: In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic two. Formulas are given for the length ...
Progress Towards Counting D_5 Quintic Fields
Progress Towards D_5 Quintic Fields Number Theory
2011/9/15
Abstract: Let $N(5,D_5,X)$ be the number of quintic number fields whose Galois closure has Galois group $D_5$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(5,D_...
Representation of powers by polynomials over function fields and a problem of Logic
problem of Logic polynomials over function fields Number Theory
2011/9/15
Abstract: We solve a generalization of B\"uchi's problem in any exponent for function fields, and briefly discuss some consequences on undecidability. This provides the first example where this proble...
Additive decompositions induced by multiplicative characters over finite fields
Characters Residuacity Finite Fields
2011/8/26
Abstract: In 1952, Perron showed that quadratic residues in a field of prime order satisfy certain ad- ditive properties. This result has been generalized in different directions, and our contribution...
Upper Bounds for the Number of Number Fields with Alternating Galois Group
Number Fields Alternating Galois Group Number Theory
2011/8/26
Abstract: We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect th...
The Gross - Kuz'min conjecture for CM fields
The Gross - Kuz'min conjecture CM fields Number Theory
2011/8/26
Abstract: Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension $\K_{\infty}/\K$ of a CM number field $\K$....
Exact Covering Systems in Number Fields
Exact covering systems Lattice parallelotopes Chinese Remainder Theorem
2011/2/22
It is well known that in an exact covering system in Z, the biggest modulus must be repeated. Very recently, S. Kim proved an analogous result for certain quadratic fields. In this paper, we prove tha...
The theory of valued difference fields $(K, \sigma, v)$ depends on how the valuation $v$ interacts with the automorphism $\sigma$. Two special cases have already been worked out - the isometric case,...