The class number one problem for the dihedral and dicyclic CM-fields
We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.
The class number one problem for the non-abelian normal CM-fields of degree 24 and 40
The class number one problem for the non-normal sextic CM-fields. Part 2
The computation of certain numbers using a ruler and compass.
The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves
The determination of parabolic points in modular and extended modular groups by continued fractions.
The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives
We connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.
The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields
1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of . We know by the Ferrero-Washington...
The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated
Let be a numerical semigroup. In this work we show that is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set for a given As a consequence, we obtain another algorithm that computes all the elements of with a fixed genus.
The Ljunggren equation revisited
We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.
The minimal resultant locus
Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map factors through a function on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in , or on a segment, and the minimal resultant locus is contained in the tree in spanned by the fixed points and poles...
The narrow class groups of some ℤₚ-extensions over the rationals
The non-abelian normal CM-fields of degree 36 with class number one
The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class groups of exponent
The number Γ(k) in Waring's problem
The -adic valuations of sequences counting alternating sign matrices.
The R₂ measure for totally positive algebraic integers
Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates are positive real numbers. We study the set ₂ of the quantities . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.
The extensions of degree 6 with minimum discriminant.
The size of the fundamental solutions of consecutive Pell equations.
The splitting of primes in division fields of elliptic curves.