Class Groups of Number Fields and Related Topics

In this article, we present streamlined proofs of results of Ankeny, Artin, and Chowla concerning the fundamental unit of the real quadratic field \(\mathbb {Q}(\sqrt{p})\) for primes \(p\equiv 1 \left( \textrm{mod}\ 4\right) \) while providing a generalization of their conjecture. Using our generalization, we relate Fermat quotients of quadratic non-residues \(\left( \textrm{mod}\ p\right) \) to sums of harmonic numbers.

In this note, we study the mod p behavior of Kato’s Euler systems and fine Selmer groups for an elliptic curve with good reduction at a prime \(p \ge 5\). We show that we observe a version of the \(\lambda \)-invariant formula for fine Selmer groups for congruent elliptic curves holds, as in the work of Greenberg–Vatsal, and formulate a mod p version of Kato’s main conjecture.

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In [5] we generalized these methods and results to Pell’s equation. We find a similar group structure and count on the number of solutions for a given z to \(x^2 + Dy^2 = z^2\) when D is 1 or 2 modulo 4 and the class group of \(\mathbb {Q}[\sqrt{-D}]\) is a free \(\ensuremath {\mathbb {Z}}_2\) module, which always happens if the class number is at most 2. In this paper we discuss the main results of [5] using some concrete examples in the case of \(D=105\).

It is widely believed that the parity of the partition function p(n) is “random”. Contrary to this expectation, in this note we prove the existence of infinitely many congruence relations modulo 4 among its values. For each square-free integer \(1<D\equiv 23\pmod {24},\) we construct a weight 2 meromorphic modular form that is congruent modulo 4 to a certain twisted generating function for the numbers \(p\big (\frac{Dm^2+1}{24}\big )\pmod 4\). We prove the existence of infinitely many linear dependence congruences modulo 4 among suitable sets of holomorphic normalizations of these series. These results rely on the theory of class numbers and Hilbert class polynomials, and generalized twisted Borcherds products developed by Bruinier and the author.

For an integer n, a set of distinct nonzero integers \(\{a_1, a_2, ... , a_m\}\) such that \(a_i a_j + n\) is a perfect square for all \(1 \le i < j \le m\), is called a Diophantine m-tuple with the property D(n) or simply a D(n)-set. D(1)-sets are also called Diophantine m-tuples. The first Diophantine quadruple, the set \(\{1,3,8,120\}\) was found by Fermat. He, Togbé and Ziegler proved in 2019 that there does not exist a Diophantine quintuple. On the other hand, it is known that there exist infinitely many rational Diophantine sextuples. When considering D(n)-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property D(n) for several different n’s. For example, \(\{8,21,55\}\) is a D(1)-triple and D(4321)-triple. In joint work with Adžaga, Kreso and Tadić, we presented several families of Diophantine triples, which are D(n)-sets for two distinct n’s with \(n \ne 1\). In joint work with Petričević we proved that there are infinitely many (essentially different) quadruples which are simultaneously \(D(n_1)\)-quadruples and \(D(n_2)\)-quadruples with \(n_1 \ne n_2\). Moreover, the elements in some of these quadruples are squares, so they are also D(0)-quadruples. E.g. \(\{54^2, 100^2, 168^2, 364^2\}\) is a \(D(8190^2)\), \(D(40320^2)\) and D(0)-quadruple. In recent joint work with Kazalicki and Petričević, we considered D(n)-quintuples with square elements (so they are also D(0)-quintuples). We proved that there are infinitely many such quintuples. One example is a \(D(480480^2)\)-quintuple \(\{225^2, 286^2, 819^2, 1408^2, 2548^2\}\). In this survey paper, we describe methods used in constructions of mentioned triples, quadruples and quintuples.

The investigation of the ideal class group \(Cl_K\) of an algebraic number field K is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of research on many topics linked to quadratic field class groups notably an intriguing aspect is the divisibility of the class numbers. This article discusses a few recent results on the divisibility of class numbers and the Iizuka conjecture. We also discuss the quantitative aspect of the Iizuka conjecture.

In this article, we first discuss congruent numbers and certain recent results concerning their generalizations. Then we review some recent results concerning the 2-part of the class group of \(\mathbb {Q}(\sqrt{\pm n})\) for a congruent number n. We conclude with a sufficient congruence condition for a prime p so that 2p is non-congruent.

Let \(p\equiv 5\pmod 8\) be a prime number and d any positive square-free integer. The main goal of this note is to give explicitly the 2-rank of the class group of the imaginary triquadratic number fields \(\mathbb {K}=\mathbb {Q}(i,\sqrt{p}, \sqrt{d})\).

Let \( P_{n} \), and \( Q_{n} \) be the \(n\mathrm{{th}}\) Pell and Pell-Lucas numbers, respectively. In this paper, we prove that \(P_{k}\) never divides neither \(P^{2}_{x}+P_{x}+1\) nor \(Q^{2}_{x}+Q_{x}+1\), for \(k\ge 2\) and that \(Q_{k}\) never divides \(P^{2}_{x}+P_{x}+1\), for any nonnegative integer k and for any nonnegative integer x.

We completely solve the Diophantine equation \(x^2+5^a7^b 11^c 19^d=4y^n\) under the condition \(n\ge 3\), \(x,\,y>0\), and \(\gcd (x,\,y)=1\).

The Pólya group of a number field is a particular subgroup of the ideal class group of that number field. In this article, we discuss some recent results on Pólya groups of number fields, their connection with the ring of integer-valued polynomials, and touches upon some results on number fields having large Pólya groups. We include the proof of Zantema’s theorem which laid the foundation to determine the Pólya groups of many finite Galois extensions over \(\mathbb {Q}\). At the end, we provide an elementary proof of a weaker version of a recent result of Cherubini et al.

Let K be any non-totally real number field and \(\mathcal {O}_K\) be its ring of integers. Let \(m_K\) be the smallest positive integer such that every element of \(m_K \mathcal {O}_K \) can be written as sums of integral squares. We have shown that \(m_K \le 4\).

Let \(k\ge 2\). A generalization of the well-known Lucas sequence is the k-Lucas sequences. For this sequence, the first k terms are \(0,\ldots ,0,2,1\) and each term afterwards is the sum of the preceding k terms. In this paper, we show that there is no k-Lucas written as product of two Fermat numbers.

This survey article deals with the Diophantine m-tuples in positive integers as well as in some commutative rings with unity. In the first section, we give a brief history of the Diophantine m-tuples with the property D(n). We focus on these two problems: how many ways we can extend a Diophantine pair or triple to Diophantine quadruple; and how large a Diophantine m-tuple can be. In the last section, we generalize the concept of Diophantine m-tuples from positive integers to any commutative ring of unity and discuss some results on the line of the above problems.

In this paper, we explicitly construct Carmichael integers with 8, 9 and 10 prime factors, and Carmichael integers of degree 1 and 2 with 3 and 4 prime factors, respectively. We also study Carmichael ideals in number fields and address a few quantitative results.