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 $\mathrm {GL}_2$-type with real multiplication over $\mathbb Z_p$-extensions of number fields, where v is a prime lying above p. Building on the work of Iovita and Pollack in the case of elliptic curves, under precise ramification and splitting conditions on p, we construct explicit systems of local points using the theory of Lubin–Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell–Weil groups are bounded over any subextensions of the ${\mathbb Z}_p$-extension and provide an asymptotic formula for the growth of the v-primary part of the Tate–Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on p-supersingular elliptic curves.

Let $n\ge 1$, $r\ge 0$, and $s\ge 0$ be integers satisfying $4+r+3 s\le 3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb {Q}(\sqrt {f_1(D)}), \ldots , \mathbb {Q}(\sqrt {f_r(D)})$ and $\mathbb {Q}(\sqrt {-f_{r+1}(D)}), \ldots , \mathbb {Q}(\sqrt {-f_{r+s}(D)})$ is simultaneously less than n. For a positive integer $k $, let $g_1,\dots ,g_k\in \mathbb {Q}[x]$ be polynomials taking integer values at integers with $g_i(0)=0$. We also prove that there exist positive integers $a,d$, such that each $a+g_i(d)$ is a fundamental discriminant and the 3-rank of the class group of each quadratic field $\mathbb {Q}(\sqrt {a+g_1(d)}), \ldots ,\mathbb {Q}(\sqrt {a+g_k(d)})$ is simultaneously less than n. Moreover, these discriminants can be chosen all positive or all negative, giving either all real or all imaginary quadratic fields $\mathbb {Q}(\sqrt {a+g_i(d)})$.

In 1972, Heilbronn introduced the notion of virtual characters and used it to study simple real zeros of Dedekind zeta functions. One of the consequences of his elegant work is the following. Let $\mathrm{K}/ \mathrm{F}$ be a Galois extension of number fields of odd degree. Then any real simple zero of $\zeta_{\mathrm{K}}$ is necessarily a simple zero of $\zeta_{\mathrm{F}}$. The ethos of this paper is to carry out investigation for arbitrary odd order real zeros. While the Riemann zeta function is conjectured to have only simple zeros, the same does not hold for arbitrary Dedekind zeta functions. One of the consequences of our work is that any Galois number field K of odd degree cannot have a non-trivial odd order real zero. Such parity is at least in conformity with extended Riemann hypothesis as the order of vanishing of the Dedekind zeta function $\zeta_{\mathrm{K}}$ at $1/2$ is necessarily even. We also indicate, via a number of illustrative examples (see Remarks 1·1 and 1·2), that in some sense our results are optimal.

In this paper, we study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$. Motivated by analogies with Goldfeld’s conjecture on ranks in quadratic twist families of elliptic curves, we investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg’s local conditions under congruences of residual Galois representations. Let X be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form f of weight 2 and optimal level. We count the number of level-raising modular forms g of weight 2 that are congruent to f modulo p, with level $N_g\leq X$, such that the p-rank of the Selmer groups of g equals that of f. Under some mild assumptions on $\bar{\rho}$, we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$, for an explicit constant $\alpha \gt 0$. The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of a transfer homomorphism implies that the order and the associated Dedekind domain share the same arithmetic properties. This is not the case for arbitrary orders in Dedekind domains.

Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$, with good ordinary reduction at $p$. We compare the $\Lambda$-ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$-extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$. This generalizes and complements previous work over ${\mathbb{Z}}_p$-extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$-extensions of $K$ which are contained in $\mathbb{L}_\infty$.

Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn’s criterion applies to a polynomial f if it applies to the number field $K=\mathbb {Q}[x]/(f)$ generated by f. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd (p-1,n)=1$. Let $\mathcal {F}_{p,n}$ denote the collection of monic polynomials $f\in \mathbb {Z}[x]$ of degree n that are Eisenstein at the prime p. Order the polynomials by the natural height $\operatorname {\mathrm {Ht}}(f)$. Define $\delta _{p,n}(X)$ to be the proportion of polynomials $f\in \mathcal {F}_{p,n}$ with $\operatorname {\mathrm {Ht}}(f)\leq X$ for which Heilbronn’s criterion applies. Then,

$$ \begin{align*} \liminf_{X\to\infty} \delta_{p,n}(X)\geq \max\{\tfrac{2}{27},\;1-\varepsilon(p)\}, \end{align*} $$

where $\varepsilon (p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to \infty $ uniformly in n. We also give a version of this result where we weaken the condition on $\gcd (p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree n fail to generate norm-Euclidean fields.

The notion of $\theta $-congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $-congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $-congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103(2) (2021), 218–229] established criteria for numbers to be $\theta $-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $-congruent and properly $\theta $-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$.

Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$. We provide a method to use the t-rank of the class group of K to obtain a lower bound on the t-rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K. As an application, for any odd integer $t> 1$, we provide a family of cyclotomic fields whose class groups have t-rank at least $2$. Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$, where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$-th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.

This paper studies the $4$-ranks of narrow class groups in certain families of quadratic fields. We prove that for any positive integer n, there exists an integer $s_\lambda (n)$ depending on n and the sign $\lambda $ of the fundamental discriminant D, such that for any choice of $s_\lambda (n)$ integers $t_1, \ldots , t_{s_\lambda (n)}$, there are infinitely many D for which the narrow class group of $\mathbb {Q}(\sqrt {D + t_i})$ has $4$-rank bounded by n for all i. This result extends previous work on $3$-ranks to the case of $4$-ranks.

Given a nonzero integer n, Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull., to appear] classified all polynomials $x^2+ax+b\in {\mathbb {Z}}[x]$ for which the polynomial Pell equation $P^2-(x^2+ax+b)Q^2=n$ has solutions ${P,Q\in {\mathbb {Z}}[x]}$ with $Q\neq 0$. We generalise their work to the equation $P^2-(f^2+af+b)Q^2=nR$, where f is a fixed polynomial in ${\mathbb {Z}}[x]$. As an application of our results, we study the equation $P^2-D(f)Q^2=n$, where D is a monic, quartic and non square-free polynomial in ${\mathbb {Z}}[x]$. This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].

Let A be an abelian variety defined over a global function field F and let p be a prime distinct from the characteristic of F. Let $F_\infty $ be a p-adic Lie extension of F that contains the cyclotomic $\mathbb {Z}_p$-extension $F^{\mathrm {cyc}}$ of F. In this paper, we investigate the structure of the p-primary Selmer group $\mathrm {Sel}(A/F_\infty )$ of A over $F_\infty $. We prove the $\mathfrak {M}_H(G)$-conjecture for $A/F_\infty $. Furthermore, we show that both the $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F^{\mathrm {cyc}})$ and the generalized $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F_\infty )$ are zero, thereby proving Mazur’s conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Assuming the finiteness of the Tate–Shafarevich group, we establish that this corank equals the order of vanishing of the L-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi—originally proved for elliptic curves without complex multiplication—to abelian varieties over global function fields. This is achieved by adapting the notion of generalized Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalized Euler characteristic of $\mathrm {Sel}(A/F_\infty )$ to the Euler characteristic of $\mathrm {Sel}(A/F^{\mathrm {cyc}})$.

Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F. We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$-extensions of a complex multiplication (CM) extension of F. We propose an extension of Mazur’s growth number conjecture for elliptic curves to this new setting. We provide evidence supporting an affirmative answer by studying special cases of this problem, generalising previous results on elliptic curves and imaginary quadratic fields.

In this paper, we study partitions of totally positive integral elements $ \alpha $ in a real quadratic field $ K $. We prove that for a fixed integer $ m \geq 1 $, an element with $ m $ partition exists in almost all $ K $. We also obtain an upper bound for the norm of $\alpha$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $\alpha$’s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $ m $ partitions for $ 1 \leq m \leq 7 $.

The descent method is one of the approaches to study the Brauer–Manin obstruction to the local–global principle and to weak approximation on varieties over number fields, by reducing the problem to ‘descent varieties’. In recent lecture notes by Wittenberg, he formulated a ‘descent conjecture’ for torsors under linear algebraic groups. The present article gives a proof of this conjecture in the case of connected groups, generalizing the toric case from the previous work of Harpaz–Wittenberg. As an application, we deduce directly from Sansuc’s work the theorem of Borovoi for homogeneous spaces of connected linear algebraic groups with connected stabilizers. We are also able to reduce the general case to the case of finite (étale) torsors. When the set of rational points is replaced by the Chow group of zero-cycles, an analogue of the above conjecture for arbitrary linear algebraic groups is proved.

Let K be an imaginary quadratic field, and let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication by $\mathcal {O}_K$. Let $K_\infty /K$ be the anticyclotomic $\mathbb {Z}_p$-extension of K and $K_n$ be the intermediate layers. Under additional assumptions on Kobayashi’s signed Selmer groups, we prove an asymptotic formula for .

Except for a limited number of cases, a complete classification of the Diophantine (i.e., positive existentially definable) sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine. For instance, we show that the relation of equality of degrees is not Diophantine over the field of complex rational functions in one variable and, in the same structure, we show that certain family of relations that approximates the valuation ring at infinity is not Diophantine either.

The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$. To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$. This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$, empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

A 2009 article of Allcock and Vaaler explored the $\mathbb {Q}$-vector space $\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$, showing how to represent it as part of a function space on the places of $\overline {\mathbb {Q}}$. We establish a representation theorem for the $\mathbb {R}$-vector space of $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {R}$, enabling us to classify extensions to $\mathcal {G}$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {Q}$, i.e., elements of the algebraic dual of $\mathcal {G}$. Our results make heavy use of Dirichlet’s S-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.

Let K be an infinite field. If $\alpha $ and $\beta $ are algebraic and separable elements over K, then by the primitive element theorem, it is well known that $\alpha +u\beta $ is a primitive element for $K(\alpha , \beta )$ for all but finitely many elements $u\in K$. If we let

$$ \begin{align*}\xi_K(\alpha, \beta) = \{u\in K : K(\alpha, \beta) \ne K(\alpha+u\beta)\}\end{align*} $$

be the exceptional set, then by the primitive element theorem, $|\xi _K(\alpha , \beta )| < \infty $. Dubickas [‘An effective version of the primitive element theorem’, Indian J. Pure Appl. Math. 53(3) (2022), 720–726] estimated the size of this set when $K = \mathbb {Q}$. We take K to be a finite extension over $\mathbb {Q}$ or $\mathbb {Q}_p$, the field of p-adic numbers for some prime p, and estimate the size of the exceptional set.