The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.

Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

Let $\mathbb {F}_q$ be the finite field of q elements. In this paper, we study the vanishing behavior of multizeta values over $\mathbb {F}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums $S_d(k)$ which are polynomials in t. By studying the t-valuation of $S_d(s)$ for $s < 0$ , we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of “greedy element” by Carlitz.

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

We investigate unramified extensions of number fields with prescribed solvable Galois group G and certain extra conditions. In particular, we are interested in the minimal degree of a number field K, Galois over $\mathbb {Q}$ , such that K possesses an unramified G-extension. We improve the best known bounds for the degree of such number fields K for certain classes of solvable groups, in particular for nilpotent groups.

Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Aigner showed in 1934 that nontrivial quadratic solutions to $x^4 + y^4 = 1$ exist only in $\mathbb Q(\sqrt {-7})$ . Following a method of Mordell, we show that nontrivial quadratic solutions to $x^4 + 2^ny^4 = 1$ arise from integer solutions to the equations $X^4 \pm 2^nY^4 = Z^2$ investigated in 1853 by V. A. Lebesgue.

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

The notion of $\theta $ -congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $ -congruent over certain real number fields.

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Waldspurger’s formula gives an identity between the norm of a torus period and an $L$-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding $L$-functions agree, (the norms of) these periods—which occur on different quaternion algebras—are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.