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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.
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.
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 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.
In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.
In this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple
$\mathbb{Z}_{p}$
-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the
$\unicode[STIX]{x1D707}=0$
conjecture for cyclotomic
$\mathbb{Z}_{p}$
-extensions. We show that in certain non-cyclotomic
$\mathbb{Z}_{p}$
-towers, the
$\unicode[STIX]{x1D707}$
-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified
$p$
-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-
$p$
-adic analytic.
It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.
Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.