We show that if V and V′ are two p-adic representations of Gal(Q¯p/Qp) whose tensor product is trianguline, then V and V′ are both potentially trianguline.

In this paper, we show that if (Xn,Yn) is the nth solution of the Pell equation X2−dY2=±1 for some non-square d, then given any integer c, the equation c=Xn−2m has at most 2 integer solutions (n,m) with n≥0 and m≥0, except for the only pair (c,d)=(−1,2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as x2+y2+z2 for some

Following the work of Brown, we can canonically associate a family of motivic periods — called the motivic Feynman amplitude — to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown’s “small graphs principle”

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the ‘probabilities’ are integers modulo a prime $p$. The entropy, too, is an integer $\operatorname{mod} p$. Entropy $\operatorname{mod} p$ is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish

Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$‑adic string theory and $p$‑adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed $\operatorname{modulo} \mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to $8$, and general families of identities in

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions

In this paper k always denotes a finite field of characteristic p>0. A variety over k means a reduced scheme separated and of finite type over k. An étale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9]).

The p-rationality of a totally real abelian number field can be checked from the values L(2−p,χ) of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of Q(ζ2l+1)+, the maximal real subfield of Q(ζ2l+1), for Sophie Germain primes l and odd primes p that

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight wPJ2(x,t;α,β)=xα(1−x)βe−tx(1−x), where t∈[0,∞), α>0, β>0 and 0

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference of the two preceding terms where the pluses and minuses follow a certain pattern. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern

Let p≥5 be a prime number, F a finite field of characteristic p and let χ¯ be the mod-p cyclotomic character. Let ρ¯:GQ→GL2(F) be a Galois representation such that the local representation ρ¯↾GQp is flat and irreducible. Further, assume that detρ¯=χ¯. The celebrated theorem of Khare and Wintenberger asserts that if ρ¯ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform

Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially large integer set S. This implies that if S is a set of small multiplicative doubling then the size of any arithmetic progression in S, beginning at zero, is at most O(log⁡|S|⋅log⁡log⁡|S|). This result can be considered as an integer analogue of Vinogradov's question about the least

Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C‾4k,k(n) is almost always divisible

We propose a function-field analog of Pisot's d-th root conjecture on linear recurrences, and prove it under some “non-triviality” assumption. Besides a recent result of Pasten-Wang on Büchi's d-th power problem, our main tool, which is also developed in this paper, is a function-field analog of a GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such a GCD estimate, we

Let p be a prime number and k a perfect field of characteristic p. In the present paper, we study deformations of finite flat commutative group schemes over k to the ring W of Witt vectors with coefficients in k. We prove that, for a given principally quasi-polarizable p-torsion finite flat commutative group scheme over k, it holds that the group scheme is pseudo-rigid — i.e., roughly speaking, has

In a paper of Kedlaya and Medvedovsky [KM19], the number of distinct dihedral mod 2 modular representations of prime level N was calculated, and a conjecture on the dimension of the space of level N weight 2 modular forms giving rise to each representation was stated. In this paper we prove this conjecture.

The purpose of the present paper is to give an effective version of the noncritical p-tame Belyĭ theorem. That is to say, we compute an explicit bound on the minimal degree of tamely ramified Belyĭ maps in positive characteristic which are unramified at a prescribed finite set of points.

We study the Selmer group associated to a p-ordinary newform f∈S2r(Γ0(N)) over the anticyclotomic Zp-extension of an imaginary quadratic field K/Q. Under certain assumptions, we prove that this Selmer group has no proper Λ-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants

A famous conjecture of Parkin-Shanks predicts that p(n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the ‘‘striking” fact that, if for any A≡±1(mod6) the multipartition function pA(n) has positive odd density

We close a gap in the literature by showing that the full covering groups of the Mathieu group M22 and of its automorphism group Aut(M22) occur as the Galois groups of Q-regular Galois extensions of Q(t).

In 1972 Delange [9] observed in analogy of the classical Erdős-Wintner theorem that q-additive functions f(n) has a distribution function if and only if the two series ∑f(dqj), ∑f(dqj)2 converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the q-ary and Cantor case we focus on the Zeckendorf

Given a minimal model of an elliptic curve, E/K, over a finite extension, K, of Qp for any rational prime, p, and any point P∈E(K) of infinite order, we determine precisely min⁡(v(ϕn(P)),v(ψn2(P))), where v is a normalised valuation on K and ϕn(P) and ψn(P) are polynomials arising from multiplication by n on this model of the curve.

We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4]. This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein

We study the parity of coefficients of classical mock theta functions. Suppose g is a formal power series with integer coefficients, and let c(g;n) be the coefficient of qn in its series expansion. We say that g is of parity type (a,1−a) if c(g;n) takes even values with probability a for n≥0. We show that among the 44 classical mock theta functions, 21 of them are of parity type (1,0). We further conjecture

For any prime p, let Rp be the set of all quadratic residues modulo p. In 2012, Sárközy proved that if p is a sufficiently large prime, then Rp has no 3-decomposition A+B+C=Rp with |A|,|B|,|C|≥2. In this paper, we prove that for any prime p, Rp has no additive 3-decomposition A+B+C=Rp with |A|,|B|,|C|≥2. Furthermore, for any prime p, if A+B=Rp is a 2-decomposition, then 0.17p+1<|A|,|B|<2.8p−6.63. We

In this paper, we extend the recent theorem of G. Heier and A. Levin [HL21] on the generalization of Schmidt's subspace theorem and Cartan's Second Main Theorem in Nevanlinna theory to closed subschemes located in l-subgeneral position, using the generic linear combination technique due to Quang (see [Quang19]).

Let K be a finite Galois extension of Q. We count primes in short intervals represented by the norm of a prime ideal of K satisfying a small sector condition determined by Hecke characters. We also show that such primes are well-distributed in arithmetic progressions in the sense of Bombieri-Vinogradov. This extends previous work of Duke and Coleman.

In [3] André showed that the minimal differential equations of Э-functions and E-functions have a basis of holomorphic solutions at every point except for zero and infinity. In addition, in [5] he observed that if f1(z) is an E-function with rational coefficients such that f1(1)=0, then f1(z)/(z−1) is also an E-function. With this additional result André derived transcendence results for the values

For x irrational, we study the convergence of series of the form ∑n−sf(nx) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal

Let η(s)=∑n=1∞(−1)n+1n−s be the alternating zeta function. For a real number τ we define certain complex numbers bM,m(τ) and consider finite Dirichlet series υM(τ,s)=∑m=1MbM,m(τ)m−s and ηN(τ,s)=∑M=1NυM(τ,s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that ηN(τ,s) approximates η(s) with

We study the maximum gap g (maximum of the differences between any two consecutive exponents) of cyclotomic polynomials. In 2012, Hong, Lee, Lee and Park showed that g(Φp1p2)=p1−1 for primes p2>p1. In 2017, based on numerous calculations, the following generalization was conjectured: g(Φmp)=φ(m) for square free odd m and prime p>m. The main contribution of this paper is a proof of this conjecture.

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population

In this paper, a ridgelized Hotelling’s T2 test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling’s t2 test for pathway analysis in proteomic studies, J. Am. Stat. Assoc.106(496) (2011) 1345–1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment

In this note we derive a non-commutative version of the Wüstholz analytic subgroup theorem in transcendence theory from the original (commutative) theorem and provide an application.

For q an odd prime power, A=Fq[T], and F=Fq(T), let ψ:A→F{τ} be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p=pA be a prime of A of good reduction for ψ, with residue field Fp. We study the growth of the absolute value |Δp| of the discriminant of the Fp-endomorphism ring of the reduction of ψ modulo p and prove that, for all p, |Δp| grows with |p|. Moreover, we

Consider two random vectors C11/2x∈ℝp and C21/2y∈Rq, where the entries of x and y are i.i.d. random variables with mean zero and variance one, and C1 and C2 are respectively, p×p and q×q deterministic population covariance matrices. With n independent samples of (C11/2x,C21/2y), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Zp, in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular

In [CS04], Calegari and Stein studied the congruences between classical cusp forms Sk(Γ0(p)) of prime level and made several conjectures about them. In [AB07] (resp., [BP11]) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.

Suppose an elliptic curve E/Q has good supersingular reduction at a prime p, and satisfies 1+p−#E(Z/pZ)=0, which is always true if p is supersingular and p>3 by the Hasse inequality. Expanding Kobayashi's plus/minus-local points over the cyclotomic Zp-extension of Qp, where π is any element in Cp with positive p-adic valuation, we construct doubly indexed plus/minus local points over Qp(πpn−m,μpn)

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number −1.

Let K=Q(N) be the Nth layer in the cyclotomic Zˆ-extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington, …) prove results on the p-class groups CK. We enlarge “Weber's problem” to the Tate–Shafarevich groups IIIK1≃CKSp (Sp-class group) and IIIK2 having same p-rank as the more easily computable torsion group, TK, of the Galois group

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell–Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin–Mumford conjecture for curves. The proof is built up on our previous work [DGH21].

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields. In this paper, we determine an explicit basis

In this paper we apply Arakelov theory to study the distribution of the Petersson norms of classical cusp forms as well as the distribution of the sup norms of rational functions on adelic subsets of curves. The method in both cases is to study the limiting distribution of the successive minima of norms of global sections of powers of a metrized ample line bundle as one takes increasing powers of the

In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of Q. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of GL4(AQ) which contribute to the cuspidal cohomology of GL4 and are obtained from symmetric cube transfer of automorphic representations of GL2(AQ) of a given weight and with varying level structure. This generalises the recent work of C. Ambi about the similar problem for GL3.

LetPn(y1,…,yn):=∏1≤i

In this paper, we study moments of the central values of quartic Dirichlet L-functions and establish quantitative non-vanishing result for these L-values.

Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture and its blockwise version hold for the alternating groups.

Let A be an abelian variety over a global function field K of characteristic p. We study the μ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his

A Drinfeld module has a 𝔭-adic Tate module not only for every finite place 𝔭 of the coefficient ring but also for 𝔭 = ∞. This was discovered by J.-K. Yu in the form of a representation of the Weil group. Following an insight of Taelman we construct the ∞-adic Tate module by means of the theory of isocrystals. This applies more generally to pure A-motives and to pure F-isocrystals of p-adic cohomology

Let H be a weak Hopf algebra that is a finitely generated module over its affine center. We show that H has finite self-injective dimension and so the Brown–Goodearl conjecture holds in this special weak Hopf setting.

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral D2n or (generalized) quaternion ℍ2n of two-power order. In the spirit of recent work of Fiorilli and Jouve (2020), we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical

We prove that for any proper smooth formal scheme 𝔛 over 𝒪K, where 𝒪K is the ring of integers in a complete discretely valued nonarchimedean extension K of ℚp with perfect residue field k and ramification degree e, the i-th Breuil–Kisin cohomology group and its Hodge–Tate specialization admit nice decompositions when ie < p − 1. Thanks to the comparison theorems in the recent works of Bhatt, Morrow

For a Galois extension K/Q, Pólya group Po(K) of K coincides with the subgroup of strongly ambiguous ideal classes of K, which is controlled by ramification and Galois cohomology. Using Setzer's result [9] on the first cohomology group of units in the real biquadratic fields, we describe the structure of Po(K) for an infinite family of real biquadratic fields K with five ramifications and no quadratic

For any positive integers k,m,n with m≥2 and k≤n, let A(m,n,k) be the sum of all the multiple alternating zeta values of weight mn and depth k with arguments being multiples of m, i.e.,A(m,n,k)=∑s1+s2+…+sk=nsj∈Nζ(ms1‾,ms2‾,…,msk‾). In this note, we obtain the evaluationA(m,n,k)=∑a+b=na,b∈N0(−1)a−k(ak)⋅ζ({m‾}a)⋅ζ⋆({m‾}b), where ζ({m‾}a) and ζ⋆({m‾}b) denote the multiple alternating zeta values and multiple