We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes.
This paper aims at studying the Iwasawa $λ$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension, where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refi…
The Kida's formula in classical Iwasawa theory relates the Iwasawa $λ$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $λ$-invariants of Selmer groups under an appropriate $μ=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $μ\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.
Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M.~Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on $λ$-invariants in a $p$-extension of $\mathbb{Z}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa's second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched $\mathbb{Z}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the…
We prove a slight generalization of Iwasawa's `Riemann-Hurwitz' formula for number fields and use it to generalize Ferrero's and Kida's well-known computations of Iwasawa λ-invariants for the cyclotomic Z_2-extensions of imaginary quadratic number fields. In particular, we show that if p is a Fermat prime, then similar computations of Iwasawa λ-invariants hold for certain imaginary quadratic extensions of the unique subfield k of Q(ζ_{p^2}) such that [k:Q] = p. In fact, we actually prove more by explicitly computing cohomology groups of principal ideals. The computation of lambda invariants ob…
We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weak…
The paper proposes a polynomial formula for solution quadratic congruences in $\mathbb{Z}_p$. This formula gives the correct answer for quadratic residue and zeroes for quadratic nonresidue. The general form of the formula for $p=3 \;\rm{mod}\,4$, $p=5 \;\rm{mod}\,8$ and for $p=9 \;\rm{mod}\,16$ are suggested.
In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups $Γ_0(N)$ and $Γ_1(N)$, obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature, and hold without any restriction on the index of the operators. We give two applications of the trace formula for $Γ_1(N)$: we determine explicit trace forms for $Γ_0(4)$ wi…
Much like the important work of Hardy and Ramanujan proving the asymptotic formula for the partition function, Auluck and Wright gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences on a two-parameter family of mixed congruence relations, with parts on one side up to the peak satisfying r (mod m) and parts on the other side -r (mod m). Techniques used in the proofs include Wright's circle method, modular transformations, and bounding of complex integrals.
We give a reciprocity formula for a two-variable sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is approximable from below and show that the reciprocity formula is a simple consequence of this fact.
We solve a few open problems related to a peculiar property of the integer tetration ${^{b}a}$, which is the constancy of its congruence speed for any sufficiently large $b=b(a)$. Assuming radix-$10$ (the well-known decimal numeral system), we provide an explicit formula for the congruence speed $V(a) \in \mathbb{N}_0$ of any $a \in \mathbb{N}-\{0\}$ that is not a multiple of $10$. In particular, for any given $n \in \mathbb{N}$, we prove to be true Ripà's conjecture on the smallest $a$ such that $V(a)=n$. Moreover, for any $a \neq 1 : a \not\equiv 0 \pmod {10}$, we show the existence of infin…
Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maass cusp forms of weight 0 or 1 for the congruence subgroups $Γ_0(q)$, $Γ_1(q)$, and $Γ(q)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.
The aim of this paper is to investigate the trivial zeros of the Katz $p$-adic $L$-functions by the CM congruence. We prove the existence of trivial zeros of the Katz $p$-adic $L$-functions for general CM fields and establish a first derivative formula of the cyclotomic $p$-adic $L$-functions at trivial zeros under some Leopoldt hypothesis. The crucial ingredients in our proof are a special case of $p$-adic Kronecker limit formula for CM fields and a leading term formula of anticyclotomic $p$-adic $L$-functions at trivial zeros via the explicit congruences between CM and non-CM Hida families o…
For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied to obtain new information on the exceptional set of the multiplication table problem in a residue ring modulo $m$ and a new bound for a double trigonometric sum with an exponential function.
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when varying the weight or the level in the trace formula.
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for p-Bernoulli numbers.
We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectu…
We show that there is no simple congruence formula for the number of points of the mod p reduction of a regular model of a smooth proper variety defined over a local field with $\ell$-adic cohomology supported in codimension $\ge κ$.
We define a weighted multiplicity function for closed geodesics of given length on a finite area Riemann surface. These weighted multiplicities appear naturally in the Selberg trace formula, and in particular their mean square plays an important role in the study of statistics of the eigenvalues of the Laplacian on the surface. In the case of the modular domain, E. Bogomolny, F. Leyvraz and C. Schmit gave a formula for the mean square, which was rigorously proved by M. Peter. In this paper we calculate the mean square of weighted multiplicities for some surfaces associated to congruence subgro…
We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves $E$. We show that the number of twists of $E$ up to twisting discriminant $X$ of analytic rank zero (resp. one) is $\gg X/\log^{5/6}X$, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We also prove th…
Let $N_k(n,r,\boldsymbol{a})$ denote the number of incongruent solutions of the quadratic congruence $a_1x_1^2+\ldots+a_kx_k^2\equiv n$ (mod $r$), where $\boldsymbol{a}=(a_1,\ldots,a_k)\in {\Bbb Z}^k$, $n\in {\Bbb Z}$, $r\in {\Bbb N}$. We give short direct proofs for certain less known compact formulas on $N_k(n,r,\boldsymbol{a})$, valid for $r$ odd, which go back to the work of Minkowski, Bachmann and Cohen. We also deduce some other related identities and asymptotic formulas which do not seem to appear in the literature.
Rosen M. gave a determinant formula for relative class numbers for the P-th cyclotomic function fields in the case of the monic irreducible polynomial P, which is regarded as an analogue of the classical Maillet determinant. In this paper, we will give a determinant formula for the relative congruence zeta functions for cyclotomic function fields. Our formula is regarded as a generalization of the determinant formula for the relative class number.
The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n of Pascal's triangle is captured by the so-called Gaussian Formula. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients.
We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Tay…
In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean balls of increasing radius. This formula involves an error term that depends on the first nonzero eigenvalue of the Laplace-Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations, with the error term mentioned above, for several choices of quadratic and her…
Firstly we prove that the Voronoi formula of Miller-Schmid type applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is a multiple of $N$. As an application, we produce a result about the functional equation of $L$-function of the automorphic form on GL(3) twisted by Dirichlet characters. Secondly we prove that a similar formula applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is coprime with $N$.
Let $d(n;\ell_1,M_1,\ell_2,M_2)$ denote the number of factorizations $n=n_1n_2$, where each of the factors $n_i\in\mathbb{N}$ belongs to a prescribed congruence class $\ell_i\bmod M_i\,(i=1,2)$. Let $Δ(x;\ell_1,M_1,\ell_2,M_2)$ be the error term of the asymptotic formula of $\sum\limits_{n\leqslant x}d(n;\ell_1,M_1,\ell_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $Δ(M_1M_2x;\ell_1,M_1,\ell_2,M_2)$ and prove that \begin{equation*} \int_1^TΔ^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)\mathrm{d}x=\frac{1}{32π^4}C_4\Big(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\Big) T…
Let $p>5$ be a fixed prime and assume that $α_1,α_2,α_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $α_1x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $α_1,α_2,α_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula hol…
This article considers the eta power $\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.
We show that Riemann-Hurwitz-style translation formulas obtained by Kuz'min, Kida, Iwasawa, Wingberg et alii for the lambda invariant attached to certain Iwasawa moduli in cyclotomic Z{\ell}-extension of number fields are essentially equivalent. More precisely, we prove that all these formulas,including those stated in terms of representations, resul tidentically for purely algebraic reasons from the arithmetic computation of a suitable Herbrand quotient which it suffices to carry out in the cyclic case of prime degree {\ell}.
We study a mod $p^c$ analog of the notion of transfer for automorphic forms. Instead of existence of eigenforms, such transfers yield congruences between eigenforms but, like transfers, we show that they can be established by a comparison of trace formulas. This rests on the properties of mod $p^c$ reduced multiplicities which count congruences between eigenforms. As an application we construct finite slope $p$-adic {\it continuous} families of Siegel eigenforms using a comparison of trace formulas.
An explicit formula for a weight enumerator of linear-congruence codes is provided. This extends the work of Bibak and Milenkovic [IEEE ISIT (2018) 431-435] addressing the binary case to the non-binary case. Furthermore, the extension simplifies their proof and provides a complete solution to a problem posed by them.
Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes
Institute of Mathematics University of the Philippines Diliman（菲律宾大学达曼数学研究所）；Department of Mathematics and Statistics University of Ottawa（渥太华大学数学与统计系）
Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators …
Let $p\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$. We consider supersingular elliptic curves with a basis of $2$-torsion over $\overline{\mathbf{F}}_p$, or equivalently supersingular Legendre $λ$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $λ$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\frac{F_p(X,X^{p})}{p}$ is in $\mathbf{Z}[X]$. We give a formula for $R(λ)$ if $λ$ is a supersingular. This formula is related to the Manin--Dri…
In this paper we obtain asymptotic formulas for the Fourier coefficients of an infinite family of crank generating functions. Moreover we use this result to show that the crank obeys certain inequalities. This implies that the crank can not explain any partition congruences in the usual way beside the three deduced by Ramanujan.
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are …
