Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $Ψ$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $κ$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{κΨ},k+j)$ is a constant independent of $κ$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.
In this article, we follow Hida's approach to establish an analogue of Washington's theorem on the non-vanishing modulo p of Hecke L-values for CM fields with anticyclotomic twists.
Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n). We show that a normalized form of the generating function of spt(n) is an eigenform modulo 32 for the Hecke operators T(l^2) for primes l >= 5 with l congruent to 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for l congruent to? 13, 23 (mod 24).
Modular forms mod 2 : structure of the Hecke ring We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x, y]], where x = T3 and y = T5.
We continue the study of strong, weak, and $dc$-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level $1$ modulo $128$, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level $1$ modulo $9$, reducing the question to the finiteness of a single e…
The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization of T^{N,chi}_{p,k}(x) mod l as k varies, and we explicitly calculate those factorizations for N=1 and small l. These factorizations are used to deduce the irreducibility of certain T^{1,1}_{q,k}(x) from the irreducibility of T^{1,1}_{2,k}(x).
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
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 one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$.
We present a collection of results on a conjecture of Jannsen about the $p$-adic realizations associated to Hecke characters over an imaginary quadratic field $K$ of class number 1. The conjecture is easy to check for Galois groups purely of local type. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field $K$ at $p$, which is related to the property that a global Galois group is purely of local type. Without this regularity assumption at $p$, we present a review of the known situations in the critical case and in the non-critical case for the realiza…
The nilpotence order of the mod 2 Hecke operators. Let $Δ=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form f modulo 2 of level 1 is a polynomial in $Δ$. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $Δ$ whose degree is smaller than the degree of f, so that Tp is nilpotent. The order of nilpotence of f is defined as the smallest integer g = g(f) such that, for every family of g odd primes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds. We show how one can compute expl…
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized form of the generating function for spt(n) is an eigenform modulo 72 for the Hecke operators T(p^2) for primes p > 3, and an eigenform modulo t for t = 5, 7 or 13 provided that (t, 6p) = 1. The result for the modulus 3 was observed earlier by the author and considered by Ono and Folsom. Similar congruences for higher powers of t (namely 5^6, 7^4 and 13^2) occur…
We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.
Let $-D < -4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\Bbb Q(\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\Bbb Q(\sqrt{-D})$. Let $χ$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\Bbb Q(\sqrt{-D})$ by the Kronecker's symbol $n\mapsto(\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,χ)$ at its central point $s=k$ is determined by its root number when $|d| \leq c(ε)D^{{1/24}-ε}$…
We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level density for Hecke--Maass forms over $F$ of square-free level $\mathfrak{q}$, as $\mathrm{N}(\mathfrak{q})$ tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over $F$ of full level, we prove similar statements for the $1$- and $2$-level densities, as the Laplace eigenvalues tends to infinity.
We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.
Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…
In this paper we prove the weak local Tamagawa number conjecture for the remaining non-critical cases for the motives associated to Hecke characters $ψ_θ:\mathbb{A}_K\to K^*$ of the author's previous paper, where $K$ is an imaginary quadratic field with $cl(K)=1$, under certain restrictions which originate mainly from the Iwasawa theory of imaginary quadratic fields.
We use a method of Buzzard to study p-adic families of different types of modular forms - classical, over imaginary quadratic fields and totally real fields. In the case of totally real fields of even degree, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.
We investigate the distribution of $αp$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[ω]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVertαp\rVert_ω< \mathrm{N}(p)^{-1/8+ε}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_ω$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method …
Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension $K/\Q$. Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ an odd unramified prime. We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$ over the $\Z_l$-anticylotomic extension of $K$. The result is an evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of two, it provides a refinement of the results of Cornut and Vatsal on the non-triviality of Heegner points over the …
Let $NT(m, k, n)$ denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results, Chern obtain congruences for $NT(m, k, n)$ modulo 11 and 13. More recently, the second author use the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes $\ell \ge 7$. In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.
We study the arithmetic of Eisenstein cohomology classes (in the sense of G. Harder) for symmetric spaces associated to GL_2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of a special L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. We also prove under some additional assumptions that the restriction of the classes to the boundary of the Borel-Serre compactification of the spaces is integral. Such classes are interesting for their use in congruences with cuspidal class…
We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\bar{p}(n)$ and two smallest parts functions: $\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$ for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function $p(n)$ in 1966 and Garvan for the smallest parts function $\operatorname{spt}(n)$ in 2010. The proofs depend on congruences between the generating functions for $\bar{p}(n)$, $\bar{\operatorname{spt1}}(n),$ and $\operatorname{M2spt}(n)$ and eigenfor…
The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,χ}_{l,k} for all primes l, all weights k>1 and all characters χtaking values in {+1,-1} splits completely modulo p has density 0, unconditionally for p=2 and under the Cohen-Lenstra heuristics for odd p. The method of proof is based on the construction of suitable dihedral modular forms.
Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates.
We generalize Sczech's Eisenstein cocycle for $\mathrm{GL}(n)$ over totally real extensions of $\mathbb{Q}$ to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L-functions.
Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,χ_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $σ>1/2$, we prove an asymptotic distribution function $F_σ$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,χ_c)L(s,\overlineχ_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_σ$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$.
Let $K$ be an imaginary quadratic number field and let $L(s,ξ_{\ell})$ denote the Hecke $L$-function to an angular character $ξ_{\ell}$ with frequency $\ell$. We detect values of $\log |L(\tfrac12,ξ_{\ell})|$ with size at least $(\sqrt{2} + o_K(1))(\log X / \log \log X)^{1/2}$ along each dyadic range $X \leqslant \ell \leqslant 2X$. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the complex embedding of $K$.
In this paper, we will talk about the titled elliptic curve defined over imaginary quadratic fields such as $\mathbb{Q}(\sqrt{-q})$, where $q$ is congruent to 3 modulo 8 and $(p,q)=1$.
We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck constant N=1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are << N^epsilon for all…
For the anticyclotomic p-adic Rankin--Selberg L-function attached to a fixed Hecke eigenform and an imaginary quadratic field we introduce the second p-adic variable by considering Hida families of Hecke eigenforms parametrized by the weight.
We study the supersingular curves on Picard modular surfaces modulo a prime $p$ which is inert in the underlying quadratic imaginary field. We analyze the automorphic vector bundles in characteristic $p$, and as an application derive a formula relating the number of irreducible components in the supersingular locus to the second Chern class of the surface.
Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number formula for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$.
In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrtε)/\mathbb{Q}(\sqrt{d})$ (where $ε$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb…
