Energy bounds for modular roots and their applications
University of New South Wales（新南威尔士大学）；London Institute for Mathematical Sciences（伦敦数学科学研究所）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Simon Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）
Bryce Kerr, Ilya D. Shkredov, Igor E. Shparlinski, Alexandru Zaharescu
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.
Let $p$ be a prime. We prove that if a modular unit has a $p^{th}$ root that is again a modular unit then the level of that root is at most $p$ times the level of the original unit.
We construct a meromorphic function on the eigencurve that interpolates a square root of the ratio of the central values of two quadratic twists of the $L$-function at classical points.
Let $p$ be a prime number, $p=2^nq+1$, where $q$ is odd. D. Shanks described an algorithm to compute square roots $\pmod{p}$ which needs $O(\log q + n^2)$ modular multiplications. In this note we describe two modifications of this algorithm. The first needs only $O(\log q + n^{3/2})$ modular multiplications, while the second is a parallel algorithm which needs $n$ processors and takes $O(\log q+n)$ time.
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.
Let $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $φ(N)$ and the class number of $\mathbb{Q}(\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.
In this article we perform an extensive study of the spaces of automorphic forms for GL(2) of weight two and level N, for N an ideal in the ring of integers of the quartic CM field generated by the twelfth roots of unity. This study is conducted through the computation of the group cohomology of the congruence subgroup of upper triangular matrices modulo N, and the corresponding Hecke action which this space admits. Combining this Hecke data with the Faltings-Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curv…
In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $\mathbb…
The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime $p$, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at $p$ by analyzing the change of sign under a suitable twist. We also explain the case $p=2$, where twisting is not enough in general.
Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $\rm{SL}_2(\mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $\mathbb H$, and the rationals $\mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta func…
Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalen…
A modular form on an even lattice $M$ of signature $(l,2)$ is called reflective if it vanishes only on quadratic divisors orthogonal to roots of $M$. In this paper we show that every reflective modular form on a lattice of type $2U\oplus L$ induces a root system satisfying certain constrains. As applications, (1) we prove that there is no lattice of signature $(21,2)$ with a reflective modular form and that $2U\oplus D_{20}$ is the unique lattice of signature $(22,2)$ and type $U\oplus K$ which has a reflective Borcherds product; (2) we give an automorphic proof of Shvartsman and Vinberg's the…
Let $f$ be a newform of even weight $2κ$ for $D^\times$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend our theory of $p$-adic measures attached to the power series expansions of $f$ around the Galois orbit of the CM point corresponding to an embedding $K\hookrightarrow D$ to forms with any nebentypus and to $p$ dividing the level of $f$. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic $p$-level str…
We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.
The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a different construction of the $E_8$ root lattice. It also gives an explicit construction of the modular curve $X(13)$.
This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many…
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
For 25 orthogonal groups of signature $(2,n)$ related to the root lattices $A_1$, $2A_1$, $3A_1$, $4A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$, $A_7$, $D_4$, $D_5$, $D_6$, $D_7$, $D_8$, $E_6$, $E_7$, we prove that the algebras of modular forms on symmetric domains of type IV are freely generated by the additive lifts of some special Jacobi forms. The proof is universal and elementary.
We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear sums and to some questions regarding smooth and square-free polynomials in residue classes.
Let p be a prime number, and let f, g, and h be three modular forms of weights $κ$, $λ$, and $μ$ for $SL(2,\Bbb{Z})$. We suppose $κ\geq λ+ μ$. In joint work with Kudla, one of the authors obtained a formula for the normalized {\it square root} of the value at $s = {1/2}(κ+ λ+ μ- 2)$ (the {\it central critical value}) of the triple product $L(s,f,g,h)$. We apply this formula, letting $f$ (and thus $κ$) vary in a $p$-adic analytic family ${\bold f}$ of ordinary modular forms (a Hida family). By modifying Hida's construction of the $p$-adic Rankin-Selberg convolution, we obtain a generalized $p$-…
We study the Witt classes of the modular categories $SO(2r)_{2r}$ associated with quantum groups of type $D_r$ at $4r-2$th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent modulo the subgroup generated by the pointed modular categories. In particular we produce an example of a simple, completely anisotropic modular category that is not pointed whose Witt class has order 2, answering a question of Davydov, Müger, Nikshych and Ostrik. Our results show that the trivial Witt class $[Vec]$ has infinitely many square roots modulo the …
We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.
We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is $\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ and the level is the cube of certain rational integers. This complements a previous result of the second author and extends the bias phenomenon to the number fields. Our method is based on Jacquet-Zagier's trace formula, and the explicit calculation works generally for all real quadratic fields of narrow class number one and for rational cubic levels.
Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant. We give a condition that the modular invariant is a multiple root of the modular polynomial. Consequently, we give a method to reduce the amount of computation in the process of counting th…
We consider double scaled little string theory on $K3$. These theories are labelled by a positive integer $k \ge 2$ and an $ADE$ root lattice with Coxeter number $k$. We count BPS fundamental string states in the holographic dual of this theory using the superconformal field theory $K3 \times \left( \frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/ \mathbb{Z}_k$. We show that the BPS fundamental string states that are counted by the second helicity supertrace of this theory give rise to weight two mixed mock modular forms. We compute the helicity supertraces using two s…
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Appli…
In this paper, we obtain two effective bounds for the $j$-invariant of integral points on certain modular curves which has positive genus and less than three cusps.
In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.
In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to Dyson's rank (mock theta) function and the Andrews--Garvan crank (modular) function---the match with explicit formulae for implied $O(1)$ constants. In this note we give another elementary proof of Ramanujan's original claim and outline some heuristic…
We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+ε})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.
Sturm obtained the bounds for the number of the first Fourier coefficients of elliptic modular form $f$ to determine vanishing of $f$ modulo a prime $p$. In this paper, we study analogues of Sturm's bound for Siegel modular forms of genus 2. We show the resulting bound is sharp. As an application, we study congruences involving Atkin's $U(p)$-operator for the Fourier coefficients of Siegel mdoular forms of genus 2.
We prove explicit bounds on canonical Green functions of Riemann surfaces obtained as compactifications of quotients of the upper half-plane by Fuchsian groups.
We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g.
Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup norm of a half integral weight cusp form is bounded in terms of the level.
Recently, D. Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $Γ_0(N)$. Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $X_0(N)$ is $0$ or $1$.
We study the $2k$-th moment of the family of twisted modular $L$-functions to a fixed prime power modulus at the central values. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.
We give a formula for divisors of modular units on $X_1(N)$ and use it to prove that the $\mathbb{Q}$-gonality of the modular curve $X_1(N)$ is bounded above by $\left[\frac{11N^2}{840}\right]$, where $[\bullet]$ denotes the nearest integer.
Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant. We provide a simple geometric interpretation for this invariant, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant introduced in this paper coincides with an invariant introduced in a recent paper by S.-W. Zhang.
We investigate geometric and combinatorial aspects of the mysterious relationship between the action of the motivic Galois group on the motivic fundamental group of the projective line punctured at zero, infinity, and N-th roots of unity, and the geometry of modular manifolds for the congruence subgroup Y(m;N) of GL(m,Z) fixing (0, ... , 0, 1) modulo N. To achieve this, we consider a canonical collection of elements in the image of the motivic Galois Lie algebra, spanning it over Q. These elements are the motivic correlators at N-th roots of unity. We assign to them chains in the complex compu…
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi-Perron type algorithms.
In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattic…
