Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell--Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq \mathbb{Z}^r \oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins' Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins' Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two.
Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.
Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.
In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over $\mathbb{F}_q(T)$ after extending constant scalars, and every quadratic twist of a modular elliptic curve over $\mathbb{F}_q(T)$ by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known f…
In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many almost primes. We show, assuming a strong version of the Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly bounded density in the set of primes. Studying the Cremona and Stein-Watkins databases of elliptic curves we conjecture that the set of conductors should be of density zero in the set of natural numbers and that the set of prime…
We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein-Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.
The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria for m_E to be odd. Specializing to N prime our results imply a conjecture of Mark Watkins. As a technical tool we also prove a certain multiplicity one result for p=2 that may be of independent interest.
This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic fo…
In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^α$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and…
We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of $J_0(N)$. We show the conductor of such Abelian varieties must be of a special type, for example if $N$ is odd then $N=p^α$ or $N=pq$ for some prime $p$ and $q$. We then focus our attention to modular elliptic curves, and using result of Agashe, Ribet, and Stein, we try to classify all elliptic curves of odd modular degree. Our studies prove many cases of the Stein and Watkins's conjecture on elliptic curves with odd modular degree.
We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.
In a previous paper we proved that if an $L$-function $F$ from the Selberg class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler product. In this paper we extend this result to $L$-functions of degree 2 with square-free conductor $q_F$, which are weakly twist-regular at all primes $p\nmid q_F$
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is…
Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.
In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)$ over the family of even primitive Dirichlet characters $χ$ of conductor $q$ for more general $q$.
This Note answers, and generalizes, a question of Kaisa Matomäki. We show that give two cuspidal automorphic representations $π_1$ and $π_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $χ$ such that $π_1\otimesχ\simeqπ_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $π_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$
Consider an irreducible, admissible representation $π$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $π_+ + π_-$. If $π=r_θ$, the Weil representation of GL(2,$F$) attached to a character $θ$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $χ\in \hat{K^*}$ with $(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$ occurs in ${r_θ}_+$ if and only if $ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$ and in ${r_θ}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $χ$ will appear in $π$? We…
We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes χ)$ and $L(\tfrac{1}{2}, χ)$ do not vanish. We expect the same method to work for any large integer $q$.
Extending Gross's result, we prove that a certain factorizaton of measures holds for all $p$ and any finite even Dirichlet character $χ$ of any conductor, rather than only for split $p$ and $χ$ with conductor a power of $p$. Using this generalization, we find lower bounds on the proportion of imaginary quadratic fields $K$ for which (under certain assumptions on the elliptic curve) a chosen quadratic twist of an elliptic curve $E$ over $K$ has rank $1$. We also find lower and upper bounds for the proportion of quadratic twists with rank $1$ when we vary $D$, the factor we twist by, under the a…
We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve $E/\Fp_q(C)$ over a function field over a finite field that have rank $\geq 2$, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form and $χ$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ L\left(\frac{1}{2}+it,π\times χ\right)\ll_{π,\varepsilon} p^{3/4}\big(\mathfrak{q}(1+|t|)\big)^{3/4-3/40+\varepsilon}, $$ for any $\varepsilon >0$ and $t \in \mathbb{R}$.
We prove a general result relating the shape of the Euler product of an $L$-function to the analytic properties of certain linear twists of the $L$-function itself. Then, by a sharp form of the transformation formula for linear twists, we check the required analytic properties in the case of $L$-functions of degree 2 and conductor 1 in the Selberg class. Finally we prove a converse theorem, showing that $ζ(s)^2$ is the only member of the Selberg class satisfying the above conditions and, moreover, having a pole at $s=1$.
Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.
Let E be an elliptic curve and ρan Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters χ, ramified only at p, such that the Artin-twisted L-values L(E,ρχ,β) are non-zero when βlies in a specified region in the critical strip (assuming the conjectural continuations and functional equations for these L-functions). The new contribution of our paper is that we may choose our characters to be ramified only at one prime, which may divide the conductor of ρ.
In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our r…
The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $α\frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $α$ is depending on the conductor of the chosen elliptic curves an…
It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-…
London Institute for Mathematical Sciences（伦敦数学科学研究所）；Merton College, Oxford（牛津大学默顿学院）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；University of Westminster（威斯敏斯特大学）；Northeastern University（东北大学）
We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb Q$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.
Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplif…
The Asai (or twisted tensor) $L$-function of a Bianchi modular form $Ψ$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $Ψ$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the A…
Let E be an elliptic curved defined over $\Q$ and let $K/\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\times_{\Q} K)(1)$ viewed as a motive over $\Q$ with coefficients in $\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\Q$.
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…
It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the conductor, i.e. the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented.
Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character χof G. Our main result is that if the special value of the χ-twisted L-function of E/K is non-zero then the tensor product (with respect to χ) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite ext…
We describe a systematic investigation into the existence of congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod $p$ congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes $p\ge7$ to the elliptic curves in the LMFDB database, which currently includes all el…
Let $E/\mathbb{Q}$ be an optimal elliptic curve, $-D$ be a negative fundamental discriminant coprime to the conductor $N$ of $E/\mathbb{Q}$ and let $E^{-D}/\mathbb{Q}$ be the twist of $E/\mathbb{Q}$ by $-D$. A conjecture of Agashe predicts that if $E^{-D}/\mathbb{Q}$ has analytic rank $0$, then the square of the order of the torsion subgroup of $E^{-D}/\mathbb{Q}$ divides the product of the order of the Shafarevich-Tate group of $E^{-D}/\mathbb{Q}$ and the orders of the arithmetic component groups of $E^{-D}/\mathbb{Q}$, up to a power of $2$. This conjecture can be viewed as evidence for the s…
We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling).Our purpose here is to provide a random matrix model for Miller's surprising discovery. We consider the fami…
Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions
American Institute of Mathematics（美国数学研究所）；Institute for Advanced Study（普林斯顿高等研究院）；Data Science Institute, Shandong University（山东大学数据科学研究院）；State Key Laboratory of Cryptography and Digital Economy Security, Shandong University（山东大学密码与数字经济安全国家重点实验室）；Carleton College（卡尔顿学院）
