J. Brian Conrey, Atul Pokharel, Michael O. Rubinstein, Mark Watkins
We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.
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We present theoretical and numerical evidence for a random matrix theoretic approach to a conjecture about vanishings of quadratic twists of certain L-functions
Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,χ) of the twisted L-function as χranges over Dirichlet characters of given order.
Non-vanishing theorems for prime twists of some modular $L$-functions
本文研究了特定椭圆模形式的模 $L$-函数经虚二次域素数扭曲后的中心值非零性问题，并证明在根数为正时这些中心值恒不为零。
In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.
In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.
Secondary terms for first moments of Selmer groups of twists of elliptic curves over global function fields
本文在显式条件下，确定了全局函数域上非平凡椭圆曲线循环素数扭转族中素数Selmer群第一矩的次级项。
We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.
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.
In this paper, we prove that if the Fourier coefficients of a $\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maaß cusp form $π$ are not too correlated with additive characters, then there exists infinitely many Dirichlet characters such that \begin{align*} L\left(\frac{1}{2},π\otimesχ\right)L\left(\frac{1}{2},χ\right)\neq 0. \end{align*} To prove this result, we compute the first twisted moment of these $L$ function averaged over a well chosen set of conductors.
We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt th…
Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters chi of the ideal class group Cl_K such that L(1/2, f_K \times chi) \neq 0 and L(1/2, g_K \times chi) \neq 0. The proof is based on a non-vanishing result for Fourier coefficients of Siegel modular forms combined with the theory of Yoshida liftings.
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.
Let $E$ be an elliptic curve over $\mathbb{Q}$, with L-function $L_E(s)$. For any primitive Dirichlet character $χ$, let $L_E(s, χ)$ be the L-function of $E$ twisted by $χ$. In this paper, we use random matrix theory to study vanishing of the twisted L-functions $L_E(s, χ)$ at the central value $s=1$. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that $L_E(1, χ)=0$, but that for any fixed prime $k \geq 7$, there are only finitely many character of order $k$ such that $L_E(1, χ)$ vanishes. With the Birch and Swinnerton-Dyer Conjectu…
Let pi be an automorphic representation on GL(r, A_Q) for r=1, 2, or 3. Let d be a fundamental discriminant and chi_d the corresponding quadratic Dirichlet character. We consider the question of the least d, relative to the data (level, weight or eigenvalue) of pi, such that the central value of the twisted L-series is nonzero, i.e. L(1/2, pi x chi_d) != 0. For example, let N be the level of pi. Using multiple Dirichlet series, we prove the nonvanishing of a central twisted $L$-value with d << N^{1/2} for GL(1), d << N^1 for GL(2), and d << N^2 for GL(3), the last case assuming that a certain …
Simultaneous non-vanishing of twists of $\mathrm{GL}_3$ and $\mathrm{GL}_1$ $L$-functions
该研究针对固定的SL₃(ℤ)赫克-马阿斯尖型F，证明存在无穷多本原狄利克雷特征χ使对应GL₃与GL₁的扭L-函数不同时为零，还得到了相关中心值乘积的渐近公式。
Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,χ)$ as $χ$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on $E$. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Earlier work by David, Fearnley and Kisilevsky formulates analogous conjectures for characters of any odd prime order. In the composite order case, however, we need to justify …
We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.
Horizontal $p$-adic $L$-functions
本文定义了关联有理数域上椭圆曲线p幂阶特征扭结L值的水平p进L函数，研究其性质后得到多类扭结中心L值非零性的定量下界，改进了已有结果并应用于丢番图稳定性。
The theme of this work is the study of the Nekovář-Selmer group H^1_f(K,T) attached to a twisted Hida family T of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted L-function of a suitable modular form in the Hida family has order of vanishing r at most 1 at the central critical point then the rank of H^1_f(K,T) as a module over a certain local Hida-Hecke algebra is equal to r. Under the above assumption, we also show that infinitely many twisted L-functions of modular forms in the Hida family have the same order of vanishi…
Non-Vanishing of Cubic Twists of $GL_n(\mathbb{Q})$ $L$-functions
本文研究了三次Dirichlet字符与$GL_n(\mathbb{Q})$自动表示的$L$-函数非消失问题，证明了在特定实部条件下存在无限多个原始三次Dirichlet字符使$L$-函数不为零。
The main objective of this article is to compute a first moment for product of Dirichlet and twisted self-dual $GL(3)$ $L$-functions. We discuss the possible simultaneous non vanishing at the central point. We use properties of symmetric squares $L$-functions.
We use relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change $L$-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal representations. This generalizes the work of Feigon and Whitehouse.
Non-vanishing of twists of $\text{GL}_4(\mathbb{A}_{\mathbb{Q}})$ $L$-functions
针对GL₄(ℚ的阿代尔)酉尖点自守表示，证明存在无穷多个导子与给定f互素的本原偶或奇Dirichlet特征，使中心L值非零，并应用于GSp₄的p进L函数构造、BSD猜想及重数一等问题。
The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $Π$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent and separate works of all the three authors. By construction our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. More…
Let F(z) be a newform of weight 2k and level one with a trivial character, and assume that F(z) is a non-zero eigenform of all Hecke operators. In this paper, we study nonvanishing for central values of twisted modular L-function of F.
Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields
本文通过推导一般权 Waldspurger 公式并结合 Heegner 点等分布性，计算了虚二次域 Hecke 特征扭曲的 Rankin--Selberg L-函数中心值的宽矩，并由此得到弱同时非零结果及三重卷积 L-函数非零性的新方法。
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be another elliptic curve over a real quadratic number field. We construct a $\mathbb{Q}$-motive of rank $8$, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch-Kato Selmer group of the motive is $0$; and the non-vani…
We consider $L$-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these $L$-functions by characters of order prime to the characteristic of the ground field and by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of $L$-functions, and some monodromy calculations of Katz. As an …
We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.
Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions
针对标准和对称平方L函数的素二次扭曲，研究其一致非零性，证明满足特定条件的素数比例趋于零，还得到相关基的生成与密度一结果，核心方法涉及随机欧拉乘积的非集中定理。
The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors.…
We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.
We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves via the modularity theorem and the work of Gross and Zagier.
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…
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