We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its l-function does not vanish under certain conditions. As an application, we generalize our result to Ko...
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We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its l-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen's plus space and prove an analogous result for Jacobi forms.
Consider the family of automorphic l-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of sh...
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Consider the family of automorphic l-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted l-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.
Given a half-integral weight holomorphic Kohnen newform f on Pp(4), we prove an asymptotic formula for large primes p with power saving error term for X x (mod p) jl(1/2, f, x)jt. Our result is unconditional, it does ...
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Given a half-integral weight holomorphic Kohnen newform f on Pp(4), we prove an asymptotic formula for large primes p with power saving error term for X x (mod p) jl(1/2, f, x)jt. Our result is unconditional, it does not rely on the Ramanujan-Petersson conjecture for the form f. This gives a very sharp lindel & ouml;f-on-average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The lindel & ouml;f hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Sali & eacute;sums in the P & oacute;lya-Vinogradov range. Half- integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Sali & eacute;sum to relate our problem to the sequence ant (mod 1). Our treatment of this sequence is inspired by work of Rudnick-Sarnak and the second author on the local spacings of ant modulo 1.
We describe an algorithm for computing, for all primes p 1isanexpanded version of Harvey's "generic prime" construction, making it possible toincorporate certainp-adic transcendentalfunctions into the c...
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We describe an algorithm for computing, for all primes p <= X, the trace of Frobenius at p of a hypergeometric motive over Q in time quasilinear inX. This involves computing the trace modulo p(e )for suitablee;as in our previous work treating the case e = 1, we combine the Beukers-Cohen-Mellit trace formula with average polynomial time techniques of Harvey and Harvey-Sutherland. The key new ingredient fore>1isanexpanded version of Harvey's "generic prime" construction, making it possible toincorporate certainp-adic transcendentalfunctions into the computation;one of theseis thep-adic Gamma function, whose average polynomial time computation is anintermediate step which may be of independent interest. We also provide animplementation in Sage and discuss the remaining computational issues around tabulating hypergeometric l-series.
This volume brings together research and survey articles in algebraic number theory and arithmetic geometry. All of them share as a general theme the mysterious connections between special values of l -functions and a...
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ISBN:
(数字)9783985475841
ISBN:
(纸本)9783985470846
This volume brings together research and survey articles in algebraic number theory and arithmetic geometry. All of them share as a general theme the mysterious connections between special values of l -functions and algebraic invariants of Galois representations. These relationships are explored primarily through the lenses of Iwasawa theory and other Galois-equivariant points of view. The topics covered include the Galois module structure of ideal class groups, reciprocity laws in Iwasawa theory, Euler systems, p -adic l -functions, and étale cohomology – each of which has had a remarkable importance in the study of p -adic Galois representations over the last few decades. In addition, the final chapters of this volume serve as an introduction to the emerging subject of speciall -values in positive characteristic. This is a new direction in the general area of global function field arithmetic that is concerned with the invariants of Galois representations valued in positive characteristic, as provided by Drinfeld modules or t -modules. Serving as the proceedings of an international conference held at ICMAT (Madrid) in May 2023, this volume is a useful resource for important techniques and approaches, as well as a source of concrete results and bibliographic references. It is of interest both to established researchers and to graduate students interested in algebraic number theory or in arithmetic geometry.
let C be a smooth curve over R = O/p'O, O being the valuation ring of an unramified extension of the field Q(p) of P-adic numbers, with residue field k = F-q. let f be a function over C, and Psi be in additive cha...
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let C be a smooth curve over R = O/p'O, O being the valuation ring of an unramified extension of the field Q(p) of P-adic numbers, with residue field k = F-q. let f be a function over C, and Psi be in additive character of order p' over R: in this paper we study the exponential sums associated to f and Psi over C, and their l-functions. We show the rationality of the l-functions ill a more general setting, then in the case of curves we express them as products of l-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the l-functions are, till to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums. (C) 2009 Elsevier Inc. All rights reserved.
We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated l-functions, prove functional equations for twisted versions ...
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We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated l-functions, prove functional equations for twisted versions of these l-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion.
We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients...
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We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients consisting of p-adic analytic functions with polyhedral growth prescribed by the relative polytope. Using this we compute relative cohomology for such families and calculate sharp estimates for the relative Frobenius map. In applications one is interested in l-functions associated with linear algebra operations (symmetric powers, tensor powers, exterior powers and combinations thereof) applied to relative Frobenius. Using methods pioneered by Ax, Katz and Bombieri we prove estimates for the degree and total degree of the associated l-function and p-divisibility of the reciprocal zeros and poles. Similar estimates are then established for affine families and pure Archimedean weight families (in the simplicial case). (C) 2014 Elsevier Inc. All rights reserved.
Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of glob...
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Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the l-functions and local theta correspondence.
This paper is devoted to upper and lower bounds of fractional moments of l-functions attached to certain cusp forms. The upper bound is proved under the analogue of the Riemann hypothesis.
This paper is devoted to upper and lower bounds of fractional moments of l-functions attached to certain cusp forms. The upper bound is proved under the analogue of the Riemann hypothesis.
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