Lower order terms for the one-level density of elliptic curve L-functions

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-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.     

A unitary test of the Ratios Conjecture

The Ratios Conjecture of Conrey, Farmer and Zirnbauer (2008) [CFZ1], (preprint) [CFZ2] predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal (Huynh and Miller (preprint) [HuyMil], Miller (2009) [Mil5], Miller and Montague (in press) [MilMo]) and symplectic (Miller (2008) [Mil3], Stopple (2009) [St]) families of L-functions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet L-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (−1,1), and for support up to (−2,2) we show agreement up to a power savings in the family's cardinality.     

On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205-250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.     

Riemann Zeros and Random Matrix Theory

In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Zeros of derivatives of sum of Dirichlet L-functions

A linear combination L(s) of two Dirichlet L-functions has infinitely many complex zeros in Res<0. In this note we prove an infinity of complex zeros of L(k)(s) in the same region.     

Determining L-functions by twists

Suppose that L₁(s) and L₂(s) are two L-functions whose twists by a set of Dirichlet characters simultaneously vanish (vanish mod p) at a critical point. We examine the extent to which this property determines the L-functions in the cases of L-functions of elliptic curves, of number fields, and of curves over finite fields.     

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ₀(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once.     

A large sieve for a class of non-abelianL-functions

Letq be a fixed odd prime. We consider the sequence of Kummer fields asa varies. Estimates are given for the global density of zeroes of ArtinL-functions of these fields. These results are obtained by deducing a series representation for the ArtinL-functions that arises naturally in the arithmetic ofQ.     

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

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The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by , where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.     

The Sixth Power Moment of Dirichlet L-Functions

We prove a formula, with power savings, for the sixth moment of Dirichlet L- functions averaged over all primitive characters χ (mod q) with q ≤?Q, and over the critical line. Our formula agrees precisely with predictions motivated by random matrix theory. In particular, the constant 42 appears as a factor in the leading order term, exactly as is predicted for the sixth moment of the Riemann zeta-function.     

An elliptic curve test of the L-Functions Ratios Conjecture

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We compare the L-Function Ratios Conjecture?s prediction with number theory for quadratic twists of a fixed elliptic curve, showing agreement in the 1-level density up to for test functions supported in (−σ,σ), giving a power-savings for σ<1. This test introduces complications not seen in previous cases (due to the level of the elliptic curve). The results here are a key ingredients in Dueñez et al. (preprint) [DHKMS2], which determine the effective matrix size for modeling zeros near the central point. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining data observed in Miller (2006) [Mil3]. A key ingredient is generalizing Jutila?s bound for quadratic character sums restricted to fundamental discriminant congruent to non-zero squares modulo a square-free integer. Another application is determining the main term in the 1-level density of quadratic twists of a fixed GLn form; this generalization was implicitly assumed in Rubinstein (2001) [Rub].

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=-Cbj1n5y-WE.     

A note on recent papers of Ramachandra and Huxley

The methods of the two authors on the zeros of zeta and L-functions are compared.     

Applications of the L-functions ratios conjectures

In upcoming papers by Conrey, Farmer and Zirnbauer there appearconjectural formulas for averages, over a family, of ratiosof products of shifted L-functions. In this paper we will presentvarious applications of these ratios conjectures to a wide varietyof problems that are of interest in number theory, such as lowerorder terms in the zero statistics of L-functions, mollifiedmoments of L-functions and discrete averages over zeros of theRiemann zeta function. In particular, using the ratios conjectureswe easily derive the answers to a number of notoriously difficultcomputations.     

Low-lying zeros of families of elliptic curves

There is a growing body of work giving strong evidence that zeros of families of -functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of -functions. We study these low-lying zeros for families of elliptic curve -functions. For these -functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and Swinnerton-Dyer and the impressive partial results towards resolving the conjecture).

We calculate the density of the low-lying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. An important challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great arithmetical interest. An application of our results is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann Hypothesis (GRH)). The upper bound obtained is less than , which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the ``extra" zero at the central point).

     

Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

Analytic monoids and factorization problems

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of \(GL_d({\mathbb {A}}_{\mathbb {Q}})\) satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.     

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves “affine” class groups. We show that, in general, these two criterions are not linked.     

On zeros of Hecke <Emphasis Type="Italic">L</Emphasis>-functions and their linear combinations on the critical line

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.