Let F be a global function field of characteristic p with ring of integers A and let Φbe a Hayes module on the Hilbert class field H(A) of F. We prove an Iwasawa Main Conjecture for the Z_p^\infty-extension F/F generated by the \mathfrak{p}-power torsion of Φ(\mathfrak{p} a prime of A). The main tool is a Stickelberger series whose specialization provides a generator for the Fitting ideal of the class group of F. Moreover we prove that the same series, evaluated at complex or p-adic characters, interpolates the Goss Zeta-function or some p-adic L-function, thus providing the link between the algebraic structure (class groups) and the analytic functions, which is the crucial part of Iwasawa Main Conjecture.
Let $L/K$ be a finite Galois CM-extension of number fields with Galois group $G$. In an earlier paper, the author has defined a module $SKu(L/K)$ over the center of the group ring $\mathbb Z[G]$ which coincides with the Sinnott-Kurihara ideal if $G$ is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of $SKu(L/K)$ implies the minus part of the equivariant Tamagawa number conjecture at an odd prime $p$ for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above $p$…
Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$, and $\mathfrak{C}(d)$ the ray class group of modulus $d$. Let $ρ$ be an irreducible character of $G=\mathrm{Gal}(k/\mathbb{Q})$ over $\mathbb{Q}_p$ and $e_ρ \in \mathbb{Z}_p[G]$ the corresponding idempotent. We show that if the ramification index of $p$ in $k$ is less than $p-1$, then $|e_ρ \mathrm{Syl}_p(E_d/C_d) | = |e_ρ \mathrm{Syl}_p(\mathfrak{C}_d)|$ wh…
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Sta…
The Stickelberger splitting map in the case of abelian extensions $F / \Q$ was defined in [Ba1, Chap. IV]. The construction used Stickelebrger's theorem. For abelian extensions $F / K$ with an arbitrary totally real base field $K$ the construction of \cite{Ba1} cannot be generalized since Brumer's conjecture (the analogue of Stickelberger's theorem) is not proved yet at that level of generality. In this paper, we construct a general Stickelberger splitting map under the assumption that the first Stickelberger elements annihilate the Quillen $K$--groups groups $K_2 ({\mathcal O}_{F_{l^k}})$ for…
For a CM abelian extension $F/K$ of an arbitrary totally real number field $K$, we construct the Stickelberger splitting maps (in the sense of \cite{Ba1}) for both the étale and the Quillen $K$--theory of $F$ and we use these maps to construct Euler systems in the even Quillen $K$--theory of $F$. The Stickelberger splitting maps give an immediate proof of the annihilation of the groups of divisible elements $div K_{2n}(F)_l$ of the even $K$--theory of the top field by higher Stickelberger elements, for all odd primes $l$. This generalizes the results of \cite{Ba1}, which only deals with CM abe…
We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of Rubin-Stark Conjecture and Burns Conjecture.
We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(θ)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[θ]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $β(j)$ (i.e., on the values of the Goss $ζ$-function at…
Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\text{Cl}^T(H) \…
We formulate a generalization of a `refined class number formula' of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the `order of vanishing' and the `leading term'. Using the theory of Kolyvagin systems we prove a large part of this conjecture when the order of vanishing of the corresponding complex $L$-function is $1$.
We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those $L$-functions of the extension which are non-zero at the special point $s = 0$, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include $L$-functions with a simple zero at $s = 0$, and was shown b…
We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$ (equivalently, $s=0$). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form. The conjecture considers a totally real field $F$ and a finite abelian C…
We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.
In this paper, we construct a higher rank Euler system for the multiplicative group over a totally real field by using the Iwasawa main conjecture proved by Wiles. A key ingredient of the construction is to generalize the notion of the characteristic ideal. Under certain technical assumptions, we prove that all higher Fitting ideals of a certain $p$-ramified Iwasawa module are described by analytic invariants canonically associated with Stickelberger elements.
We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where $x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.
Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadra…
Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberge…
Suppose that $EE$ is a totally real number field which is the composite of all of its subfields $E$ that are relative quadratic extensions of a base field $F$. For each such $E$ with ring of integers $Ø_E$, assume the truth of the Birch-Tate conjecture (which is almost fully established) relating the order of the tame kernel $K_2(Ø_E)$ to the value of the Dedekind zeta function of $E$ at $s=-1$, and assume the same for $F$ as well. Excluding a certain rare situation, we prove the annihilation of $K_2(\Oc_EE)$ by a generalized Stickelberger element in the group ring of the Galois group of $EE/F…
This expository paper is based on the author's series of lectures delivered at the January 1999 Mini-course in Number Theory, held at Sogang University (Seoul). The aim is to give an elementary and self-contained introduction to the theory of Newton polygons (namely, the $p$-adic Riemann hypothesis) for L-functions of exponential sums over a finite field. The main idea of our approach is to establish a suitable local to global principal which would allowus to reduce the harder ``global'' case to various easier well understood "local" cases for which Stickelberger's theorem applies. For this pu…
Fix a relative quadratic extension E/F of totally real number rields and let G denote the Galois group of order 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E and let O_E^S denote the ring of S_E-integers of E. Assume the truth of the 2-part of the Birch-Tate conjecture relating the order of the tame kernel K_2(O_E^S) to the value of the Dedekind zeta function of E at s=-1, and assume the same for F as well. We then prove that the Fitting ideal of K_2(O_E^S) as a Z[G]-module is equal to a generalized Sticke…
We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z_p-tower of abelian fields and {\mathfrak J} is linked by explicit reciprocity to a third ideal {\mathfrak S} studied more generally in a previous work. This leads to a new and unifying framework for the Iwasawa Theory of such fields including a real analogue of Stickelberger's Theorem, links with certain Fitting ideals and Λ-torsion submodules, …
We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known re…
We obtain an asymptotic for the regularized fourth moment of the Eisenstein series for the full modular group, in agreement with the Random Wave Conjecture.
Let $\k$ be an arbitrary field. For any fixed badly approximable power series $Θ$ in $\k((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $Φ$ for which the pair $(Θ, Φ)$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.
Some results and conjectures on $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|ζ(1/2+it)$ and the error term in the fourth moment of $|ζ(1/2+it)$ are discussed.
We formulate a version of the Random Wave Conjecture for the fourth moment of Eisenstein series which is based on Zagier's regularized inner product. We prove an asymptotic formula expressing the regularized fourth moment as a mean value of L-functions. This is an advantage over previous work in the literature, which has approached the fourth moment problem through truncated Eisenstein series and not yielded a suitable expression in terms of L-functions.
Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$).
Gross and Smith have put forward generalizations of Hardy - Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series that arises in these conjectures, up to lower order terms. More exactly, we find asymptotic formulas for smoothed sums of the singular series minus one. Based upon Gross and Smith's conjectures, we use our result to suggest that for large enough 'short intervals' in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramer model prediction by a universal fact…
In this note, we propose a conjecture stating that some series involving primitive sequences are convergent. Then, we show (by a counterexample) that the analogue of a conjecture of Erdős, for those series, is false.
In this article, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above $p$ to satisfy the $\M_H(G)$-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the $\M_H(G)$-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series.…
Recent results of Zlobin and Cresson-Fischler-Rivoal allow one to decompose any suitable $p$-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most $p$; in some cases, only the multiple zeta values with 2's and 3's are involved (as in Hoffman's conjecture). In this text, we study the depth $p$ part of this linear combination, namely the contribution of the multiple zeta values of depth exactly $p$. We prove that it satisfies some symmetry property as soon as the $p$-uple series does, and make some conjectures on the depth $p-1…
In a previous article, we constructed an entire power series over $p$-adic weight space (the 'ghost series') and conjectured, in the $Γ_0(N)$-regular case, that this series encodes the slopes of overconvergent modular forms of any $p$-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at $p$. Ample numerical evi…
In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a suitable series of positive terms related to the logarithmic integral function. Using this proposed method we discuss the Polignac's conjecture and present some consequences.
In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the length of the interval $H$ is sufficiently large as a function of $k$). This result of Gallagher played a key role in our original joint work with D. A. Goldston and C. Y. Yildirim, where we showed the existence of infinitely many small gaps between consecutive primes. In the present work some weaker variants of Gallagher`s result are also proved (in an even ea…
We present a conjecture about the asymptotic representation of certain series. The conjecture implies the Riemann hypothesis and it would also indicate the simplicity of the non-trivial zeros of the zeta-function.
A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erdős showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ is a primitive sequence different from $\{1\}$, are all convergent and their sums are bounded above by an absolute constant. Besides, he conjectured that the upper bound of the preceding sums is reached when $\mathscr{A}$ is the sequence of the prime numbers. The purpose of this paper is to study the Erdős conjecture. In the first part of the paper, we give two sig…
In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $p-$adic gamma function, and are called Ramanujan type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the prove…
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $π$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In addition we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation.
Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.
Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition on the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds for the first coefficients in a large number of examples.
We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences in this theory. By using this approach we obtain a variety of new results, ranging from a proof, mod…
In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.
In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+α)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,α\in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,α\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is t…
In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their relation to the weights. We check the validity of the conjecture on several numerical examples.
We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is full (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of this group. The proof uses ramification theory and some $p$-adic Hodge theory.
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
Let $K$ be a number field. This paper is devoted to a $p$-adic study of the algebraic differential equation of $K[x,d/dx]$ with minimal degree in $x$ annihilating a given $E$-function. Precisely, we prove a conjecture of yves André (conjecture 4.7 of math.NT/0003238)
We consider a Laurent series defined by infinite products $g_u(t) = \prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and $\mathbb{F}$ is a field. We show that for all $u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it satisfies the $t$-adic Littlewood conjecture.
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.
