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We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings., Comment: 26 pages, LaTeX Revision includes additional results on consequences of algebracity of Stieltjes transform

In connection with each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author (math.NT/0407535). Most notably, all examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i.~e., shtukas not associated to any Drinfeld module., Comment: 25 pages, LaTeX

Let $k$ be a local field of positive characteristic. Let $L$ be a cocompact discrete subgroup of $k$. Let $U$ be an open compact subgroup of $k$. Let $\ell$, $u$ and $a$ be elements of $k$, with $a$ nonzero. We study the behavior of the product $\prod_{0\neq x\in (\ell+L)\cap a(u+U)}x$ as $a$ varies. Our main result, which blends harmonic analysis and local class field theory, provides local confirmation of a conjecture concerning global function fields recently made by the author (math.NT/0407535)., Comment: LaTeX, 50 pages. v2: Introduction and last section rewritten, otherwise only some typos corrected. This is the final version, to appear in Crelle

Our main result is elementary and concerns the relationship between the multiplicative groups of the coordinate and endomorphism rings of the formal additive group over a field of characteristic $p>0$. The proof involves the combinatorics of base $p$ representations of positive integers in a striking way. We apply the main result to construct a canonical quotient of the module of Coleman power series over the Iwasawa algebra when the base local field is of characteristic $p$. This gives information in a situation which apparently has never previously been investigated., Comment: LaTeX, 15 pages

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincare type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices., Comment: To appear, Prob. Theory Rel. Fields

We propose a conjecture refining the Stark conjecture St(K/k,S) (Tate's formulation) in the function field case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is that it involves two- rather than one-variable algebraic functions over finite fields. Because the conjecture is formulated in the language and framework of Tate's thesis we have powerful standard techniques at our disposal to study it. We build a case for our conjecture by (i) proving a parallel result in the framework of adelic harmonic analysis which we dub the adelic Stirling formula, (ii) proving the conjecture in the genus zero case and (iii) explaining in detail how to deduce St(K/k,S) from our conjecture. In the genus zero case the class of two-variable algebraic functions thus arising includes all the F_q[T]-solitons previously studied and applied by the author, collaborators and others. Ultimately the inspiration for this paper comes from striking examples given some years ago by R. Coleman and D. Thakur., Comment: v1: 64 pages, LaTeX. v2: 55 pages, LaTeX, smaller font; revised following referee's suggestions (locally rewritten, typos corrected, explanatory remarks added, no global reorganization); to appear in Compositio

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the curvature of the initial metric. We show that the norm of the solution of the Lichnerowicz Laplacian heat equation (coupled to the Ricci flow) is bounded by a constant (depending on time) times the scalar curvature plus a constant. This relies on a Bochner type formula and establishing the nonnegativity of a degree 4 homogeneous polynomial in 6 variables., Comment: 12 pages, no figures

By the Lefschetz embedding theorem, a principally polarized $g$-dimensional abelian variety is embedded into projective space by the linear system of $4^g$ half-characteristic theta functions. Suppose we {\em edit} this linear system by dropping all the theta functions vanishing at the origin to order greater than parity requires. We prove that for Jacobians the edited $4\Theta$ linear system still defines an embedding into projective space. Moreover, we prove that the projective models of Jacobians arising from the elementary construction of Jacobians recently given by the author are (after passage to linear hulls) copies of the edited $4\Theta$ model. Thus, for all compact Riemann surfaces, we tie together algebraic and analytic Jacobians in a new way.

The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carr\`{e}re symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-DeConcini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained by the standard functional equations., Comment: 62 pages

In this note we give the Kolyvagin recursion in cyclotomic Euler systens a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of the chosen field satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution a la Kubert. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion., Comment: 17 pages

The {\em abeliant} is a polynomial rule for producing an $n$ by $n$ matrix with entries in a given ring from an $n$ by $n$ by $n+2$ array of elements of that ring. The theory of abeliants, first introduced in an earlier paper of the author, is redeveloped here in a simpler way. Then this theory is exploited to give an explicit elementary construction of the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. The standard of usefulness and aptness we strive toward is that set by Mumford's elementary construction of the Jacobian of a hyperelliptic curve. This paper has appeared as Advances in Math 172 (2002) 169-205.

We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+\epsilon}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+\epsilon}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+\epsilon}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $\Gamma$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.

We explore the reach of luminosity upgrades of the Fermilab Tevatron collider for SU(5) supergravity models in which non-universal GUT-scale gaugino masses arise via a vacuum expectation value for the auxiliary component of a superfield that transforms as a 24, 75 or 200 dimensional representation of SU(5). This results in a different pattern of sparticle masses and mixing angles from what is expected in the minimal supergravity model (mSUGRA) with universal GUT scale gaugino masses. We find that the resulting signal cross sections, and hence the reach of the Tevatron, are sensitive to the gaugino masses at the GUT scale. In the 24 model, the large splitting amongst the two lightest neutralinos leads to SUSY events containing many isolated leptons, including events with a real leptonic Z boson plus jets plus missing energy signal which is visible over much of parameter space. In contrast, in the 75 and 200 models, the reach via leptonic SUSY signals is greatly reduced relative to mSUGRA, and the signal is usually visible only via the canonical $\eslt +$jets channel., Comment: 38 page REVTEX file including 21 PS figures

We propose a simple model in which the cosmological dark matter consists of particles whose mass increases with the scale factor of the universe. The particle mass is generated by the expectation value of a scalar field which does not have a stable vacuum state, but which is effectively stabilized by the rest energy of the ambient particles. As the universe expands, the density of particles decreases, leading to an increase in the vacuum expectation value of the scalar (and hence the mass of the particle). The energy density of the coupled system of variable-mass particles (``vamps'') redshifts more slowly than that of ordinary matter. Consequently, the age of the universe is larger than in conventional scenarios., Comment: 14 pages, 2 figures; based on a talk by SMC at Cosmo-97, September 1997, Ambleside, England. Important references added

We give a new method for solving a problem originally solved about 20 years ago by Sinnott and Kubert, namely that of computing the cohomology of the universal ordinary distribution with respect to the action of the two-element group generated by complex conjugation. We develop the method in sufficient generality so as to be able to calculate analogous cohomology groups in the function field setting which have not previously been calculated. In particular, we are able to confirm a conjecture of L.~S.~Yin conditional on which Yin was able to obtain results on unit indices generalizing those of Sinnott in the classical cyclotomic case and Galovich-Rosen in the Carlitz cyclotomic case. The Farrell-Tate cohomology theory for groups of finite virtual cohomological dimension plays a key role in our proof of Yin's conjecture. The methods developed in the paper have recently been used by P.~Das to illuminate the structure of the Galois group of the algebraic extension of the rational number field generated by the roots of unity and the algebraic $\Gamma$-monomials. This paper has appeared as Contemp. Math. 224 (1999) 1-27.

We quantify the extent to which naturalness is lost as experimental lower bounds on the Higgs boson mass increase, and we compute the natural upper bound on the lightest supersymmetric Higgs boson mass. We find that it would be unnatural for the mass of the lightest supersymmetric Higgs boson to saturate it's maximal upper bound. In the absence of significant fine-tuning, the lightest Higgs boson mass should lie below $120$ GeV, and in the most natural cases it should be lighter than $108$ GeV. For modest $tan \beta$, these bounds are significantly lower. Our results imply that a failure to observe a light Higgs boson in pre-LHC experiments could provide a serious challenge to the principal motivation for weak-scale supersymmetry., Comment: 11 pages, LaTex, 3 ps.figures

Fluctuations of scalar fields produced at the stage of preheating after inflation are so large that they can break supersymmetry much stronger than inflation itself. These fluctuations may lead to symmetry restoration along flat directions of the effective potential even in the theories where the usual high temperature corrections are exponentially suppressed. Our results show that nonthermal phase transitions after preheating may play a crucial role in the generation of the primordial baryon asymmetry by the Affleck-Dine mechanism. In particular, the baryon asymmetry may be generated at the very early stage of the evolution of the Universe, at the preheating era, and not when the Hubble parameter becomes of order the gravitino mass., Comment: 4 pages, no figures

Experimental searches for supersymmetry are entering a new era. As future experiments explore the mass range above the current lower bounds on superpartner masses, a failure to observe signals of superpartner production will begin to erode the central motivation for supersymmetry at the weak scale. In this article we present a detailed examination of which regions of supersymmetric parameter space are most natural and the extent to which weak-scale supersymmetry becomes unnatural if no superpartners are observed at LEP-II, the Tevatron, possible upgrades of these machines, and the LHC., Comment: 14 pages, LaTex, 10 figures, uuencoded, gz-compressed file. Minor revisions bring archived manuscript in line with version to be published

Superpartner masses cannot be arbitrarily heavy if supersymmetric extensions of the standard model explain the stability of the gauge hierarchy. This ancient and hallowed motivation for weak scale supersymmetry is often quoted, yet no reliable determination of this upper limit on superpartner masses exists. In this paper we compute upper bounds on superpartner masses in the minimal supersymmetric model, and we identify which values of the superpartner masses correspond to the most natural explanation of the hierarchy stability. We compare the most natural value of these masses and their upper limits to the physics reach of current and future colliders. As a result, we find that supersymmetry could explain weak scale stability naturally even if no superpartners are discovered at LEP II or the Tevatron (even with the Main Injector upgrade). However, we find that supersymmetry cannot provide a complete explanation of weak scale stability, if squarks and gluinos have masses beyond the physics reach of the LHC. Moreover, in the most natural scenarios, many sparticles, for example, charginos, squarks, and gluinos, lie within the physics reach of either LEP II or the Tevatron. Our analysis determines the most natural value of the chargino (squark) ((gluino)) mass consistent with current experimental constraints is $\sim$ 50 (250) ((250)) GeV and the corresponding theoretical upper bound is $\sim$ 250 (700) ((800)) GeV., Comment: 14 pages, LaTex, 17 figures uuencoded, gz-compressed file. Minor revisions bring archived manuscript in line with the published version

Fine-tuning criteria are frequently used to place upper limits on the masses of superpartners in supersymmetric extensions of the standard model. However, commonly used prescriptions for quantifying naturalness have some important shortcomings. Motivated by this, we propose new criteria for quantifying fine tuning that can be used to place upper limits on superpartner masses with greater fidelity. In addition, our analysis attempts to make explicit the assumptions implicit in quantifications of naturalness. We apply our criteria to the minimal supersymmetric extension of the standard model, and we find that the scale of supersymmetry breaking can be larger than previous methods indicate., Comment: 15 pages, LaTex, 5 figures uuencoded, gz-compressed file. Minor revisions bring the archived manuscript into agreement with published version

In the standard scenario, the electroweak phase transition is a first order phase transition which completes by the nucleation of critical bubbles. Recently, there has been speculation that the standard picture of the electroweak phase transition is incorrect. Instead, it has been proposed that throughout the phase transition appreciable amounts of both broken and unbroken phases of $SU(2)$ coexist in equilibrium. I argue that this can not be the case. General principles insure that the universe will remain in a homogenous state of unbroken $SU(2)$ until the onset of critical bubble production., Comment: 7 pages plus three figures. OHSTPY-HEP-T-92-016 A topdrawer file of the figures is appended to the end

The fermion mass and mixing angle predictions of a recently proposed framework are investigated for large b and $\tau$ Yukawa couplings. A new allowed region of parameters is found for this large $\tan \beta$ case. The two predictions which are substantially altered, $m_t$ and $\tan \beta$, are displayed, including the dependence on the inputs $|V_{cb}|$, $m_c$, $m_b$ and $\alpha_s$. A simple restriction on this framework yields an additional prediction, for $|V_{cb}|$. If the b,t, and $\tau$ Yukawas are equal at the GUT scale then $|\Vcb|$ is predicted and the top quark mass is constrained to lie in the range $\mt = 179. \pm 4.$ GeV., Comment: (beta), 13 pages pages of text plus figures and captions. OHSTPY-HEP-92-018, LBL-332817, UCB-PTH-92/28, Stanford-92-25

In the standard scenario, the electroweak phase transition is a first order phase transition which completes by the nucleation of critical bubbles. Recently, there has been speculation that the standard picture of the electroweak phase transition is incorrect. Instead, it has been proposed that throughout the phase transition appreciable amounts of both broken and unbroken phases of $SU(2)$ coexist in equilibrium. I argue that this can not be the case. General principles insure that the universe will remain in a homogenous state of unbroken $SU(2)$ until the onset of critical bubble production. In addition, an analytic treatment of the one Higgs doublet, electroweak phase transition in the standard model and minimal extensions is reviewed. Results from the thin wall approximation are compared to results obtained using the Lindes' action. Perhaps the most important quantitative result we can get from an analysis of the phase transition is determination of $\vevphi$ when the phase transition completes. For Higgs boson masses above the current experimental limit, the thin wall approximation determines the value of $\vevphi$ at the end of the phase transition to an accuracy of better than three percent., Comment: 11