Greg W. Anderson

A Short Proof of Selberg's Generalized Beta Formula.

in a quite nonobvious way. After an initial period in the shadows, Selbergs formula has come to play an important role in mathematics not only because of the interesting applications which have been made of it, but also because of the many conjectures and generalizations inspired by it. To give an extremely tiny sample of the vast literature on this subject, we cite papers of Askey [3], Evans [4], Evans-Root [6], Kadell [7] and Macdonald [8]. Selbergs proof of his formula was rather complicated. The purpose of this note is present a short proof of Selbergs formula based upon the comparison of two methods of evaluating a double integral. Our proof is not distinguished merely by its

Convergence of the largest singular value of a polynomial in independent Wigner matrices

For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form “no eigenvalues outside the support of the limiting eigenvalue distribution.” We build on ideas of Haagerup–Schultz–Thorbjornsen on the one hand and Bai–Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincare-type inequalities, we use a variety of matrix identities and Lp estimates. The Schwinger–Dyson equation controls much of the analysis.

Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring

Abstract 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-Carrère 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-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

Kronecker-Weber plus epsilon

We say that a group is 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 Q of rational numbers in the field of complex numbers. Let Gab + be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups and let Qab + /Q be the corresponding Galois extension. We prove that Qab + is generated by the roots of unity, the fourth roots of the rational primes and the square roots of certain algebraic sine-monomials. The inspiration for the paper came from recent studies of algebraic Γ-monomials by P. Das and by S. Seo.

A CLT for regularized sample covariance matrices

We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our knowledge is new, even in the Gaussian setup.

Abeliants and their application to an elementary construction of Jacobians

Abstract The abeliant is a polynomial rule which to each n×n by n+2 array with entries in a commutative ring with unit associates an n×n matrix with entries in the same ring. The theory of abeliants, first introduced in an earlier paper of the author, is simplified and extended here. Now let J be the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. With the aid of the theory of abeliants we obtain explicit defining equations for J and its group law.

A two-variable refinement of the Stark conjecture in the function-field case

We propose a conjecture refining the Stark conjecture St(

A NOTE ON CYCLOTOMIC EULER SYSTEMS AND THE DOUBLE COMPLEX METHOD

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Integral kašin splittings

) in the function-field case. Of course St(

Normalization of the Hyperadelic Gamma Function

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