Dinesh S. Thakur

Shtukas and Jacobi sums

SummaryWe show how the analogues of Jacobi sums, in the context of function fields, introduced and studied in [T1, T2, T3] can be obtained from shtukas introduced and studied in [D2, D3, M]. We apply this to obtain some results on the prime factorization of analogues of Gauss sums and to prove an analogue of the Gross-Koblitz formula for general function field, generalizing the results in [T2]. For this purpose, we also introduce and interpolate a new analogue of gamma function.

Transcendence of Gamma Values for #q( T )

We prove that many values at proper fractions of the gamma function for Fq[T] (introduced by Carlitz and Goss) are transcendental over Fq(T). In particular, (n - 1/b)! is transcendental for any integer n and a positive integer b > 1, prime to q. Our proof is based on the transcendence criterion of Christol.

Multiplicative relations between coefficients of logarithmic derivatives of 𝔽q-linear functions and applications

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from

Power sums of polynomials over finite fields and applications

\mathbb{F}_q

Fermat versus Wilson congruences, arithmetic derivatives and zeta values☆

-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as well as applications to function field zeta and multizeta values evaluations and relations between them. Using specialization/generalization arguments, we provide much more general identities on linear forms providing a switch between power sums for positive and negative powers.

Binomial and factorial congruences for Fq[t]

In this brief expository survey, we explain some results and conjectures on various aspects of the study of the sums of integral powers of monic polynomials of a given degree over a finite field. The aspects include non-vanishing criteria, formulas and bounds for degree and valuation at finite primes, explicit formulas of various kind for the sums themselves, factorizations of such sums, generating functions for them, relations between them, special type of interpolations of the sums by algebraic functions, and the resulting connections between the motives constructed from them and the zeta and multizeta special values. We mention several applications to the function field arithmetic.

Integrable systems and number theory in finite characteristic

Abstract We look at two analogs each for the well-known congruences of Fermat and Wilson in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, we find interesting relations linking them together, as well as linking them with derivatives and zeta values. The link with the zeta value carries over to the number field case, with the zeta value at 1 being replaced by Eulers constant.

A note on numerators of Bernoulli numbers

Abstract We present several elementary theorems, observations and questions related to the theme of congruences satisfied by binomial coefficients and factorials modulo primes (or prime powers) in the setting of polynomial ring over a finite field. When we look at the factorial of n or the binomial coefficient ‘ n choose m ’ in this setting, though the values are in a function field, n and m can be usual integers, polynomials or mixed. Thus there are several interesting analogs of the well-known theorems of Lucas, Wilson etc. with quite different proofs and new phenomena.

Computational classification of numbers and algebraic properties

The purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic. The applications include explicit class field theory and Langlands conjectures for function fields, effect of the geometry of the theta divisor on factorization of analogs of Gauss sums, special values of function field Gamma, zeta and L-functions, analogs of theorems of Weil and Stickelberger, control of the intersection of the Jacobian torsion with the theta divisor. The techniques are the Krichever–Drinfeld dictionaries and the theory of solitons, Akhiezer–Baker and tau functions developed in this context of arithmetic geometry by Anderson.

Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields

The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.