On a problem of Chowla

The equation $\text{Σ}\text{f(n)}\text{n}\text{= 0}$ is studied for periodic algebraically-valued functions f and, in particular, a well known problem of Chowla in this context is resolved. The work depends on an application of a theorem of the first author concerning linear forms in the logarithms of algebraic numbers.     

On a conjecture of Chowla and Chowla

The equation y² ≡ x(x + a₁)(x + a₂) … (x + a_r) (mod p), where a₁, a₂, …, a_r are integers is shown to have a solution in integers x, y with 1 ≦ x ≦C, where C is a constant depending only on a₁, a₂, …, a_r.     

On a theorem of Chowla

Let be a finite field with q=p^felements, where p is a prime number and f is a positive integer. For a nonprincipal multiplicative character χ and a nontrivial additive character ψ on , it is well known that Gauss sum has absolute value . In this paper, we investigate when is a root of unity.     

A problem of Chowla revisited

In 1964, S. Chowla asked if there is a non-zero integer-valued function f with prime period p such that f(p)=0 and

     

On a conjecture of Chowla and Walum

We show the asymptotic behaviour of the mean square of the sum n c√x naPk(x/n),where Pk(x) = Bk({x}) and Bk(x) denotes the Bernoulli polynomial of degree k and c 0 is a real number such that c2 is rational.Our result implies that a conjecture of Chowla and Walum is true on average.     

On a conjecture of S. Chowla and of S. Chowla and H. Walum,I

As an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, as x → ∞, $\text{Σ}{}_{\mathrm{n\; \le \; \surd x}}\text{n}{}^{\mathrm{a}}\text{B}{}_{\mathrm{r}}\text{({}\text{x}\text{n}\text{}) = O(x}{}^{\mathrm{<mtext>a</mtext><mtext>2</mtext><mtext>\; +\; </mtext><mtext>1</mtext><mtext>4</mtext><mtext>\; +\; \epsilon </mtext>}}\text{)}$ holds for each ε > 0. Here integers a ≥ 0 and r ≥ 1 are given. B_r(x) denotes the rth Bernoulli polynomial and {x} denotes the fractional part of x. The special case a = 0, r = 2 of this conjecture was also mentioned by S. Chowla. In this paper we prove this conjecture for all $\text{e ≥}\text{1}\text{2}$ and r ≥ 2 with ε = 0 (with x^ε replaced by log x in case $\text{a =}\text{1}\text{2}$).     

On a conjecture of S. Chowla and of S. Chowla and H. Walum,II

In 1965, Chowla and Walum conjectured that, $\text{G}{}_{\mathrm{a,k}}\text{(x):=}\text{Σ}{}_{\mathrm{n\; \le \; \surd x}}\text{n}{}^{\mathrm{a}}\text{P}{}_{\mathrm{k}}\text{(}\text{x}\text{n}\text{) = O(x}{}^{\mathrm{<mtext>a</mtext><mtext>2</mtext><mtext>\; +\; </mtext><mtext>1</mtext><mtext>4</mtext><mtext>\; +\; \epsilon </mtext>}}\text{)}$ holds for each ε > 0 and x → ∞, where integers a ≥ 0 and k ≥ 1 are given and P_k is the periodic Bernoulli function of order k. Recently, the authors established this conjecture in case k ≥ 2 and $\text{a >}\text{1}\text{2}$ with ε = 0 while if $\text{a =}\text{1}\text{2}$, with a log factor. In this paper, it is proven that the conjecture, in case $\text{|a| <}\text{1}\text{2}$, k = 2, is true “on average”.     

On a function due to S. Chowla

Let θ(k, pⁿ) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ ? + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0}$ (mod pⁿ) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then $\text{θ(k, p}{}^{\mathrm{n}}\text{) ≤ k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n.     

On two conjectures of P. Chowla and S. Chowla concerning continued fractions

Summary The alternating sum of the partial quotients in the primitive period of a continued fraction expansion of √D is determined mod 2 and mod 3. To Professor BeniaminoSegre on his 70-th birthday. Pervenuto il 7 maggio 1973.     

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

     

Moduli problem and points on some twisted Shimura varieties of PEL type

In this article we describe the moduli problem of a “twist” of some simple Shimura varieties of PEL type that appear in Kottwitz's papers [R. Kottwitz, Shimura varieties and λ-adic representations, in: Automorphic Forms, Shimura Varieties and L-Functions, part 1, in: Perspect. Math., vol. 10, Academic Press, San Diego, CA, 1990, pp. 161-209; R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (2) (1992) 373-444] and [R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992) 653-665] and then, using the moduli problem, we compute the cardinality of the set of points over finite fields of the twisted Shimura varieties. Using this result, we compute the zeta function of the twisted varieties. The twist of the Shimura varieties is done by a mod q representation of the absolute Galois group of the reflex field of the Shimura varieties.     

On the characterization of complex Shimura varieties

In this paper we recall basic properties of complex Shimura varieties and show that they actually characterize them. This characterization immediately implies the explicit form of Kazhdan's theorem on the conjugation of Shimura varieties. It also implies the existence of unique equivariant models over the reflex field of Shimura varieties corresponding to adjoint groups and the existence of a p-adic uniformization of certain unitary Shimura varieties. In the appendix we give a modern formulation and a proof of Weil's descent theorem.     

On the Group of Automorphisms of Shimura Curves and Applications

Let V _D be the Shimura curve over attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of V _D and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over and over and we use it to study the set of rational points on V _D over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of V _D .     

On the Hasse principle for Shimura curves

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X ₀ ^D (N)_/ℚ or X ₁ ^D (N)_/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.     

Proof of a conjecture of S. Chowla

Let θ(k, p) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ … + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0(}\text{mod}\text{p)}$ has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: $\text{θ(k) = O(k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}\text{)}$.     

On the slope stratification of certain Shimura varieties

In this paper we study the slope stratification on the good reduction of the type C family Shimura varieties. We show that there is an open dense subset U of the moduli space such that any point in U can be deformed to a point with a given lower admissible Newton polygon. For the Siegel moduli spaces, this is obtained by F. Oort which plays an important role in his proof of the strong Grothendieck conjecture concerning the slope stratification. We also investigate the p-divisible groups and their isogeny classes arising from the abelian varieties in question. Received: 10 November 2004; 13 February 2005 The research is partially supported by NSC 93-2119-M-001-018.