In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p1 and m\geq 3 if q is even and deg f' \leq 1. We show that this estimation fails in the neglected cases. Let \pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \FF_q. For relatively prime polynomials f,D\in \FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\pi_q(k)/\phi(D)|\leq c(k)\pi_q(k)q^{-1/2}/\phi(D), as long as 1\leq \deg D\leq k-3 (or \leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types.Comment: Changes in the introduction, accepted to Duk

We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in ℓ-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work by Kohel in his thesis. We also give a criterion under which the ring generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. The most difficult part of these computations is the calculation of the trace of these cycles. We show that a generalization of Schoof's algorithm can accomplish this computation efficiently. that led to a significant improvement in the running time analysis of the generalization of Schoof's algorithm, and for outlining the proofs of Proposition 2.4 and Lemma 6.9. Finally, we thank the Women in Numbers 4 conference and BIRS, for enabling us to start this project in a productive environment. K.E. and T.M. were partially supported by National Science Foundation awards DMS-1056703 and CNS-1617802. Isogeny graphs2.1. Definitions and properties. In this section, we recall several definitions and notation that are used throughout. We refer the reader to [Sil09] and [Koh96] for a detailed overview on some of the below. Let k be a field of characteristic p > 3.

Landau's theorem asserts that the asymptotic density of sums of two squares in the interval 1 ≤ n ≤ x is K/ √ log x, where K is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals |n − x| ≤ x ǫ for a fixed ǫ and x → ∞.This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f 0 ∈ F q [T ] of degree n and take ǫ with 1 > ǫ ≥ 2 n . Then the asymptotic density of polynomials f in the 'interval' deg(fn as q → ∞. This density agrees with the asymptotic density of such monic f 's of degree n as q → ∞, as was shown by the second author, Smilanski, and Wolf.A key point in the proof is the calculation of the Galois group of f (−T 2 ), where f is a polynomial of degree n with a few variable coefficients: The Galois group is the hyperoctahedral group of order 2 n n!.

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime polynomial values of n linear functions, in the limit of a large finite field.

We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval "of size E" in this setting is any additive translate of the space of global sections of a sufficiently positive divisor E by a suitable rational function f. Our main theorem gives an asymptotic count of irreducible elements in short intervals on a curve in the "large q" limit, uniformly in f and E. This result provides a function field analogue of an unresolved short interval conjecture over number fields, and extends a theorem of Bary-Soroker, Rosenzweig, and the first author, which can be understood as an instance of our result for the special case of a divisor E supported at a single rational point on the projective line. CONTENTS