An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field 𝔽 q with characteristic greater than 3 , Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is 1 q + O ( q - 2 ) . We improve this result, showing that the density is exactly q - 1 q 2 - q + 1 . We consider the problem of determining the density of pairs of conics satisfying the Poncelet n -gon condition for larger n . We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is 1 q + O ( q - 3 / 2 ) . We show that when n is an odd integer coprime to q , the density of pairs of smooth conics satisfying this condition is d ( n ) - 1 q + O ( q - 3 / 2 ) , where d ( n ) is the number of divisors of n . More generally, we conjecture that the density of pairs of conics satisfying the Poncelet n -gon condition is d ′ ( n ) / q in general, where d ′ ( n ) is the number of divisors of n not equal to 1 or 2 . Our argument involves analyzing the n -torsion points on a certain elliptic curve over the function field K = 𝔽 q ( λ ) .
Motivated by the list of problems for moments of L -functions due to Hamieh et al. [7], we prove an asymptotic formula for the cubic moment of central L -values associated to holomorphic cusp forms of level one, where the weights k lie in a short interval K - Δ < k < K + Δ with K ε ≤ Δ ≤ K 1 - ε , as K → ∞ .
Let k ⊂ ℂ be a number field and ℰ be an elliptic curve defined over k ( t ) that is isomorphic to the generic fiber of an elliptic surface π : 𝒮 ℰ → ℙ k 1 . For any extension 𝒦 ⊆ ℂ of k , the set ℰ ( 𝒦 ( t ) ) of 𝒦 ( t ) -rational points of ℰ is known to be a finitely generated abelian group. The splitting field of ℰ defined over k ( t ) is the smallest finite extension 𝒦 ⊂ ℂ of k such that ℰ ( ℂ ( t ) ) ≅ ℰ ( 𝒦 ( t ) ) . In this paper, we consider the elliptic K 3 surfaces defined over k = ℚ with the generic fiber given by the Weierstrass equation ℰ n : y 2 = x 3 + t n + 1 / t n , 1 ≤ n ≤ 6 , and determine the splitting field 𝒦 n , and find an explicit set of linearly independent generators for ℰ n ( 𝒦 𝓃 ( t ) ) for 1 ≤ n ≤ 6 .
A critical value on an abelian group G of odd order d is a value λ such that the functional equation f ☆ f ( 2 t ) = λ f ( t ) 2 on G has a nonzero solution f . We construct many critical values by using abelian varieties with complex multiplication.
Let p >= 5 be a prime and k be an integer in [2p + 2, p(2)-p + 3]. Let b be in [2, p] such that k-2 equivalent to b mod p-1 and c := k-2-b/p-1 . We give an explicit radius of local constancy in the weight space of the mod p reduction of the two dimensional crystalline representations V-k,V-ap of Gal(Q(p)(-)/Q(p)), where the slope v(a(p)) is constrained to be in (1, c) and non-integral. We use the mod p local Langlands correspondence for GL(2)(Q(p)) to compute the mod p reductions explicitly under additional conditions on the slope. We show that the reduction in the disk depends only on k and [v(a(p))]. As an application, we obtain explicit mod p reductions at many new values of k and a(p). We also obtain an explicit radius of local constancy in the ap space (for a fixed k as above) which is bigger than the explicit radius given in a result of Berger.
