International Journal of Number Theory

Let S₁ be a Del Pezzo surface of degree 1 over a number field k. We establish a criterion for the existence of a nontrivial element of order 5 in the Brauer group of S₁ in terms of certain Galois-stable configurations of exceptional divisors on this surface.

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.

We investigate the transcendental nature of the sum

In [1], the authors prove a conjecture of Calegari and Stein regarding mod p congruences between cusp forms of weight four for Γ₀(p) and the derivatives of cusp forms of weight two for the same congruence subgroup. In this paper, we investigate whether or not the result remains valid for cusp forms of level N_p.

Let Γ ⊂ SL₂(ℝ) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m ≥ 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L¹-Poincaré series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms for general Γ having at least one cusp. We obtain explicit results for congruence subgroups Γ(N), Γ₀(N), and Γ₁(N) (N ≥ 1).

Let be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method, we show that L^(k)(½, χ) ≠ 0 for almost all the characters when k and q are large. Here L(s, χ) is the Dirichlet L-function associated to the character χ.

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic C. This can be expressed as a relation between the period spectrum and the ortholength spectrum of C. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along C as well estimates on the lengths of geodesic segments which start and end orthogonally on C. Variant trace formulas also lead to several simultaneous nonvanishing results for different periods.

Very recently, Nesterenko proved a p-adic analogue of his famous dimension estimate from 1985. The main aim of our present paper is to use this criterion to obtain lower bounds for the dimension of ℚ-vector spaces spanned by the values at certain rational points of p-adic solutions of a class of linear q-difference equations. For the application of Nesterenko's new estimate, we first need a p-adic analogue of Töpfer's results on entire solutions of such functional equations, and secondly, very precise evaluations of certain p-adic Schnirelman integrals.

In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp₂(F) (the rank 2 symplectic group) and its metaplectic cover for F a finite extension of ℚ_p with p ≠ 2. We determine a set of "good test vectors" for the irreducible constituents of unramified principal series representations for these groups as well as a set of "good theta test sections" in a family of degenerate principal series representations of Sp₄(F) and . Determining "good test data" that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type (Sp₂(F), O(V)) (respectively ) where V is a quadratic space of an even (respectively odd) dimension.

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K^×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.