The Hasse Norm Principle For Biquadratic Extensions

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.Comment: 19 pages. Proof of Theorem 1 improved/simplified. Accepted by Journal de Th\'eorie des Nombres de Bordeau

Oscillatory integrals with uniformity in parameters

We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.Comment: Final version. To appear in Journal de Th\'eorie des Nombres de Bordeaux. Portions of this work originally appeared in arXiv:1608.06854 (Petrow-Young) and arXiv:1701.07507 (Kiral-Young). arXiv admin note: text overlap with arXiv:1701.0750

A local-global principle for rational isogenies of prime degree

Let K be a number field. We consider a local-global principle for elliptic curves E/K that admit (or do not admit) a rational isogeny of prime degree n. For suitable K (including K=Q), we prove that this principle holds when n = 1 mod 4, and for n < 7, but find a counterexample when n = 7 for an elliptic curve with j-invariant 2268945/128. For K = Q we show that, up to isomorphism, this is the only counterexample.Comment: 11 pages, minor edits, to appear in Journal de Th\'eorie des Nombres de Bordeau

Higher congruences between newforms and Eisenstein series of squarefree level

Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $\Gamma_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert--Samuel multiplicities.Comment: 19 pages. Minor revisions. Accepted for publication in The Journal de Th\'eorie des Nombres de Bordeau

An Infinitesimal pp-adic Multiplicative Manin-Mumford Conjecture

Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(\zeta_1-1,\ldots,\zeta_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just the formal multiplicative group and we extend the results to this setting.Comment: 14 pages, minor corrections, slightly strengthened the statement of the main theorem. Accepted for publication in Journal de Th\'eorie des Nombres de Bordeau

Criteria for irreducibility of mod p representations of Frey curves

Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the representation on E[p] is irreducible. Our sufficient conditions are often satisfied for Frey elliptic curves associated to solutions of Diophantine equations; in that context, the irreducibility of the mod p representation is a hypothesis needed for applying level-lowering theorems. We illustrate our approach by improving on an existing result for Fermat-type equations of signature (13, 13, p).Comment: Some minor misprints have been corrected. The paper will appear in Journal de Th\'eorie des Nombres de Bordeau

On Salem numbers, expansive polynomials and Stieltjes continued fractions

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $\beta$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over a Salem polynomial having simple roots, not cancelling at $\pm 1$, coming from monic expansive polynomials of constant term equal to their Mahler measure, has a semigroup structure. The sets of corresponding generalized Garsia numbers inherit this semi-group structure.Comment: 35 pages, Journal de Th{\'e}orie des nombres de Bordeaux, Soumissio