Function Fields Of P-adic Curves Research Articles

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

Reciprocity obstruction to strong approximation over p-adic function fields

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SLn with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.

Diagonal forms of higher degree over function fields of p-adic curves

We investigate diagonal forms of degree [Formula: see text] over the function field [Formula: see text] of a smooth projective [Formula: see text]-adic curve: if a form is isotropic over the completion of [Formula: see text] with respect to each discrete valuation of [Formula: see text], then it is isotropic over certain fields [Formula: see text], [Formula: see text] and [Formula: see text]. These fields appear naturally when applying the methodology of patching; [Formula: see text] is the inverse limit of the finite inverse system of fields [Formula: see text]. Our observations complement some known bounds on the higher [Formula: see text]-invariant of diagonal forms of degree [Formula: see text]. We only consider diagonal forms of degree [Formula: see text] over fields of characteristic not dividing [Formula: see text].

Hermitian 𝑢-invariants over function fields of 𝑝-adic curves

Letppbe an odd prime. LetFFbe the function field of app-adic curve. LetAAbe a central simple algebra of period 2 overFFwith an involutionσ\sigma. There are known upper bounds for theuu-invariant of hermitian forms over(A,σ)(A, \sigma ). In this article we compute the exact values of theuu-invariant of hermitian forms over(A,σ)(A, \sigma ).

Hasse principle for hermitian spaces over semi-global fields

In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local–global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves.

Tame division algebras of prime period over function fields of p-adic curves

Let F be a field finitely generated and of transcendence degree one over a p-adic field, and let ℓ ≠ p be a prime. Results of Merkurjev and Saltman show that H2(F, µℓ) is generated by ℤ/ℓ-cyclic classes. We prove the “ℤ/ℓ-length” in H2(F, µℓ) is less than the ℓ-Brauer dimension, which Salt-man showed to be three. It follows that all F-division algebras of period ℓ are crossed products, either cyclic (by Saltman’s cyclicity result) or tensor products of two cyclic F-division algebras. Our result was originally proved by Suresh when F contains the ℓ-th roots of unity µℓ.

Admissibility of groups over function fields of p-adic curves

Let K be a field and G a finite group. The question of ‘admissibility’ of G over K was originally posed by Schacher, who gave partial results in the case K=Q. In this paper, we give necessary conditions for admissibility of a finite group G over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over Qp(t). We also prove a certain Hasse principle for division algebras over such fields.

Theu-invariant of the function fields ofp-adic curves

The<i>u</i>-invariant of the function fields of<i>p</i>-adic curves

The u-invariant of the function fields of p-adic curves

The u-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the u-invariant of function fields of p-aidc curves is 8. In this paper, we answer this question in the affirmative for function fields of nondyadic p-adic curves.

Isotropy of quadratic forms over function fields ofp-adic curves

Isotropy of quadratic forms over function fields ofp-adic curves