Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

Factors influencing the flush response and flight initiation distance of three owl species in the Andaman Islands

Effects of anthropogenic pressures on birds of the Andaman Islands have been documented to some extent, however studies on the effect of human activities on the behavioural response of these birds are limited. This study assessed the anti-predatory behaviour (flush response - FR and flight initiation distance - FID) of three owl species (Otus sunia, Otus balli, and Ninox obscura) in response to human stimuli and factors influencing it on the Andaman Islands. In total, 63 % of owls flushed from their roost sites in response to approaching human, and such a response varied between species. Similarly, FID varied widely among the species ranging from 4.23 to 6.73 m. The FR of N. obscura was influenced by the count of climbers, presence of spine, and branch status, while roost height, ambient temperature, and lower count of climbers contributed to a higher FID. For the two Otus species, camouflage and pairing were found to influence their FR while FID of O. balli was influenced by roost height, pairing, and presence of spines. Our results indicated that the anti-predatory behaviour of owls on the Andaman Islands was species- and site-specific and prolonged disturbance to their roost sites may affect the survival and reproductive rate of these owls.

Erratum: “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”

We wish to point out errors in the paper “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the “ramified case” of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the $(S,n)$-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected. Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let $I$ be an ideal such that $I^n$ is principal in ${\mathcal{O}}_{F,S}$. Write $I^n=(f^{-1})$. Then the Kummer cocycles $k_n(f)$ will be in $Z^1(U, {{\mathbb{Z}}/{n}{\mathbb{Z}}})$. For any $a\in F$, denote by $a_S$ its image in $\prod _{v\in S} F_v$. Thus, we get an element $$\begin{equation*}[f]_{S,n}:=[(k_n(f), k_{n^2}(f_S), 0)] \in Z^1(U, {{{\mathbb{Z}}}/{n}{{\mathbb{Z}}}} \times_S{\mathbb{Z}}/n^2{\mathbb{Z}}),\end{equation*}$$which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both $\mu _{n^2}$ and $\mu _n$.)

DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

Decomposition of the Geometric Kernel

This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights by standard results in Arakelov theory is achieved. The chapter proceeds by reviewing the definition of the Néeron–Tate height and shows how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. The chapter proves that the contribution of the Hodge bundles in the height series is zero. It also compares two kernel functions and states the computational result. It concludes by deducing the kernel identity.

Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms

We study the interplay between canonical heights and endomorphisms of an abelian variety A over a number field k. In particular we show that whenever the ring of endomorphisms defined over k is strictly larger than there will be -linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of A(k).