arXiv subjects

Daniel Macias Castillo

Publications and source records attributed to Daniel Macias Castillo.

12 recordsLinked to original sources

arXiv2026

On Selmer complexes, Stark systems and derived $p$-adic heights

We develop the theory of Nekovář's Selmer complexes. We prove that, under mild hypotheses, Nekovář's Selmer complexes are canonically quasi-isomorphic to ``Poitou-Tate complexes", which arise from Poitou-Tate global duality exact sequences. We give two applications. Firstly, we prove that the determinant of a Selmer complex is canonically isomorphic to the module of Stark systems and, by using this result, we construct a canonical ``Heegner point Stark system" which controls Selmer groups. Secondly, we prove that the derived $p$-adic height pairing of Bertolini-Darmon concides with that of Nekovář.

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arXiv2025

The refined class number formula for Drinfeld modules

Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive explicit consequences for the Galois structure of the Taelman class group of $E$ over $K$.

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arXiv2023

On Weil-Stark elements, I: general properties

We construct a canonical family of elements in the reduced exterior power lattices of the unit groups of global fields. We prove that this family recovers the theory of cyclotomic elements in real abelian fields and also establish detailed arithmetic properties of its elements in the general case.

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arXiv2023

Annihilating class groups in $p$-elementary extensions

We derive new cases of conjectures of Rubin and of Burns--Kurihara--Sano concerning derivatives of Dirichlet $L$-series at $s = 0$ in $p$-elementary extensions of number fields for arbitrary prime numbers $p$. In naturally arising examples of such extensions one therefore obtains annihilators of class groups from $S$-truncated Dirichlet $L$-series for `large-enough' sets of places $S$.

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arXiv2023

On Weil-Stark elements, II: refined Stark conjectures

The theory of Weil-Stark elements is used to develop an axiomatic approach to the formulation of refined versions of Stark's Conjecture. This gives concrete new results concerning leading terms of Artin $L$-series and arithmetic properties of Stark elements.

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arXiv2023

Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves

Let $A/\mathbb{Q}$ be a Jacobian variety and let $F$ be a totally real, tamely ramified, abelian number field. Given a character $ψ$ of $F/\mathbb{Q}$, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value $\mathcal{L}(A,ψ,1)$ at $z=1$ of the Hasse-Weil-Artin $L$-function of the $ψ$-twist of $A$. We formulate a conjecture regarding the integrality properties of the family of normalised $L$-values $(\mathcal{L}(A,ψ,1))_ψ$, and its relation to the Tate-Shafarevich group of $A$ over $F$. We numerically investigate our conjecture through $p$-adic congruence relations between these values.

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arXiv2022

On non-abelian higher special elements of $p$-adic representations

We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.

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arXiv2021

On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series

We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of Deligne and Gross and delicate p-adic congruence relations that involve the bi-extension height pairing of Mazur and Tate and are related to key aspects of non-commutative Iwasawa theory. In important special cases we provide strong evidence, both theoretical and numerical, in support of the conjectures. We also point out an inconsistency in a conjecture of Bradshaw and Stein regarding Zhang's Theorem on Heegner points and suggest a possible correction.

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arXiv2021

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.

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arXiv2015

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell-Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s=1 of derivatives of the Hasse-Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate-Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

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arXiv2015

On the Galois structure of Selmer groups

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F. We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.

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arXiv2014

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

Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate.

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