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Christopher Skinner

Publications and source records attributed to Christopher Skinner.

At least 19 recordsLinked to original sources
arXiv2026

A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under $p$-adic logarithms of abelian varieties. We largely consider situations most applicable to ${\mathrm GL}_2$-type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the $p$-adic analytic subgroup theorem.

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arXiv2024

Zeta elements for elliptic curves and applications

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two different $p$-adic $L$-functions associated to $E$ over $L$ via explicit reciprocity laws at the primes above $p$. We formulate a main conjecture for $E$ over $L$ in terms of the zeta element, mediating different main conjectures in which the $p$-adic $L$-functions appear, and prove some results toward them. The zeta element has various applications to the arithmetic of elliptic curves. This includes a proof of main conjecture for semistable elliptic curves $E$ over $\mathbb{Q}$ at supersingular primes $p$, as conjectured by Kobayashi in 2002. It leads to the $p$-part of the conjectural Birch and Swinnerton-Dyer (BSD) formula for such curves of analytic rank zero or one, and enables us to present the first infinite families of non-CM elliptic curves for which the BSD conjecture is true. We provide further evidence towards the BSD conjecture: new cases of $p$-converse to the Gross--Zagier and Kolyvagin theorem, and $p$-part of the BSD formula for ordinary primes $p$. Along the way, we give a proof of a conjecture of Perrin-Riou connecting Beilinson--Kato elements with rational points.

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arXiv2024

Anti-cyclotomic Euler system of diagonal cycles

We construct split anti-cyclotomic Euler systems for Galois representations attached to certain RACSDC automorphic representations on the group $\mathrm{GL}_n\times\mathrm{GL}_{n+1}$. As a result, we make progress towards certain rank 1 cases of the Beilinson--Bloch--Kato conjecture for those representations.

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arXiv2024

Base change and Iwasawa Main Conjectures for ${\rm GL}_2$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $p$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $K$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa Main Conjectures for the $\mathbb{Z}_p$-cyclotomic and $\mathbb{Z}_p$-anticyclotomic deformations of $E$ over $\mathbb{Q}$ and $K$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $E$ over $K$, via which the sought after main conjectures are deduced from Wan's divisibility towards a three-variable main conjecture for $E$ over a quartic CM field containing $K$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $E$ over $K$. The aforementioned one-variable main conjectures imply the $p$-part of the conjectural Birch and Swinnerton-Dyer formula for $E$ if ${\rm ord}_{s=1}L(E,s)\leq 1$. They are also an ingredient in the proof of Kolyvagin's conjecture and its cyclotomic variant in our joint work with Grossi.

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arXiv2023

Non-vanishing of Kolyvagin systems and Iwasawa theory

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.

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arXiv2023

Mazur's main conjecture at Eisenstein primes

Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(\phi)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $\phi\vert_{G_p}\neq 1,\omega$, where $G_p$ is a decomposition group at $p$ and $\omega$ is the Teichm\"uller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes.

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arXiv2020

An Euler system for GU(2, 1)

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of GL(3) over imaginary quadratic fields, using the cohomology of Shimura varieties for GU(2, 1).

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arXiv2020

On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dyer formula in analytic rank $1$ for Eisenstein primes $p$.

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arXiv2020

p-adic L-functions for unitary groups

This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three named authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent results, including the first named author's construction of Eisenstein measures and $p$-adic differential operators, Part II of the present paper provides the calculations of local $ζ$-integrals occurring in the Euler product (including at $p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

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arXiv2019

Higher Hida theory and p-adic L-functions for GSp(4)

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).

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arXiv2018

On the Iwasawa main conjectures for modular forms at non-ordinary primes

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by Büyükboduk--Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-type for non-ordinary primes $p>2$.

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arXiv2017

Syntomic regulators of Asai--Flach classes

In this paper, we derive a formula for the p-adic syntomic regulators of Asai--Flach classes. These are cohomology classes forming an Euler system associated to a Hilbert modular form over a quadratic field, introduced in an earlier paper (arXiv:1607.07813) by Antonio Lei and the first and third authors. The formula we develop here is expressed in terms of differential operators acting on overconvergent Hilbert modular forms; it is analogous to existing formulae for the regulators of Beilinson--Flach classes, but a novel feature is the appearance of a projection operator associated to a critical-slope Eisenstein series. We conclude the paper with numerical calculations giving strong evidence for the non-vanishing of these regulators in an explicit example.

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arXiv2015

The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One

Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{Ω_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.

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arXiv2014

Indivisibility of Heegner points in the multiplicative case

For certain elliptic curves $E$ over $\mathbb{Q}$ with multiplicative reduction at a prime $p\geq 5$, we prove the $p$-indivisibility of the derived Heegner classes defined with respect to an imaginary quadratic field $K$, as conjectured by Kolyvagin. The conditions on $E$ include that $E[p]$ be irreducible and not finite at $p$ and that $p$ split in the imaginary quadratic field $K$, along with certain $p$-indivisibility conditions on various Tamagawa factors. The proof extends the arguments of the second author for the case where $E$ has good ordinary reduction at~$p$.

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arXiv2014

Multiplicative reduction and the cyclotomic main conjecture for $\mathrm{GL}_2$

We show that the cyclotomic Iwasawa--Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at $p$, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument.

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arXiv2014

A converse to a theorem of Gross, Zagier, and Kolyvagin

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s)=1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

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