arXiv subjects

Rodolfo Venerucci

Publications and source records attributed to Rodolfo Venerucci.

4 recordsLinked to original sources

arXiv2025

Syntomic formalism with coefficients

This paper provides the technical tools needed in ongoing work of the authors to compute p-adic \'etale Abel-Jacobi maps in order to obtain explicit reciprocity laws for GSp4. In particular, we define and study syntomic polynomial cohomology for filtered Frobenius log-isocrystals over proper and semistable schemes over the ring of integers of a local field, with smooth generic fiber, endowed with horizontal divisors. We introduce syntomic polynomial cohomology with support, we define Kunneth morphisms, trace maps and cup products, Gysin maps with respect to divisors and we study some of their properties. We establish the relation with Hyodo-Kato cohomology of the special fiber and de Rham cohomology of the generic fiber. We also introduce overconvergent variants with and without support by restricting to open smooth formal subschemes. Most of all, in case that the filtered log-isocrystal is associated to a p-adic local system on the generic fiber, we establish comparison morphisms between \'etale and syntomic cohomology and compatibilities with Hochschild-Serre morphisms and between Gysin morphisms.

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arXiv2023

The anticyclotomic main conjectures for elliptic curves

The goal of this article is to obtain a proof of the Main conjectures of Iwasawa theory for rational elliptic curves over anticyclotomic extensions of imaginary quadratic fields, under mild arithmetic assumptions, both in the case where the rational prime $p$ is good ordinary or supersingular.

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arXiv2015

On the p-converse of the Kolyvagin-Gross-Zagier theorem

Let $A/\mathbb{Q}$ be an elliptic curve having split multiplicative reduction at an odd prime $p$. Under some mild technical assumptions, we prove the statement: $$rank_{\mathbb{Z}}A(\mathbb{Q})=1 \ \ and\ \ \ \#\Big(\underline{III}(A/\mathbb{Q})[p^\infty]\Big)<\infty\ \ \implies\ \ \mathrm{ord}_{s=1}L(A/\mathbb{Q},s)=1,$$ thus providing a "$p$-converse" to a celebrated theorem of Kolyvagin-Gross-Zagier.

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arXiv2015

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\infty}$. We prove a $p$-adic Gross$-$Zagier formula, expressing the quadratic term of the Taylor expansion of $L_{p}(f_{\infty},k,s)$ at $(k,s)=(2,1)$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $A(\mathbb{Q})$.

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