arXiv subjects

Guhan Venkat

Publications and source records attributed to Guhan Venkat.

7 recordsLinked to original sources

arXiv2025

A $p$-Converse theorem for Real Quadratic Fields

Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an application to a $p$-converse theorem over $\mathbb{Q}$.

math.NT↗

arXiv2023

The rationality of Stark-Heegner cycles attached to Bianchi modular forms -- The base-change scenario

We study Stark-Heegner cycles attached to Bianchi modular forms, that is automorphic forms for GL(2) over an imaginary quadratic field F . The Stark-Heegner cycles are local cohomology classes in the p-adic Galois representation associated to the Bianchi eigenform. They are conjectured to be the restriction (at a prime p) of global cohomology classes in the (semistable) Bloch-Kato Selmer group defined over ring class fields of a relative quadratic extension K/F. In this paper, we show that these conjectures hold when the Bianchi eigenform is the base-change of a classical elliptic cuspform.

math.NT↗

arXiv2021

Iwasawa theory for Symmetric Square of non-$p$-ordinary eigenforms

Let $f$ be a normalized cuspidal eigen-newform of level coprime to $p$ with $a_p(f)=0$. We formulate both integral signed Iwasawa main conjectures and analytic Iwasawa man conjectures attached to the symmetric square motive of $f$ twisted by an auxiliary Dirichlet character. We show that the Beilinson--Flach elements attached to the symmetric square motive factorize into integral signed Beilinson--Flach elements, giving evidence towards the existence a rank-two Euler system predicted by Perrin-Riou. We use these integral elements to prove one inclusion in the integral and analytic Iwasawa main conjectures.

math.NT↗

arXiv2020

Stark-Heegner cycles attached to Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms.

math.NT↗

arXiv2018

Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms

Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.

math.NT↗

arXiv2018

Abelianization of p-groups with derived subgroup of prime order

In this paper, we arrive at the structure for $G^{ab}(= G/G')$ from a presentation for $G$. We also prove that the frattini subgroup of the abelianization is $G^p$ or the quotient of $G^p$ by the derived subgroup $G'$.

math.GR↗

arXiv2018

Extensions of Finite Abelian Groups

We study group extensions of Finite Abelian Groups using matrices. We also prove a Theorem for equivalence of extensions using matrices.

math.GR↗