Related papers: On the Tamagawa number conjecture for modular form…

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa…

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin…

We prove the local equivariant Tamagawa number conjecture for the motive of an abelian extension of an imaginary quadratic field with the action of the Galois group ring for all split primes p not equal to 2 or 3 at all negative integer…

Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $\theta_n(f)$, both of which encode arithmetic data about $f$ along the…

Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of…

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a…

Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero…

Let $f(z)=\sum_{n=1}^\infty \lambda_f(n)e^{2\pi i n z}\in S_{k}^{new}(\Gamma_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $\Gamma_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We…

In this paper, we prove Deligne's conjecture on the algebraicity of critical values of symmetric power $L$-functions associated to modular forms of weight greater than four. We also prove new cases of Blasius' conjecture on the algebraicity…

In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one…

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under…

We give a new construction of a $p$-adic $L$-function $\mathcal{L}(f,\Xi)$, for $f$ a holomorphic newform and $\Xi$ an anticyclotomic family of Hecke characters of $\mathbb{Q}(\sqrt{-d})$. The construction uses Ichino's triple product…

Let $p$ be an odd prime. Let $f_1$ and $f_2$ be weight-two Hecke eigen-cuspforms with isomorphic residual Galois representations at $p$. Greenberg--Vatsal and Emerton--Pollack--Weston showed that if $p$ is a good ordinary prime for the two…

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights…

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special…

In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

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…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the…