Related papers: $\mathcal{L}$-invariants and exceptional zeros of …

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We…

Let f be a modular form of weight 2 and trivial character. Fix also an imaginary quadratic field K. We use work of Bertolini-Darmon and Vatsal to study the mu-invariant of the p-adic Selmer group of f over the anticyclotomic Zp-extension of…

Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over…

We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the…

Let X be a smooth curve over a finite field of characteristic p, let l be a prime number different from p, and let L be an irreducible lisse l-adic sheaf on X whose determinant is of finite order. By a theorem of Lafforgue, for each prime…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

In this note, we describe a conjecture, that, for an odd prime p, relates special values of a cup product pairing on cyclotomic p-units in the pth cyclotomic field to the L-values of newforms satisfying modulo p congruences with Eisenstein…

For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij}…

Let $F$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $F$. Thus if $V$ is any finite dimensional $C_p$-representation…

Let $p\geq 5$ be a prime number. We consider the Iwasawa $\lambda$-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal…

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution,…

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

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…

Let $G$ be a reductive group quasi-split at $p$. Using arguments of Hansen--Thorne, we show that under the non-abelian Leopoldt conjecture (NALC), Hansen's $p$-adic overconvergent cohomology eigenvariety for $G$ is \'etale over its image in…

Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $\rho$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the \'etale cohomology of quaternion Shimura curves (thus $\rho$…

We provide a new, elementary proof of the multiplicative independence of pairwise distinct $\mathrm{GL}_2^+(\mathbb{Q})$-translates of the modular $j$-function, a result due originally to Pila and Tsimerman. We are thereby able to…

A fundamental question, first raised by Langlands, is to know whether the Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose standard…

We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar…

In this note we propose a new construction of cyclotomic p-adic L-functions attached to classical modular cuspidal eigenforms. This allows us to cover most known cases to date and provides a method which is amenable to generalizations to…