On critical values of L-functions of potentially automorphic motives

In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.Comment: 24 page

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for GL(2)/K, proving an etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a regular $p$-stabilised newform of weight $k$ at least 2 without CM by $K$. (1) We construct a two-variable $p$-adic $L$-function attached to the base-change of $f$ to $K$ under assumptions on $f$ that we conjecture always hold, in particular making no assumption on the slope of $f$. (2) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic $L$-functions of classical base-change Bianchi cusp forms in families. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space.Comment: 26 pages, with a 3 page appendix by Carl Wang-Erickson. Comments welcome! Changes for v5: added contents, minor changes to exposition. v4: corrected funding acknowledgements. v3: This version has a new introduction, has been reorganised and greatly shortened, and incorporates minor corrections. v2: minor correction

On critical values of L-functions of potentially automorphic motives

In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.Postprint (author's final draft

On pp-adic LL-functions for GL2nGL_{2n} in finite slope Shalika families

Let $p$ be a prime number and $\mathrm{Gal}_p$ be the Galois group of the maximal abelian extension unramified outside $p\infty$ of a totally real number field $F$ of degree $d$. Using automorphic cycles, we construct evaluation maps on the parahoric overconvergent cohomology for $\mathrm{GL}_{2n}/F$ in degree $d (n^2 + n -1)$, which allow us to attach a distribution on $\mathrm{Gal}_p$ of controlled growth to any finite slope overconvergent $U_p$-eigenclass. When the eigenclass comes from a non-critical refinement $\tilde\pi$ of a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model, we prove this distribution interpolates all Deligne-critical $L$-values of $\pi$, giving the first construction of $p$-adic $L$-functions in this generality beyond the $p$-ordinary setting. Further, under some mild assumptions we use our evaluation maps to show that $\tilde\pi$ belongs to a unique $(d+1)$-dimensional Shalika family in a parabolic eigenvariety for $\mathrm{GL}_{2n}/F$, and that this family is \'etale over the weight space. Finally, under a hypothesis on the local Shalika models at bad places which is empty for $\pi$ of level 1, we construct a $p$-adic $L$-function for the family.Comment: 64 pages (inc. glossary of notation), comments welcome! Changes for v2,v3: minor corrections and expositional improvement

On the 22-Selmer group of Jacobians of hyperelliptic curves

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of $\mathcal{C}$ in terms of the class group of the $K$-algebra $K[x]/(p(x))$. Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the $2$-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed $2$-Selmer group. As a second application, we study the family of octic twists of the genus $2$ curve $y^2 = x^5 + x$.Comment: 28 pages, comments welcome

Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions

Postprint (author's final draft

Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions

Postprint (author's final draft

Exceptional zeros and L-invariants of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL2 over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an Linvariant Lp, depending only on p and f, such that when the p-adic L-function of f has an exceptional zero at p, its derivative can be related to the classical L-value multiplied by Lp. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL2/Q. When p is not split and f is the base-change of a classical modular form f ˜, we relate Lp to the L-invariant of f ˜, resolving a conjecture of Trifkovi´c in this case.Postprint (author's final draft