We construct explicit bases for spaces of overconvergent $p$-adic modular forms when $p=2,3$ and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some $2$-adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.

Let ${\mathcal{K}}$ be an imaginary quadratic field. Let ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ be irreducible generic cohomological automorphic representation of $\text{GL}(n)/{\mathcal{K}}$ and $\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if ${\rm\Pi}$ is cuspidal and the weights of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ are in a standard relative position, the critical values of the Rankin–Selberg product $L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$ are essentially algebraic multiples of the product of the Whittaker periods of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal ${\rm\Pi}$ can be given a motivic interpretation, and can also be related to a critical value of the adjoint $L$-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint $L$-functions are compatible with Deligne’s conjecture.

We determine the index of five of the seven hypergeometric Calabi–Yau operators that have finite index in $\mathit{Sp}_{4}(\mathbb{Z})$ and in two cases give a complete description of the monodromy group. Furthermore, we find six nonhypergeometric Calabi–Yau operators with finite index in $\mathit{Sp}_{4}(\mathbb{Z})$, most notably a case where the index is one.

In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general $k$th power and provide an analogous estimate for the $k$th power moment of the Hecke multiplicative functions. This may be of general interest.

We analyse the $\text{mod}~p$ étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the $H_{c}^{1}$ of the Lubin–Tate tower. On the other hand, we show that in $H^{1}$ of the Lubin–Tate tower appears the $\text{mod}~p$ local Langlands correspondence and the $\text{mod}~p$ local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.

Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation

$$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$

$abc=0$

$K$

$K$

$K=\mathbb{Q}(\sqrt{d})$

$d\geqslant 2$

${\textstyle \frac{5}{6}}$

$1$

Let $K$ be a number field. For any system of semisimple mod $\ell$ Galois representations $\{{\it\phi}_{\ell }:\text{Gal}(\bar{\mathbb{Q}}/K)\rightarrow \text{GL}_{N}(\mathbb{F}_{\ell })\}_{\ell }$ arising from étale cohomology (Definition 1), there exists a finite normal extension $L$ of $K$ such that if we denote ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/K))$ and ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/L))$ by $\bar{{\rm\Gamma}}_{\ell }$ and $\bar{{\it\gamma}}_{\ell }$, respectively, for all $\ell$ and let $\bar{\mathbf{S}}_{\ell }$ be the $\mathbb{F}_{\ell }$-semisimple subgroup of $\text{GL}_{N,\mathbb{F}_{\ell }}$ associated to $\bar{{\it\gamma}}_{\ell }$ (or $\bar{{\rm\Gamma}}_{\ell }$) by Nori’s theory [On subgroups of$\text{GL}_{n}(\mathbb{F}_{p})$, Invent. Math. 88 (1987), 257–275] for sufficiently large $\ell$, then the following statements hold for all sufficiently large $\ell$.

A(i) The formal character of $\bar{\mathbf{S}}_{\ell }{\hookrightarrow}\text{GL}_{N,\mathbb{F}_{\ell }}$ (Definition 1) is independent of $\ell$ and equal to the formal character of $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}{\hookrightarrow}\text{GL}_{N,\mathbb{Q}_{\ell }}$, where $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}$ is the derived group of the identity component of $\mathbf{G}_{\ell }$, the monodromy group of the corresponding semi-simplified $\ell$-adic Galois representation ${\rm\Phi}_{\ell }^{\text{ss}}$.

A(ii) The non-cyclic composition factors of $\bar{{\it\gamma}}_{\ell }$ and $\bar{\mathbf{S}}_{\ell }(\mathbb{F}_{\ell })$ are identical. Therefore, the composition factors of $\bar{{\it\gamma}}_{\ell }$ are finite simple groups of Lie type of characteristic $\ell$ and are cyclic groups.

B(i) The total $\ell$-rank $\text{rk}_{\ell }\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) is equal to the rank of $\bar{\mathbf{S}}_{\ell }$ and is therefore independent of $\ell$.

B(ii) The $A_{n}$-type $\ell$-rank $\text{rk}_{\ell }^{A_{n}}\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) for $n\in \mathbb{N}\setminus \{1,2,3,4,5,7,8\}$ and the parity of $(\text{rk}_{\ell }^{A_{4}}\bar{{\rm\Gamma}}_{\ell })/4$ are independent of $\ell$.

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of ‘topological automorphic forms’ associated to this curve, as well as one associated to a quotient by an Atkin–Lehner involution.

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota–Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$, $K$ a finite extension of $\mathbb{Q}_{p}$, for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$, and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$, generalizing a recent result of Nicolas and Serre for the case $p=2$.

We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.

We construct $p$-adic families of Klingen–Eisenstein series and $L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$. When $r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$-adic $L$-function.

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

Let $p\geqslant 5$ be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup ${\rm\Gamma}(L)$ of $\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$ for a principal ideal $(L)\neq 0$ of $\mathbb{Z}_{p}[[T]]$ for the canonical ‘weight’ variable $t=1+T$. If $L\notin {\rm\Lambda}^{\times }$, the power series $L$ is proven to be a factor of the Kubota–Leopoldt $p$-adic $L$-function or of the square of the anticyclotomic Katz $p$-adic $L$-function or a power of $(t^{p^{m}}-1)$.

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz’s theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if ${\it\pi}$ is a polarizable, regular algebraic, cuspidal automorphic representation of $\text{GL}_{n}(\mathbb{A}_{F})$, then for a positive Dirichlet density set of rational primes $l$, the $l$-adic representations $r_{l,\imath }({\it\pi})$ associated to ${\it\pi}$ are irreducible.

Kloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the $q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

This paper proves two results on the field of rationality $\mathbb{Q}({\it\pi})$ for an automorphic representation ${\it\pi}$, which is the subfield of $\mathbb{C}$ fixed under the subgroup of $\text{Aut}(\mathbb{C})$ stabilizing the isomorphism class of the finite part of ${\it\pi}$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations ${\it\pi}$ such that ${\it\pi}$ is unramified away from a fixed finite set of places, ${\it\pi}_{\infty }$ has a fixed infinitesimal character, and $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ is bounded. The second main result is that for classical groups, $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions.