For a fixed number field and an elliptic curve defined and semi-stable over
this number field, we consider the set of prime numbers p such that the Galois
representation attached to the p-torsion points of the elliptic curve is
reducible. When the number field satisfies a certain necessary condition, we
give an explicit bound, depending only on the number field and not on the
semi-stable elliptic curve, for these primes. This generalizes previous results
of Kraus.

We study zeta-functions of weight lattices of compact connected semisimple
Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6)
and PU(4), and give some functional relations and new classes of evaluation
formulas for them.

We show that the space of Euclid's parameters for Pythagorean triples is
endowed with a natural symplectic structure and that it emerges as a spinor
space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be
conceptualized as a 4-dimensional real algebra of "kwaternions." We observe
that this makes Euclid's parameterization the earliest appearance of the
concept of spinors. We present an analogue of the "magic correspondence" for
the spinor representation of Minkowski space and show how the Hall matrices fit
into the scheme.

The is the first of three papers introducing a paradigm within which global
algebraic number theory for the reals may be formulated so as to make possible
the synthesis of algebraic and transcendental number theory into a coherent
whole. We introduce diophantine approximation groups and their associated
Kronecker foliations, using them to provide new algebraic and geometric
characterizations of K-linear and algebraic dependence.

We prove an effective upper bound on the number of effective sections of a
nef hermitian line bundle over an arithmetic surface. It is an effective
version of the arithmetic Hilbert--Samuel formula. As a consequence, we obtain
effective lower bounds on the Faltings height and on the self-intersection of
the canonical bundle in terms of the number of singular points on fibers of the
arithmetic surface.

The present paper deals with unification of the multiple twisted Euler and
Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q =
1. Some earlier results of Ozden's papers in terms of unification of the
multiple twisted Euler and Genocchi numbers and polynomials associated with
p-adic q-integral on Zp at q = 1 can be deduced. We apply the method of
generating function and p-adic q-integral representation on Zp, which are
exploited to derive further classes of Euler polynomials and Genocchi
polynomials.

We consider sets of positive integers containing no sum of two elements in
the set and also no product of two elements. We show that the upper density of
such a set is strictly smaller than 1/2 and that this is best possible.
Further, we also find the maximal order for the density of such sets that are
also periodic modulo some positive integer.

We describe a probability distribution on isomorphism classes of principally
quasi-polarized p-divisible groups over a finite field k of characteristic p
which can reasonably be thought of as "uniform distribution," and we compute
the distribution of various statistics (p-corank, a-number, etc.) of
p-divisible groups drawn from this distribution. It is then natural to ask to
what extent the p-divisible groups attached to a randomly chosen hyperelliptic
curve (resp. curve, resp. abelian variety) over k are uniformly distributed in
this sense.

We generalise a necessary and sufficient condition given by Cohn for all the
zeros of a self-inversive polynomial to be on the unit circle. Our theorem
implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We
apply our result to the study of a polynomial family closely related to
Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied
by Murty, Smyth and Wang and Lal\'in and Rogers. We prove that all polynomials
in this family have their zeros on the unit circle, a result conjectured by
Lal\'in and Rogers on computational evidence.

We show that for any tame regular discrete series parameter of GSp_4 or its
inner form GU_2(D), the L-packet attached by the local Langlands conjecture
agrees with the L-packet of depth zero supercuspidal representations
constructed by DeBacker and Reeder.

We prove a higher dimensional generalization of Gross and Zagier's theorem on
the factorization of differences of singular moduli. Their result is proved by
giving a counting formula for the number of isomorphisms between elliptic
curves with complex multiplication by two different imaginary quadratic fields
$K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime
and its powers.

This paper finds relationships between multiple logarithms with a dihedral
group action on the arguments. I generalize the combinatorics developed in
Gangl, Goncharov and Levin's R-deco polygon representation of multiple
logarithms to find these relations. By writing multiple logarithms as iterated
integrals, my arguments are valid for iterated integrals as over an arbitrary
field.

Let f be a classical holomorphic cusp form for SL_2(Z) of weight k which is a
normalized eigenfunction for the Hecke algebra, and let \lambda(n) be its
eigenvalues. In this paper we study "shifted convolution sums" of the
eigenvalues \lambda(n) after averaging over many shifts h and obtain asymptotic
estimates. The result is somewhat surprising: one encounters a transition
region depending on the ratio of the square of the length of the average over h
to the length of the shifted convolution sum.

Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is
the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field,
we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup
h(F^n(P))^(1/n) and the canonical height of P to be h_F(P) = limsup
h(F^n(P))/n^k d_F^n for an appropriately chosen integer k = k_F. In this
article we prove some elementary relations and make some deep conjectures
relating d_F, a_F(P), and h_F(P). We prove our conjectures for semisimple
monomial maps.

We consider the problem of classifying all positive-definite integer-valued
quadratic forms that represent all positive odd integers. Kaplansky considered
this problem for ternary forms, giving a list of 23 candidates, and proving
that 19 of those represent all positive odds. (Jagy later dealt with a 20th
candidate.) Assuming that the remaining three forms represent all positive
odds, we prove that an arbitrary, positive-definite quadratic form represents
all positive odds if and only if it represents the odd numbers from 1 up to
451.

Let f be a modular form of weight k>=2 and level N, let K be a quadratic
imaginary field, and assume that there is a prime p exactly dividing N. Under
certain arithmetic conditions on the level and the field K, one can attach to
this data a p-adic L-function L_p(f,K,s), as done by
Bertolini-Darmon-Iovita-Spiess. In the case of p being inert in K, this
analytic function of a p-adic variable s vanishes in the critical range
s=1,...,k-1, and therefore one is interested in the values of its derivative in
this range.

We study special values of a modular function $\Lambda$ which is one of
generalized $\lambda$ functions. We show special values of $\Lambda$ at
imaginary quadratic points are algebraic integers. Further we prove that
$\Lambda$ and the modular invariant function generate the modular function
field with respect to the modular subgroup $\Gamma_1(N)$.

We show that special cycles generate a large part of the cohomology of
locally symmetric spaces associated to orthogonal groups. We prove in
particular that classes of totally geodesic submanifolds generate the
cohomology groups of degree $n$ of compact congruence $p$-dimensional
hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than
$\frac12 [\frac{p}{2}]$. We also prove that for connected Shimura varieties
associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $<
1/2 [\frac{p+1}{2}]$.

Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22])
states that a curve of genus greater than one over any number field has only
finitely many points. Again a natural question is how many points can such a
curve have. Caporaso, Harris, and Mazur [CHM97] have shown that the weak
Bombieri-Lang conjecture implies that for any number field $F$ and any integer
$g \ge 2$ there is an absolute upper bound $B(F; g)$ on the number of points on
a genus $g$ curve over $F$.

We have looked at the evaluation of the riemann zeta function at odd
arguments and have provided a simple formula to approximate the value with
exponential convergence. We have compared it with various other formulae
present in literature. We have also evaluated an expression for the zeta
function on the plane $\Re(s) = 1$.

We prove results concerning the specialisation of torsion line bundles on a
variety $V$ defined over $\mathbb{Q}$ to ideal classes of number fields. This
gives a new general technique for constructing and counting number fields with
large class group.

Given a number field K, we consider families of critically simple rational
maps of degree d over K possessing a certain fixed-point and multiplier
structure. With suitable notions of isomorphism and good reduction between
rational maps in these families, we prove a finiteness theorem which is
analogous to Shafarevich's theorem for elliptic curves. We also define the
minimal critical discriminant, a global object which can be viewed as a measure
of arithmetic complexity of a rational map.

Let $K$ be a totally real cubic number field with fundamental discriminant.
In this note we construct a weight one modular form $f_{K}$ with level and
nebentypus depending only on the discriminant of $K$. We show that, up to
isomorphism class, the assignment $K \to f_{K}$ is injective.

In a work of T. Saito, the action of algebraic correspondences on the etale
cohomology of varieties over local fields with semistable reduction is related
to correspondences on smaller strata via weight spectral sequences. We give an
intersection theoretic construction of these correspondences. Under a
finiteness condition this enables us to compute them without involving the
blow-ups of products, and prove their compatibility with compositions. These
features are essential for the application to Shimura varieties.

We introduce and study algebraic dynamical systems generated by triangular
systems of rational functions. We obtain several results about the degree
growth and linear independence of iterates as well as about possible lengths of
trajectories generated by such dynamical systems over finite fields. Some of
these results are generalisations of those known in the polynomial case, some
are new even in this case.

We prove that if two additive functions (from a certain class) take large
values with roughly the same probability then they must be identical. This is a
consequence of a structure theorem making clear the inter-relation between the
distribution of an additive function on the integers, and its distribution on
the primes.

We study algebraic and transcendental powers of positive real numbers,
including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$,
$x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential
functions are given. The main tools used are classical theorems of
Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential
Diophantine equations.

In this article we give a modern interpretation of Kummer's ideal numbers and
show how they developed from Jacobi's work on cyclotomy, in particular the
methods for studying "Jacobi sums" which he presented in his lectures on number
theory and cyclotomy in the winter semester 1836/37.

We show that there exist real numbers $\alpha_1,\alpha_2$ linearly
independent over $\mathbb{Z}$ together with 1 such that for every non-zero
integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has
$||m_1\alpha_1+m_2\alpha_2|| \ge 2^{-300} (\max(m_1, m_2))^{-\sigma}$ with
$\sigma = 1.94696^+$.

We study the real components of modular curves. Our main result is an
abstract group-theoretic description of the real components of a modular curve
defined by a congruence subgroup of level N in terms of the corresponding
subgroup of SL_2(Z/NZ). We apply this result to several families of modular
curves (such as X_0(N), X_1(N), etc.) to obtain formulas for the number of real
components.

We analyze the multivariate generalization of Howgrave-Graham's algorithm for
the approximate common divisor problem. In the m-variable case with modulus N
and approximate common divisor of size N^beta, this improves the size of the
error tolerated from N^(beta^2) to N^(beta^((m+1)/m)), under a commonly used
heuristic assumption. This gives a more detailed analysis of the hardness
assumption underlying the recent fully homomorphic cryptosystem of van Dijk,
Gentry, Halevi, and Vaikuntanathan.

Let K be the totally real cubic field of discriminant 49, let O be its ring
of integers, and let p be the prime over 7. Let Gamma (p)\subset Gamma =
SL_2(O) be the principal congruence subgroup of level p. This paper
investigates the geometry of the Hilbert modular threefold attached to Gamma
(p) and some related varieties. In particular, we discover an octic in P^3 with
84 isolated singular points of type A_2.

We show that there are sets of integers with asymptotic density arbitrarily
close to 1 in which there is no solution to the equation ab=c, with a,b,c in
the set. We also consider some natural generalizations, as well as a specific
numerical example of a product-free set of integers with asymptotic density
greater than 1/2.

We prove the non-existence of elliptic curves having good reduction
everywhere over some real quadratic fields.

Using the theory of metaplectic forms,we study the asymptotic behavior of
cubic exponential sums over the ring of Eisenstein integers. In the first part
of the paper, some non-trivial estimates on average over arithmetic
progressions are obtained. In the second part of the paper, we prove that the
sign of cubic exponential sums changes infinitely often, as the modulus runs
over almost prime integers.

Let $\chi$ be a non-real Dirichlet character modulo a prime $q$. In this
paper we prove that the distribution of the short character sum
$S_{\chi,H}(x)=\sum_{x< n\leq x+H} \chi(n)$, as $x$ runs over the positive
integers below $q$, converges to a two-dimensional Gaussian distribution on the
complex plane, provided that $\log H=o(\log q)$ and $H\to\infty$ as
$q\to\infty$. Furthermore, we use a method of Selberg to give an upper bound on
the rate of convergence.

In this work, we set up a theory of p-adic modular forms over Shimura curves
over totally real fields which allows us to consider also non-integral weights.
In particular, we define an analogue of the sheaves of k-th invariant
differentials over the Shimura curves we are interested in, for any p-adic
character. In this way, we are able to introduce the notion of overconvergent
modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic
families over a suitable rigid-analytic space, that parametrizes the weights.
Finally, we define Hecke operators.

We prove formulas for the p-adic logarithm of quaternionic Darmon points on
p-adic tori and modular abelian varieties over Q having purely multiplicative
reduction at p. These formulas are amenable to explicit computations and are
the first to treat Stark-Heegner type points on higher-dimensional abelian
varieties.

Automorphic representations can be studied in terms of the embeddings of
abstract models of representations into spaces of functions on Lie groups that
are invariant under discrete subgroups. In this paper we describe an adelic
framework to describe them for the group GL(n,R), and provide a detailed
analysis of the automorphic distributions associated to the mirabolic
Eisenstein series. We give an explicit functional equation for some
distributional pairings involving this mirabolic Eisenstein distribution, and
the action of intertwining operators.

We give necessary and sufficient conditions for a few classes of circulant
graphs/digraphs to be singular. We also give two generalizations of the above
graphs/digraphs, namely $(r,s,t)$-digraphs for non-negative integers $r,s$ and
$t$, and the digraph $C_n^{i,j,k,l}$ with certain restrictions. A necessary and
sufficient condition for the digraphs $C_n^{i,j,k,l}$ to be singular is
obtained. Some necessary conditions are given under which the
$(r,s,t)$-digraphs are singular.

Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a
Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can
grow with n. We consider such polynomials for odd square-free n, where \phi(n)
coefficients are determined by the Jacobi symbol, but the remaining
coefficients can be freely chosen. When n is prime, these polynomials have the
smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2
among all Littlewood polynomials, namely (7/6)^{1/4}.

For $f$ and $g$ polynomials in $p$ variables, we relate the special value at
a non-positive integer $s=-N$, obtained by analytic continuation of the
Dirichlet series $$ \zeta(s;f,g)=\sum_{k_1=0}^\infty ... \sum_{k_p=0}^\infty
g(k_1,...,k_p)f(k_1,...,k_p)^{-s}\ \,(\re(s)\gg0), $$ to special values of zeta
integrals $$ Z(s;f,g)=\int_{x\in[0,\infty)^p} g(x)f(x)^{-s}\,dx \, \
(\re(s)\gg0).$$ We prove a simple relation between $\zeta(-N;f,g)$ and
$Z(-N;f_a,g_a)$, where for $a\in\C ^p,\ f_a(x)$ is the shifted polynomial
$f_a(x)=f(a+x)$.

We study a canonical basis for spaces of weakly holomorphic modular forms of
weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a
relation between the Fourier coefficients of modular forms in this canonical
basis and a generalized Ramanujan tau-function, and use this to prove that
these Fourier coefficients are often highly divisible by 2.

For an elliptic curve E over Q, the Galois action on the l-power torsion
points defines representations whose images are subgroups of GL_2(Z/l^n Z).
There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod
l^n representation does not imply that for l^(n+1). Elliptic curves with
surjective mod 3 but not mod 9 representation have been classified by Elkies.
The purpose of this note is to do this in the other two cases.

Euler's identity is shown to give a relation between the zeros of the
Riemann-zeta function and the prime numbers in terms of
$\Phi(x)=x^{-1/4}[2\sqrt{x}\Sigma e^{-p^2\pi x}\ln(p)-1]$ on $x>0$, where the
sum is over all primes $p$. If $\Phi$ is bounded on $x>0$, then the Riemann
hypothesis is true or there are infinitely many zeros Re $z_k>1/2$. The first
21 zeros give rise to asymptotic harmonic behavior in $\Phi$ defined by the
prime numbers up to one trillion.

We consider a variant of the ABC Conjecture, attempting to count the number
of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of
absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC
Conjecture is equivalent to the statement that for $a+b+c<1$, the number of
solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured
that the number of solutions is asymptotically $N^{a+b+c-1 \pm \epsilon}.$ We
prove this conjecture as long as $a+b+c \geq 2.$

Lichtenbaum conjectured in \cite{Lichtenbaum} the existence of a Weil-\'etale
cohomology in order to describe the vanishing order and the special value of
the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of
Euler-Poincar\'e characteristics. Assuming the (conjectured) finite generation
of some motivic cohomology groups we construct such a cohomology theory for
regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we
compute (unconditionally) the right Weil-\'etale cohomology of number rings and
projective spaces over number rings.

Let F be a continuous injective map from an open subset of R^n to R^n such
that the counterimage of every Lebesgue nullset is a Lebesgue nullset. Assume
that, for infinitely many positive integers k, F induces a bijection between
the rational points of denominator k in the domain and those in the image (the
denominator of (a_1/b_1,...,a_n/b_n) being the l.c.m. of b_1,...,b_n). Then F
preserves the Lebesgue measure.

This article deals with the Galois representation attached to elliptic curves
with an isogeny of prime degree over a number field. We first determine uniform
criteria for the irreducibility of Galois representations attached to elliptic
curves in some infinite families, characterised by their reduction type at some
fixed places of the base field. Then, we give an explicit form for a bound that
appear in a theorem of Momose. Finally, we use these results to precise a
previous theorem of the author about the homotheties contained in the image of
the Galois representation.

In this paper, we define a new product over $\mathbb{R}^{\infty}$, which
allows us to obtain a group isomorphic to $\mathbb R^*$ with the usual product.
This operation unexpectedly offers an interpretation of the R\'edei rational
functions, making more clear some of their properties, and leads to another
product, which generates a group structure over the Pell hyperbola. Finally, we
join together these results, in order to evaluate solutions of Pell equation in
an original way.

This is essentially a translated (and explained) version of a peper Hecke
published in 1930 where he shows, for a prime q, a relation between the class
number h(-q) and the representation of PSL(2, Z / pZ) on the space of
holomorphic differentials of X(q).

A beta expansion is the analogue of the base 10 representation of a real
number, where the base may be a non-integer. Although the greedy beta expansion
of 1 using a non-integer base is in general infinitely long and non-repeating,
it is known that if the base is a Pisot number, then this expansion will always
be finite or periodic. Some work has been done to learn more about these
expansions, but in general these expansions were not explicitly known.

Let A be a finite subset of the natural numbers containing 0, and let f(n)
denote the number of ways to write n in the form $\sum e_j2^j$, where $\e_j \in
A$. We show that there exists a computable T = T(A) so that the sequence (f(n)
mod 2) is periodic with period T. Variations and generalizations of this
problem are also discussed.

Let E be an elliptic curve over the rationals without complex multiplication.
The absolute Galois group of Q acts on the group of torsion points of E, and
this action can be expressed in terms of a Galois representation
rho_E:Gal(Qbar/Q) \to GL_2(Zhat). A renowned theorem of Serre says that the
image of rho_E is open, and hence has finite index, in GL_2(Zhat). We give the
first general bounds of this index in terms of basic invariants of E. For
example, the index can be bounded by a polynomial function of the logarithmic
height of the j-invariant of E.

In this paper we find a new lower bound on the number of imaginary quadratic
extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have
elements of a fixed odd order.

We describe various properties of continued fraction expansions of complex
numbers in terms of Gaussian integers. Numerous distinct such expansions are
possible for a complex number. They can be arrived at through various
algorithms, as also in a more general way from what we call "iteration
sequences".

Motivated by the Erdos multiplication table problem we study the following
question: Given numbers N_1,...,N_{k+1}, how many distinct products of the form
n_1...n_{k+1} with n_i<N_i for all i are there? Call A_{k+1}(N_1,...,N_{k+1})
the quantity in question. Ford established the order of magnitude of
A_2(N_1,N_2) and the author of A_{k+1}(N,...,N) for all k>1. In the present
paper we generalize these results by establishing the order of magnitude of
A_{k+1}(N_1,...,N_{k+1}) for arbitrary choices of N_1,...,N_{k+1} when k is
2,3,4 or 5.

Let E be an elliptic curve defined over a number field K. Let h be an element
of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in
infinitely many abelian surfaces up to isomorphism. This is to say that there
are infinitely many abelian surfaces J such that E\hookrightarrow J and h lies
in the kernel of the natural map H^1(K,E)\rightarrow H^1(K,J).

We give explicit formulas for Whittaker functions for the class one principal
series representations of the orthogonal groups $ SO_{2n+1}(\R) $ of odd
degree. Our formulas are similar to the recursive formulas for Whittaker
functions on $SL_n(\R)$ given by Stade and the author \cite{ISt}. Some parts of
our results are announced in \cite{I3}.

We describe (in a representation theoretic setting) a simple comparison of
trace formulas, which implies that the conjugate of a Hilbert modular form $f$
by an automorphism of ${\Bbb C}$ again is a Hilbert modular form of the same
level and conjugate weight as $f$. This is a Theorem of Shimura for which we
obtain a new proof (cf. Theorem 3.3 and Corollary 3.4

This paper treats the problem of determining conditions for the Fourier
coefficients of a Maass-Hecke newform at cusps other than infinity to be
multiplicative. To be precise, the Fourier coefficients are defined using a
choice of matrix in SL(2, Z) which maps infinity to the cusp in question. Let c
and d be the entries in the bottom row of this matrix, and let N be the level.
In earlier work with Dorian Goldfeld and Min Lee, we proved that the
coefficients will be multiplicative whenever N divides 2cd. This paper proves
that they will not be multiplicative unless N divides 576cd.

We give a precise description of how the class group of a number field
measures the failure of unique factorization in its ring of integers.
Specifically, following ideas of Kummer, we determine the structure of all
irreducible factorizations of an element in the ring of integers of a number
field, and give a combinatorial description for the number of such
factorizations. In certain cases, we show how quadratic forms can explicitly
provide all such factorizations.

In this paper we extend methods of Rubin to prove the Gras conjecture for
abelian extensions of a given imaginary quadratic field k and prime numbers p
which divide the number of roots of unity in k.

We consider a dynamical system consisting of a pair of commuting power
series, one noninvertible and another nontorsion invertible, of height one with
coefficients in the $p$-adic integers. Assuming that each point of the
dynamical system generates a Galois extension over the base field, we show that
these extensions are in fact abelian, and, using results and considerations
from the theory of the field of norms, we also show that the dynamical system
must include a torsion series of maximal order.

Let $A$ be a subset of integers and let $2\cdot A+k\cdot A=\{2a_1+ka_2 :
a_1,a_2\in A\}$. Y. O. Hamidoune and J. Ru\' e proved that if $k$ is an odd
prime and $A$ a finite set of integers such that $|A|>8k^k$, then $|2\cdot
A+k\cdot A|\ge (k+2)|A|-k^2-k+2$. In this paper, we extend this result for the
case when $k$ is a power of an odd prime.

In this paper we give an improvement of the degree of the homogeneous linear
recurrence with integer coefficients that exponential sums of symmetric Boolean
functions satisfy. This improvement is tight. We also compute the asymptotic
behavior of symmetric Boolean functions and provide a formula that allows us to
determine if a symmetric boolean function is asymptotically not balanced. In
particular, when the degree of the symmetric function is a power of two, then
the exponential sum is much smaller than $2^n$.

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an
explicit formula for the number of representations of an integer as a sum of
$k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic
factors an analogous result by Friedlander-Goldston.

We establish the functorial transfer of generic, automorphic representations
from the quasi-split general spin groups to general linear groups over
arbitrary number fields, completing an earlier project. Our results are
definitive and, in particular, we determine the image of this transfer
completely and give a number of applications.

Let $\mathscr{S}_k^+(\cn,\Phi)$ denote the space generated by Hilbert modular
newforms (over a fixed totally real field $K$) of weight $k$, level $\cn$ and
Hecke character $\Phi$. We show how to decompose $\mathscr{S}_k^+(\cn,\Phi)$
into direct sums of twists of other spaces of newforms. This sheds light on the
behavior of a newform under a character twist: the exact level of the twist of
a newform, when such a twist is itself a newform, and when a newform may be
realized as the twist of a primitive newform.

We find necessary and sufficient conditions on $n$ so that $\Phi_k(x^n)$ is
irreducible where $\Phi_k$ is the $k$-th cyclotomic polynomial.

An elementary approach is shown which derives the value of the Gauss sum of a
cubic character over a finite field $\mathbb F_{2^s}$ without using
Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if
$s$ is even its value is $-(-2)^{s/2}$).

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we
define a graph for each Hecke operator with fixed ramification. A priori, these
graphs can be seen as a convenient language to organize formulas for the action
of Hecke operators on automorphic forms. However, they will prove to be a
powerful tool for explicit calculations and proofs of finite dimensionality
results.

Starting from a classical generating series for Bessel functions due to
Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root
results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In
particular, we express the (unique) p-adic unit root of an arbitrary
exponential sum on the torus in terms of special values of the p-adic analytic
continuation of a ratio of A-hypergeometric functions.

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some
prime $p.$ We give a method to check Robin inequality $\sigma(n) < e^\gamma
n\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce
$\Psi_t,$ a generalization of Dedekind $\Psi$ function defined for any integer
$t\ge 2$ by $$\Psi_t(n):=n\prod_{p \vert n}(1+1/p+\cdots+1/p^{t-1}).$$ If $n$
is $t-$free then the sum of divisor function $\sigma(n)$ is $ \le \Psi_t(n).$
We characterize the champions for $x \mapsto \Psi_t(x)/x,$ as primorial
numbers.

By a similar idea for constructing Milnor's gamma functions, we study
``higher depth determinants'' of the Laplacian on a compact Riemann surface of
genus greater than one. We prove that, as a generalization of the determinant
expression of the Selberg zeta function, this higher depth determinant can be
expressed as a product of multiple gamma functions and what we call a
Milnor-Selberg zeta function. Moreover, it is shown that the Milnor-Selberg
zeta function admits an analytic continuation, a functional equation and,
remarkably, has an Euler product.

New congruences are found for Andrews' smallest parts partition function
spt(n). The generating function for spt(n) is related to the holomorphic part
alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a
normalized form of the generating function for spt(n) is an eigenform modulo 72
for the Hecke operators T(p^2) for primes p > 3, and an eigenform modulo t for
t = 5, 7 or 13 provided that (t, 6p) = 1. The result for the modulus 3 was
observed earlier by the author and considered by Ono and Folsom.

We study the L-functions associated to Siegel modular forms (equivalently,
automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both
theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we
perform representation theoretic calculations to cast the Langlands L-function
in classical terms. We develop a precise notion of what it means to test a
conjectured functional equation for an L-function, and we apply this to the
degree 10 adjoint L-function associated to a Siegel modular form.

Let F be a totally real number field, and let f traverse a sequence of
non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n),
trivial central character and full level. We show that the mass of f
equidistributes on the Hilbert modular variety as max(k_1,...,k_n) tends to
infinity.

For E/k an elliptic curve with CM by O, we determine a formula for (a
generalization of) the arithmetic local constant of [4] at almost all primes of
good reduction. We apply this formula to the CM curves defined over Q and are
able to describe extensions F/Q over which the O-rank of E grows.

Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$
and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We
obtain the Biro-type upper bound for the smallest such period of $B$: Let
$\varepsilon>0$. We assume that $\textrm{diam}(A)\ge n_0(\varepsilon)$ and that
$\sum_{a\in A}w_A(a)\leq (\textrm{diam}(A)+1)^{c}$, where $c$ is any constant
such that $c< 100\log2-2$. Then $B$ is periodic with period \[\log k\leq
(\textrm{diam}(A)+1)^{\frac{1}{3}+\varepsilon.

This paper contains a small improvement to the explicit bounds on the growth
of the function $S(T)$. It is shown how more substantial improvements are
possible if one has better explicit bounds on the growth of
$|\zeta(\frac{1}{2}+it)|$.

Let $Q=\{q_n\}_{n=1}^{\infty}$ be a sequence of integers greater than or
equal to $2$. We say that a real number $x$ in $[0,1)$ is {\it $Q$-distribution
normal} if the sequence $\{q_1q_2 \... q_n x\}_{n=1}^{\infty}$ is uniformly
distributed mod $1$. In \cite{Laffer}, P. Laffer asked for a construction of a
$Q$-distribution normal number for an arbitrary $Q$. Under a mild condition on
$Q$, we construct a set $\Theta_Q$ of $Q$-distribution normal numbers. This set
is perfect and nowhere dense.

Some problems involving the classical Hardy function $$ Z(t) :=
\zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) =
\chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of
$Z(t)$, the distribution of its positive and negative values and the primitive
of $Z(t)$.

We investigate recursive properties of certain p-adic Whittaker functions (of
which representation densities of quadratic forms are special values). The
proven relations can be used to compute them explicitly in arbitrary
dimensions, provided that enough information about the orbits under the
orthogonal group acting on the representations is available. These relations
have implications for the first and second special derivatives of the Euler
product over all p of these Whittaker functions.

This is an expository article, based on a lecture course given at CRM
Barcelona in December 2009. The purpose of these notes is to prove, in a
reasonably self-contained way, that finiteness of the Tate-Shafarevich group
implies the parity conjecture for elliptic curves over number fields. Along the
way, we review local and global root numbers of elliptic curves and their
classification, and discuss some peculiar consequences of the parity
conjecture.

Let $c_q(n)$ denote the Ramanujan sum modulo q, and let x and y be large
reals, with x = o(y). We obtain asymptotic formulas for the sums

\sum_{n \le y}(\sum_{q \le x} c_q(n))^k (k = 1, 2).

Let $(b_n) = (b_1, b_2, ...)$ be a sequence of integers. A primitive prime
divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any
of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a
sequence of integers $(c_n)$ where the $n$-th term is the $n$-th iterate of $f$
at 0. We consider primitive prime divisors of zero orbits of polynomials. In
this note, we show that for integers $c$ and $k$, where $k > 1$ and $c \neq \pm
1$, every iterate in the zero orbit of $f(z) = z^k + c$ contains a primitive
prime whenever zero has an infinite orbit.

We give an interpretation of the cohomology of an arithmetically defined
group as a set of equivalence classes of lattices. We use this interpretation
to give a simpler proof of the connection established by J. Rohlfs between
genus and cohomology.

We consider deformations of bounded complexes of modules for a profinite
group G over a field of positive characteristic. We prove a finiteness theorem
which provides some sufficient conditions for the versal deformation of such a
complex to be represented by a complex of G-modules that is strictly perfect
over the associated versal deformation ring.

We state conjectures on the relationships between automorphic representations
and Galois representations, and give evidence for them.

Here we prove that Benford's law holds for coefficients of an infinite class
of modular forms. Expanding the work of Bringmann and Ono on exact formulas for
harmonic Maass forms, we derive the necessary asymptotics. This implies that
the unrestricted partition function $p(n)$, as well as other natural partition
functions, satisfy Benford's law.

We discuss a clean level lowering theorem modulo prime powers for weight $2$
cusp forms. Furthermore, we illustrate how this can be used to completely solve
certain twisted Fermat equations $ax^n+by^n+cz^n=0$.

By integrating a series provided by Knopp, a new series representation of the
Euler-Mascheroni constant arises. The infinite sum representation of {\gamma}
is determined through Fourier series (sawtooth wave).

Zeckendorf proved that every positive integer has a unique representation as
a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's
natural to ask how many summands are needed. Using a continued fraction
approach, Lekkerkerker proved that the average number of such summands needed
for integers in $[F_n, F_{n+1})$ is $n / (\varphi^2 + 1) + O(1)$, where
$\varphi = \frac{1+\sqrt{5}}2$ is the golden mean. Surprisingly, no one appears
to have investigated the distribution of the number of summands; our main
result is that this converges to a Gaussian as $n\to\infty$.

The modern theory of class field towers has its origins in the study of the
p-class field tower over a quadratic imaginary number field, so it is fitting
that this problem be the first in the discipline to be nearing a solution. We
survey the state of the subject and present a new cohomological condition for a
quadratic imaginary number field to have an infinite p-class field tower (for p
odd). Under an additional hypothesis, we refine this to a necessary and
sufficient condition and describe an algorithm for evaluating this condition
for a given quadratic imaginary number field.

B\"uchi's problem asks whether there exists a positive integer $M$ such that
any sequence $(x_n)$ of at least $M$ integers, whose second difference of
squares is the constant sequence $(2)$, satisifies $x_n^2=(x+n)^2$ for some
$x\in\Z$. A positive answer to B\"uchi's problem would imply that there is no
algorithm to decide whether or not an arbitrary system of quadratic diagonal
forms over $\Z$ can represent an arbitrary given vector of integers. We give
explicitly an infinite family of polynomial parametrizations of non-trivial
length $4$ B\"uchi sequences of integers.

We present an algorithm for computing the $p$-component of the automorphic
representation arising from a cuspidal newform $f$ for a prime $p$. This is
equivalent to computing the restriction to the decomposition group at $p$ of
the $\ell$-adic Galois representations attached to $f$ for any $\ell\neq p$.
The situation is most interesting when $p^2$ divides the level of $f$, in which
case the $p$-component could be supercuspidal.

We generalise Uspensky's theorem characterising eventual exact (e.e.) covers
of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for
any m \in \N, by homogeneous sequences with irrational moduli. We also consider
inhomogeneous sequences, again with irrational moduli, and obtain a purely
arithmetical characterisation of e.e. m-covers. This generalises a result of
Graham for m = 1, but when m > 1 the arithmetical description is more
complicated.

In this survey we give an overview of recent developments on the Quantitative
Subspace Theorem. In particular, we discuss a new upper bound for the number of
subspaces containing the "large" solutions, obtained jointly with Roberto
Ferretti, and sketch the proof of the latter. Further, we prove a new gap
principle to handle the "small" solutions in the system of inequalities
considered in the Subspace Theorem.

In this paper, we prove an explicit arithmetic intersection formula between
arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert
modular surface over $\mathbb Z$. As applications, we obtain the first
`non-abelian' Chowla-Selberg formula, which is a special case of Colmez's
conjecture; an explicit arithmetic intersection formula between arithmetic
Humbert surfaces and CM cycles in the arithmetic Siegel modular variety of
genus two; Lauter's conjecture about the denominators of CM values of Igusa
invariants; and a result about bad reductions of CM genus two curves.

We give a formula for the values of automorphic Green functions on the
special rational 0-cycles (big CM points) attached to certain maximal tori in
the Shimura varieties associated to rational quadratic spaces of signature
(2d,2). Our approach depends on the fact that the Green functions in question
are constructed as regularized theta lifts of harmonic weak Mass forms, and it
involves the Siegel-Weil formula and the central derivatives of incoherent
Eisenstein series for totally real fields.

We develop a framework for solving polynomial equations with size constraints
on solutions. We obtain our results by showing how to apply a technique of
Coppersmith for finding small solutions of polynomial equations modulo integers
to analogous problems over polynomial rings, number fields, and function
fields. This gives us a unified view of several problems arising naturally in
cryptography, coding theory, and the study of lattices.

We present lower bounds on the sum and product of the distinct prime factors
of an odd perfect number, which provide a lower bound on the size of the odd
perfect number as a function of the number of its distinct prime factors.

Brylinski and Deligne have provided a framework to study central extensions
of reductive groups by K2 over a field F. Such central extensions can be used
to construct central extensions of p-adic groups by finite cyclic groups,
including the metaplectic groups. Particularly interesting is the observation
of Brylinski and Deligne that a central extension of a reductive group by K2,
over a p-adic field, yields a family of central extensions of reductive groups
by the multiplicative group over the residue field, indexed by the points of
the building.

Let $p_1,p_2,\dots,p_n, a_1,a_2,\dots,a_n \in \N$, $x_1,x_2,\dots,x_n \in
\R$, and denote the $k$th periodized Bernoulli polynomial by $\B_k(x)$. We
study expressions of the form \[

We study various properties of quasimodular forms by using their connections
with Jacobi-like forms and pseudodifferential operators. Such connections are
made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2,
\bR)$ with certain polynomials over the ring of holomorphic functions of the
Poincar\'e upper half plane that are $\G$-invariant. We consider a surjective
map from Jacobi-like forms to quasimodular forms and prove that it has a right
inverse, which may be regarded as a lifting from quasimodular forms to
Jacobi-like forms.

We obtain the approximate functional equation for the Rankin-Selberg
zeta-function on the 1/2-line.

Let X be a smooth variety over $F_p$. Let E be a number field. For each
nonarchimedean place $\lambda$ of E prime to p consider the set of isomorphism
classes of irreducible lisse $\overline{E}_{\lambda}$-sheaves on X with
determinant of finite order such that for every closed point x in X the
characteristic polynomial of the Frobenius $F_x$ has coefficents in E. We prove
that this set does not depend on $\lambda$.

The idea is to use a method developed by G.~Wiesend to reduce the problem to
the case where X is a curve. This case was treated by L. Lafforgue.

We establish a simple identity and using it we find a new proof of a result
of Kloosterman.

Let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois
extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define
$A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the
inverse different of $L/K$. For $p$ an odd prime and $L/\Q_p$ contained in
certain cyclotomic extensions, Erez has described integral normal bases for
$A_{L/\Q_p}$ that are self-dual with respect to the trace form. Assuming
$K/\Q_p$ to be unramified we generate odd abelian weakly ramified extensions of
$K$ using Lubin-Tate formal groups.

We define generalizations of classical invariants of wild ramification for
coverings on a variety of arbitrary dimension over a local field. For an l-adic
sheaf, we define its Swan class as a 0-cycle class supported on the wild
ramification locus. We prove a formula of Riemann-Roch type for the Swan
conductor of cohomology together with its relative version, assuming that the
local field is of mixed characteristic.

Let $(G,+)$ be a finite abelian group. Then, $\so(G)$ and $\eta(G)$ denote
the smallest integer $\ell$ such that each sequence over $G$ of length at least
$\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to
and at most, resp., the exponent of the group. For groups of rank two, we study
the inverse problems associated to these constants, i.e., we investigate the
structure of sequences of length $\so(G)-1$ and $\eta(G)-1$ that do not have
such a subsequence. On the one hand, we show that the structure of these
sequences is in general richer than expected.

Let $K\subseteq\mathbb{R}^2$ be a compact set such that
$K+\mathbb{Z}^2=\mathbb{R}^2$. We prove, via Algebraic Topology, that the
integer points of the difference set of $K$, $(K-K)\cap\mathbb{Z}^2$, is not
contained on the coordinate axes,
$\mathbb{Z}\times\{0\}\cup\mathbb{Z}\times\{0\}$. This result gives a negative
answer to a question posed by P. Hegarty and M. Nathanson on relatively prime
lattice points.

Let $ A$ be a subset of group $G_0$

with $|{A^{-1}A}|\le 2|A|-2.$ We show that there are an element $a\in A$ and
a non-null proper subgroup $H$ of $G$ such that one of the following holds:
\begin{itemize}

\item $x^{-1}Hy \subset A^{-1}A,$ for all $(x,y)\in A^2\setminus (Ha)^2,$

\item $xHy^{-1} \subset AA^{-1},$ for all $(x,y)\in A^2\setminus (aH)^2.$
\end{itemize} where $G$ is the subgroup generated by ${A^{-1}A}.$

We prove an asymptotic formula for the number of integer points in a family
of bounded domains in the Euclidean space with smooth boundary, which remain
unchanged along some linear subspace and stretch out in the directions,
orthogonal to this subspace. A more precise estimate for the remainder is
obtained in the case when the domains are strictly convex.

The aim of this article is to study deformation theory of trianguline B-pairs
for any p-adic field. For benign B-pairs, a special good class of trianguline
B-pairs, we prove a main theorem concerning tangent spaces of these deformation
spaces. These are generalizations of Bellaiche-Chenevier's and Chenevier's
works in the Q_p case, where they used (\varphi,\Gamma)-modules over the Robba
ring instead of using B-pairs.

In this paper we describe a generalisation and adaptation of Kedlaya's
algorithm for computing the zeta function of a hyperelliptic curve over a
finite field of odd characteristic that the author used for his implementation
of the algorithm in Magma. We generalise the algorithm to the case of an even
degree model. We also analyse the adaptation of working with the $x^idx/y^3$
rather than the $x^idx/y$ differential basis. This basis has the computational
advantage of always leading to an integral transformation matrix whereas the
latter fails to in small genus cases.

The strong recurrence is equivalent to the Riemann hypothesis. In the present
paper, we give a simple proof of the generalized strong recurrence for all
non-zero parameters.

For a negative integer $k$ let $J_k$ be the space of modified Jacobi forms of
weight $k$ and index $0$ on $\mathrm{SL}_2(\mathbb{Z})$. For each positive
integer $m$ we consider certain subspace $J_k^{m}$ of $J_k$ which satisfies
$J_k=\cup_{m=1}^\infty J_k^m$. By observing a relation between coefficients of
the Fourier development of a modified Jacobi form we show that $J_k^m$ is
finite-dimensional.

In this paper we will use the WZ algorithm to prove identities between Mahler
measures of polynomials. In particular, we will offer a new proof of a theorem
due to Lal\'{i}n. We will also show that this theorem is equivalent to a
formula for elliptic dilogarithms.

In this work we prove congruences between special values of elliptic curves
with CM that seem to play a central role in the analytic side of the
non-commutative Iwasawa theory. These congruences are the analogue for elliptic
curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive.
The proof is based on the fact that the critical values of elliptic curves with
CM, or what amounts to the same, the critical values of Gr\"{o}ssencharacters,
can be expressed as values of Hilbert-Eisenstein series at CM points.

We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27
lines are acted upon by the index two subgroup of the maximal possible Galois
group. This is the simple group of order $25\,920$. Our examples are given in
pentahedral normal form with rational coefficients. For such cubic surfaces, we
study the discriminant and show its relation to the index two subgroup. On the
corresponding parameter space, we search for rational points, discuss their
asymptotic, and construct an accumulating subvariety.

We express some basic properties of Deninger's conjectural dynamical system
in terms of morphisms of topoi. Then we show that the current definition of the
Weil-\'etale topos satisfies these properties. In particular, the flow, the
closed orbits, the fixed points of the flow and the foliation in characteristic
$p$ are well defined on the Weil-\'etale topos. This analogy extends to
arithmetic schemes. Over a prime number $p$ and over the archimedean place of
$\mathbb{Q}$, we define a morphism from a topos associated to Deninger's
dynamical system to the Weil-\'etale topos.

A ternary inclusion-exclusion polynomial is a polynomial of the form \[
Q_{{p,q,r}}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)}
{(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers
$\ge3$ and relatively prime in pairs. This class of polynomials contains, as
its principle subclass, the ternary cyclotomic polynomials corresponding to
restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here
is to continue the investigation of the relationship between the coefficients
of $Q_{{p,q,r}}$ and $Q_{{p,q,s}}$, with $r\equiv s\pmod{pq}$.

Taking a combinatorial point of view on cyclotomic polynomials leads to a
larger class of polynomials we shall call the inclusion-exclusion polynomials.
This gives a more appropriate setting for certain types of questions about the
coefficients of these polynomials. After establishing some basic properties of
inclusion-exclusion polynomials we turn to a detailed study of the structure of
ternary inclusion-exclusion polynomials. The latter subclass is exemplified by
cyclotomic polynomials $\Phi_{pqr}$, where $p<q<r$ are odd primes.

Extensive and systematic machine computations are carried out to investigate
the integral cohomology of the Euclidean Bianchi groups and their congruence
subgroups. The collected data give insight on several aspects, including the
asymptotic behaviour of the torsion in the first homology. Along with the
experimental work, some basic properties of the integral cohomology are
recorded with an eye towards the liftibility issue of Hecke eigenvalue systems.

Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod
8$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le
x$, yet is not a square. We extend this defintion to algebraic curves and say
that $n$ is an $x$-pseudopoint of a curve $f(u,v) = 0$ (where $f \in \Z[U,V]$)
if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0
\pmod p$ is satisfied for some $m$.

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference
sequences, we introduce the notion of iterated compositions of linear
operations. We prove a general result on the stability of such compositions
(with bounded coefficients) on sets of integers having a positive upper
density.

Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer
$\pi\in K$, and put $K_\infty:=K(\sqrt[p^\infty]{\pi})$. We introduce a new
technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation
rings.

Let F be the cyclotomic field of fifth roots of unity. We computationally
investigate modularity of elliptic curves over F.