1. CJM Online first
2. CJM Online first
 Sugiyama, Shingo; Tsuzuki, Masao

Existence of Hilbert cusp forms with nonvanishing $L$values
We develop a derivative version of the relative trace formula
on $\operatorname{PGL}(2)$ studied in our previous work,
and derive an asymptotic formula of an average of central values
(derivatives)
of automorphic $L$functions for Hilbert cusp forms.
As an application, we prove the existence of Hilbert cusp forms
with nonvanishing central values (derivatives)
such that the absolute degrees of their Hecke fields are arbitrarily
large.
Keywords:automorphic representations, relative trace formulas, central $L$values, derivatives of $L$functions Categories:11F67, 11F72 

3. CJM 2014 (vol 67 pp. 424)
 Samart, Detchat

Mahler Measures as Linear Combinations of $L$values of Multiple Modular Forms
We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$functions.
Keywords:Mahler measures, EisensteinKronecker series, $L$functions, hypergeometric series Categories:11F67, 33C20 

4. CJM 2014 (vol 66 pp. 1078)
 Lanphier, Dominic; Skogman, Howard

Values of Twisted Tensor $L$functions of Automorphic Forms Over Imaginary Quadratic Fields
Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$
denote the adeles of $K$.
We find special values at all of the critical points of twisted
tensor $L$functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$,
and establish Galois equivariance of the values.
To investigate the values, we determine the archimedean factors
of a class of integral representations of these $L$functions,
thus proving a conjecture due to Ghate. We also investigate
analytic properties of these $L$functions, such as their functional
equations.
Keywords:twisted tensor $L$function, cuspform, hypergeometric series Categories:11F67, 11F37 

5. CJM 2011 (vol 64 pp. 1248)
 Gärtner, Jérôme

Darmon's Points and Quaternionic Shimura Varieties
In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a GrossKohnenZagier theorem for Darmon's
points.
Keywords:elliptic curves, StarkHeegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 

6. CJM 2011 (vol 64 pp. 1122)
 Seveso, Marco Adamo

$p$adic $L$functions and the Rationality of Darmon Cycles
Darmon cycles are a higher weight analogue of StarkHeegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the BlochKato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$adic GrossZagier type formulas, relating the derivatives of $p$adic $L$functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the MazurKitagawa $p$adic $L$function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.
Categories:11F67, 14G05 

7. CJM 2009 (vol 62 pp. 400)
 Prasanna, Kartik

On pAdic Properties of Central LValues of Quadratic Twists of an Elliptic Curve
We study $p$indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semistable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
SwinnertonDyer shows that the set of quadratic discriminants $d$
splits naturally into several families $\mathcal{F}_S$, indexed by subsets $S$
of the primes dividing $N$. Let $\delta_S= \gcd_{d\in \mathcal{F}_S}
L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part
of the central $L$value, $L(1,E_d)$. Our main theorem relates the
$p$adic valuations of $\delta_S$ as $S$ varies. As a consequence we
present an application to a refined version of a question of
Kolyvagin. Finally we explain an intriguing (albeit speculative)
relation between Waldspurger packets on $\widetilde{\operatorname{SL}_2}$ and
congruences of modular forms of integral and halfintegral weight. In
this context, we formulate a conjecture on congruences of
halfintegral weight forms and explain its relevance to the problem of
$p$indivisibility of $L$values of quadratic twists.
Categories:11F40, 11F67, 11G05 

8. CJM 2009 (vol 61 pp. 1383)
 Wambach, Eric

Integral Representation for $U_{3} \times \GL_{2}$
Gelbart and PiatetskiiShapiro constructed
various integral
representations of RankinSel\berg type for groups $G \times
\GL_{n}$,
where $G$
is of split rank $n$. Here we show that their method
can equally well be applied
to the product $U_{3} \times \GL_{2}$, where $U_{3}$
denotes the quasisplit
unitary group in three variables. As an application, we describe which
cuspidal automorphic representations of $U_{3}$ occur
in the Siegel induced
residual spectrum of the quasisplit $U_{4}$.
Categories:11F70, 11F67 

9. CJM 2007 (vol 59 pp. 1323)
 Ginzburg, David; Lapid, Erez

On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases
We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67 

10. CJM 2004 (vol 56 pp. 373)
 Orton, Louisa

An Elementary Proof of a Weak Exceptional Zero Conjecture
In this paper we extend Darmon's theory of ``integration on $\uh_p\times \uh$''
to cusp forms $f$ of higher even weight. This enables us to prove a ``weak
exceptional zero conjecture'': that when the $p$adic $L$function of $f$ has
an exceptional zero at the central point, the $\mathcal{L}$invariant arising is
independent of a twist by certain Dirichlet characters.
Categories:11F11, 11F67 
