29 citations · 86 across the 20 of their papers we have counts for
25 papers
Counting modular forms by rationality field
Alex Cowan, Kimball Martin
We investigate the distribution of degrees and rationality fields of weight 2 newforms. In particular, we give heuristic upper bounds on how often degree rationality fields occ…
Root number bias for newforms
Kimball Martin
Previously we observed that newforms obey a strict bias towards root number in squarefree levels: at least half of the newforms in with root number for …
Moduli for rational genus 2 curves with real multiplication for discriminant 5
Alex Cowan, Kimball Martin
Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rat…
Rank bias for elliptic curves mod
Kimball Martin, Thomas Pharis
We conjecture that, for a fixed prime , rational elliptic curves with higher rank tend to have more points mod . We show that there is an analogous bias for modular forms wit…
Exact double averages of twisted L-values
Kimball Martin
Consider central -values of even weight elliptic or Hilbert modular forms twisted by ideal class characters of an imaginary quadratic extension . Fixing , and assu…
Zeroes of quaternionic modular forms and central -values
Kimball Martin, Jordan Wiebe
Values of quaternionic modular forms are related to twisted central -values via periods and a theorem of Waldspurger. In particular, certain twisted -values must be non-vanis…
An on-average Maeda-type conjecture in the level aspect
Kimball Martin
We present a conjecture on the average number of Galois orbits of newforms when fixing the weight and varying the level. This conjecture implies, for instance, that the central L-v…
Mass formulas and Eisenstein congruences in higher rank
Kimball Martin, Satoshi Wakatsuki
We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruenc…
Refined Goldbach conjectures with primes in progressions
Kimball Martin
We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of…
The basis problem revisited
Kimball Martin
We explicitly describe the Jacquet-Langlands correspondence at the level of modular forms. This gives a simpler and more flexible solution to Eichler's basis problem for general le…
Rationality of Darmon points over genus fields of non-maximal orders
Matteo Longo, Kimball Martin, Yan Hu
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions o…
Congruences for modular forms mod 2 and quaternionic -ideal classes
Kimball Martin
We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin--Lehner eigenvalues. The proofs involve the notion of quaternio…
Refined dimensions of cusp forms, and equidistribution and bias of signs
Kimball Martin
We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbe…
Distinguishing finite group characters and refined local-global phenomena
Kimball Martin, Nahid Walji
Serre obtained a sharp bound on how often two irreducible degree complex characters of a finite group can agree, which tells us how many local factors determine an Artin -fu…
The Jacquet-Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2
Kimball Martin
We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in w…
Strong local-global phenomena for Galois and automorphic representations
Kimball Martin
Many results are known regarding how much local information is required to determine a global object, such as a modular form, or a Galois or automorphic representation. We survey s…
A relative trace formula for a compact Riemann surface
Kimball Martin, Mark McKee, Eric Wambach
We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic . This can be expressed as a relation between the period spectrum and the ortho…
A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes
Kimball Martin, Dinakar Ramakrishnan
Let be a cyclic extension of number fields of prime degree. Let be an irreducible -dimensional representation of Artin type of the absolute Galois group of , and $π…
Distinguishing graphs with zeta functions and generalized spectra
Christina Durfee, Kimball Martin
Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here…
Test vectors and central L-values for GL(2)
Daniel File, Kimball Martin, Ameya Pitale
We determine local test vectors for Waldspurger functionals for GL(2), in the case where both the representation of GL(2) and the character of the degree two extension are ramified…
Periods and nonvanishing of central L-values for GL(2n)
Brooke Feigon, Kimball Martin, David Whitehouse
Let be a cuspidal automorphic representation of PGL() over a number field , and the quadratic idele class character attached to a quadratic extension . Guo and…
How often should you clean your room?
Kimball Martin, Krishnan Shankar
We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we inve…
Non-unique factorization and principalization in number fields
Kimball Martin
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…
Central L-values and periods for GL(2)
Kimball Martin
We give an exposition of central value formulas for twisted L-functions for GL(2) in terms of compact periods, with a focus on explaining an approach via the relative trace formula…