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Kimball Martin

Publications and source records attributed to Kimball Martin.

At least 19 recordsLinked to original sources
arXiv2025

Counting newforms with prescribed ramified supercuspidal components

We give a formula for the number of newforms in $S_k^{\mathrm{new}}(N)$ that have prescribed ramified supercuspidal components $\pi_p$ at a set $T$ of primes dividing $N$. This dimension is given in terms of the trace of the Atkin--Lehner operator at $T$ on $S_k^{\mathrm{new}}(N)$. It depends only upon the weight, the level, the ramified quadratic extensions $E_p/{\mathbb Q}_p$ attached to the $\pi_p$, and the root number of each $\pi_p$. The formula is completely explicit when $T$ consists of either a single prime or all prime factors of $N$.

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arXiv2025

Rationality fields of CM modular forms

We formulate a conjecture on the finitude of rationality fields (i.e., Fourier coefficient fields) of newforms of bounded degree, and prove this for CM forms assuming a generalized Riemann hypothesis. Then we explicitly determine what quadratic and cubic rationality fields occur for weight 2 CM forms, which is related to classifications of CM abelian varieties of GL(2) type. Several of these fields do not appear in the existing tables of newforms in the LMFDB.

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arXiv2025

Variations on murmurations

We explore several variations on the recently discovered phenomena of murmurations for elliptic curves and modular forms.

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arXiv2024

Distribution of local signs of modular forms and murmurations of Fourier coefficients

Recently, we showed that global root numbers of modular forms are biased toward +1. Together with Pharis, we also showed an initial bias of Fourier coefficients towards the sign of the root number. First, we prove analogous results with respect to local root numbers. Second, a subtle correlation between Fourier coefficients and global root numbers, termed murmurations, was recently discovered for elliptic curves and modular forms. We conjecture murmurations in a more general context of different (possibly empty) combinations of local root numbers. Last, an appendix corrects a sign error in our joint paper with Pharis.

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arXiv2024

Generic models for genus 2 curves with real multiplication

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental discriminants $D < 40$. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over $\mathbb{P}^2$. We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant $D < 100$ computed by Elkies--Kumar.

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arXiv2023

Local conductor bounds for modular abelian varieties

Brumer and Kramer gave bounds on local conductor exponents for an abelian variety $A/\mathbb Q$ in terms of the dimension of $A$ and the localization prime $p$. Here we give improved bounds in the case that $A$ has maximal real multiplication, i.e., $A$ is isogenous to a factor of the Jacobian of a modular curve $X_0(N)$. In many cases, these bounds are sharp. The proof relies on showing that the rationality field of a newform for $\Gamma_0(N)$, and thus the endomorphism algebra of $A$, contains $\mathbb Q(\zeta_{p^r})^+$ when $p$ divides $N$ to a sufficiently high power. We also deduce that certain divisibility conditions on $N$ determine the endomorphism algebra when $A$ is simple.

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arXiv2023

Counting modular forms by rationality field

We investigate the distribution of degrees and rationality fields of weight 2 newforms. In particular, we give heuristic upper bounds on how often degree $d$ rationality fields occur for squarefree levels, and predict finiteness if $d \ge 7$. When $d=2$, we make predictions about how frequently specific quadratic fields occur, prove lower bounds, and conjecture that $\mathbb{Q}(\sqrt 5)$ is the most common quadratic rationality field.

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arXiv2022

Root number bias for newforms

Previously we observed that newforms obey a strict bias towards root number $+1$ in squarefree levels: at least half of the newforms in $S_k(\Gamma_0(N))$ with root number $+1$ for $N$ squarefree, and it is strictly more than half outside of a few special cases. Subsequently, other authors treated levels which are cubes of squarefree numbers. Here we treat arbitrary levels, and find that if the level is not the square of a squarefree number, this strict bias still holds for any weight. In fact the number of such exceptional levels is finite for fixed weight, and 0 if $k < 12$. We also investigate some variants of this question to better understand the exceptional levels.

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arXiv2022

Moduli for rational genus 2 curves with real multiplication for discriminant 5

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques for reducing quadratic forms over polynomial rings.

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arXiv2021

Rank bias for elliptic curves mod $p$

We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.

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arXiv2020

Zeroes of quaternionic modular forms and central $L$-values

Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture that almost all forms have no nontrivial zeroes. In particular, almost all forms with appropriate local signs should have no zeroes. We show these conjectures follow from a conjecture on the average number of Galois orbits, and give applications to (non)vanishing of $L$-values.

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arXiv2020

Rationality of Darmon points over genus fields of non-maximal orders

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 of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor $c\geq 1$ twisted by quadratic characters of $G_c^+=Gal(H_c^+/F)$, where $H_c^+$ is the strict ring class field of $F$ of conductor $c$, come from rational points on the elliptic curve defined over $H_c^+$.

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arXiv2020

Exact double averages of twisted L-values

Consider central $L$-values of even weight elliptic or Hilbert modular forms $f$ twisted by ideal class characters $\chi$ of an imaginary quadratic extension $K$. Fixing $\chi$, and assuming $K$ is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms $f$ of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to $K$ (the "stable range"). In weight 2, we obtain exact formulas for a simultaneous average over both $f$ and $\chi$. We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of $K$ above the level. Under elementary conditions on the level, these double averages are "stable" in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central $L$-values.

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arXiv2020

The basis problem revisited

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 level than earlier work of Hijikata-Pizer-Shemanske for elliptic modular forms, and solves the basis problem for Hilbert modular forms. The approach is representation theoretic rather than the classical approach of Eichler and Hijikata-Pizer-Shemanske, and involves both a local and global theory of quaternionic newforms.

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arXiv2019

An on-average Maeda-type conjecture in the level aspect

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-values (resp. L-derivatives) are nonzero for 100% of even weight prime level newforms with root number +1 (resp. -1).

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arXiv2019

Mass formulas and Eisenstein congruences in higher rank

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 congruences between cusp forms and Eisenstein series on the quasi-split inner form. This extends recent work of the first author on weight 2 Eisenstein congruences for GL(2) to higher rank. Two issues in higher rank are that the transfer to the quasi-split form is not always cuspidal and sometimes the congruences come from lower rank (e.g., are "endoscopic"). We show our construction yields Eisenstein congruences with non-endoscopic cuspidal automorphic forms on quasi-split unitary groups by using certain unitary groups over division algebras. On the other hand, when using unitary groups over fields, or other groups of Lie type, these Eisenstein congruences typically appear to be endoscopic. This suggests a new way to see higher weight Eisenstein congruences for GL(2), and leads to various conjectures about GL(2) Eisenstein congruences. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert modular forms, and prove some special congruence mod p results between cusp forms on U(p).

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arXiv2018

Refined Goldbach conjectures with primes in progressions

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 two primes with one prime being 3 mod 4. In general, for fixed $m$ and $a, b$ coprime to $m$, any positive even $n \equiv a + b \bmod m$ outside of a finite exceptional set is expected to be a sum of two primes $p$ and $q$ with $p \equiv a \bmod m$, $q \equiv b \bmod m$. We make conjectures about the growth of these exceptional sets.

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